# HG changeset patch # User Alpar Juttner # Date 1479798916 -3600 # Node ID 20d1b0e5838f5dad6ce371ba9fe13c8e3a36b89d # Parent 76dc74f824ba4d1fbb73b0f3cb523690bc78e85e Szakd diff -r 76dc74f824ba -r 20d1b0e5838f Orig/ContactMaps.128.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/ContactMaps.128.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,78 @@ +130 0.22081 +133 0.206133 +136 0.18355 +137 0.197773 +138 0.196558 +140 0.184796 +141 0.19825 +144 0.18353 +146 0.18265 +147 0.18449 +149 0.18397 +150 0.22288 +151 0.184963 +156 0.18344 +157 0.186225 +158 0.20248 +159 0.18381 +162 0.187918 +164 0.18324 +169 0.18427 +181 0.184675 +185 0.18532 +186 0.18603 +187 0.1841 +190 0.186097 +191 0.18738 +192 0.18724 +193 0.18642 +197 0.18551 +198 0.19429 +199 0.187057 +214 0.18603 +220 0.18604 +221 0.185745 +222 0.18729 +224 0.1858 +225 0.18676 +226 0.20219 +228 0.18847 +237 0.18467 +252 0.18718 +264 0.18538 +265 0.18966 +271 0.185115 +274 0.18824 +275 0.18339 +276 0.20007 +277 0.1856 +280 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0.06413 +102 0.06942 +104 0.0717 +105 0.071015 +107 0.07732 +114 0.06407 +117 0.06332 +120 0.071185 +122 0.06583 +123 0.06372 +124 0.06389 +125 0.06419 +130 0.06533 +133 0.0644333 +136 0.06489 +137 0.06522 +138 0.0685833 +140 0.068598 +141 0.065835 +144 0.06284 +146 0.06354 +147 0.06415 +149 0.06459 +150 0.06982 +151 0.0642867 +156 0.06389 +157 0.066565 +158 0.07314 +159 0.06366 +162 0.0660544 +164 0.06373 +169 0.06494 +181 0.064515 +185 0.06557 +186 0.06613 +187 0.06552 +190 0.06936 +191 0.06751 +192 0.06777 +193 0.06904 +197 0.06383 +198 0.065475 +199 0.0696833 +214 0.06575 +220 0.066855 +221 0.065705 +222 0.06673 +224 0.06649 +225 0.06663 +226 0.06811 +228 0.06677 +237 0.06618 +252 0.06912 +264 0.06732 +265 0.06752 +271 0.065515 +274 0.06842 +275 0.06529 +276 0.06773 +277 0.118265 +280 0.0689 +286 0.07191 +302 0.07059 +327 0.07255 +330 0.06826 +334 0.07863 +336 0.07222 +359 0.07016 +360 0.06905 +385 0.071402 +387 0.070935 +388 0.070445 +393 0.06894 +395 0.06836 +414 0.077 +428 0.07548 +457 0.07198 +484 0.07438 +501 0.07351 +556 0.07125 +558 0.07053 +564 0.07919 +585 0.06738 +599 0.06781 +600 0.07072 +611 0.07688 +617 0.07725 +694 0.07614 +741 0.07769 +773 0.07191 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/ContactMaps.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/ContactMaps.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 1.1e-05 +102 1.65e-05 +104 1.2e-05 +105 1.25e-05 +107 2.15e-05 +114 1.25e-05 +117 1.35e-05 +120 0.0001175 +122 3.9e-05 +123 1.5e-05 +124 1.45e-05 +125 1.55e-05 +130 1.6e-05 +133 7.3e-05 +136 1.6e-05 +137 8.65e-05 +138 3e-05 +140 1.55e-05 +141 1.65e-05 +144 1.65e-05 +146 1.45e-05 +147 2.45e-05 +149 1.55e-05 +150 1.85e-05 +151 7.35e-05 +156 1.8e-05 +157 8.65e-05 +158 7.65e-05 +159 1.8e-05 +162 1.95e-05 +164 6.2e-05 +169 2.15e-05 +181 4.45e-05 +185 5.7e-05 +186 7.3e-05 +187 9.1e-05 +190 2.25e-05 +191 2.6e-05 +192 2.7e-05 +193 2.65e-05 +197 2.3e-05 +198 2.45e-05 +199 2.7e-05 +214 5.05e-05 +220 3.75e-05 +221 3.6e-05 +222 3.65e-05 +224 3.65e-05 +225 3.75e-05 +226 4.3e-05 +228 5.9e-05 +237 2.8e-05 +252 3.8e-05 +264 8.55e-05 +265 5.4e-05 +271 3.2e-05 +274 7e-05 +275 4.65e-05 +276 0.000166 +277 3.65e-05 +280 4.95e-05 +286 5.75e-05 +302 4.2e-05 +327 7.9e-05 +330 3.9e-05 +334 5.05e-05 +336 7.3e-05 +359 8.15e-05 +360 4.65e-05 +385 6.05e-05 +387 6.1e-05 +388 0.000126 +393 4.55e-05 +395 5.1e-05 +414 0.0001055 +428 7.75e-05 +457 7.3e-05 +484 7.3e-05 +501 0.000163 +556 7.85e-05 +558 7.35e-05 +564 0.0002265 +585 7e-05 +599 7.15e-05 +600 0.0001475 +611 0.000126 +617 0.000144 +694 0.000165 +741 0.0001345 +773 7.3e-05 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/ContactMaps.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/ContactMaps.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 5.34361e-05 +102 5.37944e-05 +104 5.72603e-05 +105 5.08056e-05 +107 5.85278e-05 +114 4.97778e-05 +117 5.03278e-05 +120 5.69722e-05 +122 5.38111e-05 +123 5.88944e-05 +124 5.25778e-05 +125 5.69556e-05 +130 0.000113842 +133 0.000112197 +136 0.000104983 +137 0.000111296 +138 0.000113233 +140 0.000109111 +141 0.000110242 +144 0.000106004 +146 0.000103967 +147 0.000106579 +149 0.000107263 +150 0.000108846 +151 0.000108735 +156 0.000108296 +157 0.000111154 +158 0.000118977 +159 0.000112633 +162 0.000113525 +164 0.000110142 +169 0.000111287 +181 0.000115817 +185 0.000119525 +186 0.000119171 +187 0.000120792 +190 0.000120919 +191 0.0001214 +192 0.000122258 +193 0.000122033 +197 0.000119675 +198 0.000123142 +199 0.000124779 +214 0.000129204 +220 0.000131454 +221 0.000130183 +222 0.000132202 +224 0.000132292 +225 0.000133004 +226 0.000133733 +228 0.000134671 +237 0.000131837 +252 0.000140367 +264 0.000281152 +265 0.000246087 +271 0.000246452 +274 0.000267333 +275 0.00028788 +276 0.000271567 +277 0.000274595 +280 0.000261947 +286 0.000264503 +302 0.000276483 +327 0.000276143 +330 0.000269307 +334 0.000276257 +336 0.00027925 +359 0.000291737 +360 0.000289417 +385 0.00029817 +387 0.000299035 +388 0.00029937 +393 0.000297727 +395 0.0002969 +414 0.000316597 +428 0.000323677 +457 0.000334139 +484 0.00034918 +501 0.00035081 +556 0.000370577 +558 0.000365067 +564 0.000386503 +585 0.000374733 +599 0.00037959 +600 0.000385833 +611 0.00041654 +617 0.000419747 +694 0.000450283 +741 0.000466307 +773 0.00044286 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/Molecules.16.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/Molecules.16.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,83 @@ +17 0.00598246 +18 0.00633118 +19 0.00615402 +20 0.00641676 +21 0.00651908 +22 0.00684749 +23 0.00685846 +24 0.007018 +25 0.00716316 +26 0.00729213 +27 0.00745774 +28 0.00772876 +29 0.00784202 +30 0.00805273 +31 0.00842222 +32 0.00847537 +33 0.00952222 +34 0.00949043 +35 0.00969127 +36 0.010043 +37 0.0103157 +38 0.0107507 +39 0.0106857 +40 0.0109116 +41 0.0106991 +42 0.0118399 +43 0.0116813 +44 0.011622 +45 0.0123581 +46 0.0126685 +47 0.0123943 +48 0.0131893 +49 0.0133457 +50 0.0127309 +51 0.0134522 +52 0.0150284 +53 0.0142254 +54 0.0146606 +55 0.0149024 +56 0.013665 +57 0.0137624 +58 0.0148932 +59 0.0197308 +60 0.0176807 +61 0.0246768 +62 0.0288649 +63 0.02005 +64 0.0175171 +65 0.0186013 +66 0.0180474 +67 0.0163694 +68 0.0162941 +69 0.0164964 +70 0.017263 +71 0.0169762 +72 0.01469 +73 0.0157778 +74 0.0167954 +75 0.0162389 +76 0.0173212 +77 0.0170606 +78 0.0175944 +79 0.0189242 +80 0.0186143 +81 0.01926 +82 0.0205238 +83 0.0207133 +84 0.0190083 +85 0.0192167 +86 0.0203556 +87 0.0188833 +88 0.0243889 +89 0.01836 +90 0.0230722 +91 0.0207462 +92 0.0223111 +93 0.0205417 +94 0.0208333 +95 0.0190333 +96 0.0198933 +97 0.02035 +98 0.0249333 +99 0.0190667 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/Molecules.32.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/Molecules.32.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,67 @@ +33 0.00967075 +34 0.00954023 +35 0.0097031 +36 0.00971922 +37 0.0104818 +38 0.0100582 +39 0.00978127 +40 0.00989455 +41 0.00985741 +42 0.00995873 +43 0.00988224 +44 0.0099366 +45 0.00996648 +46 0.0104871 +47 0.0104315 +48 0.010056 +49 0.0101574 +50 0.0111132 +51 0.009967 +52 0.010341 +53 0.00995661 +54 0.010313 +55 0.0107411 +56 0.0104566 +57 0.00993382 +58 0.0103153 +59 0.0109633 +60 0.0115063 +61 0.0106761 +62 0.0108289 +63 0.010424 +64 0.0105851 +65 0.0100064 +66 0.0106108 +67 0.0109994 +68 0.01296 +69 0.0125679 +70 0.01038 +71 0.0101857 +72 0.012406 +73 0.0102367 +74 0.0143736 +75 0.0126133 +76 0.0130855 +77 0.0127636 +78 0.0133317 +79 0.0119945 +80 0.0121757 +81 0.01076 +82 0.0118971 +83 0.01198 +84 0.01401 +85 0.0113067 +86 0.00970667 +87 0.00958 +88 0.0107733 +89 0.019072 +90 0.01185 +91 0.0112323 +92 0.00990333 +93 0.009815 +94 0.01094 +95 0.0176733 +96 0.00986 +97 0.0149433 +98 0.01082 +99 0.01169 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/Molecules.64.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/Molecules.64.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,35 @@ +65 0.0419733 +66 0.0440936 +67 0.0383759 +68 0.0331088 +69 0.0365476 +70 0.0364593 +71 0.0340333 +72 0.0344667 +73 0.0367556 +74 0.0340264 +75 0.0374222 +76 0.0372697 +77 0.0370636 +78 0.0367472 +79 0.0379667 +80 0.0390786 +81 0.04476 +82 0.0386952 +83 0.04104 +84 0.0513278 +85 0.0638222 +86 0.0395556 +87 0.03925 +88 0.0399 +89 0.03898 +90 0.0404944 +91 0.0412667 +92 0.0432 +93 0.0427083 +94 0.0488 +95 0.0416222 +96 0.0439133 +97 0.0421222 +98 0.0440333 +99 0.0404 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/Molecules.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/Molecules.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +10 0.00362222 +11 0.00398 +12 0.0039 +13 0.00376364 +14 0.00443333 +15 0.00385 +16 0.00471301 +17 0.00424035 +18 0.00488602 +19 0.00442644 +20 0.00477781 +21 0.0049817 +22 0.00530661 +23 0.00555603 +24 0.00541107 +25 0.00556826 +26 0.00552901 +27 0.00589544 +28 0.00589352 +29 0.00624727 +30 0.00600751 +31 0.00614257 +32 0.00615823 +33 0.0065738 +34 0.00686808 +35 0.00691319 +36 0.00721727 +37 0.007503 +38 0.00775833 +39 0.00737915 +40 0.00761581 +41 0.00709501 +42 0.00877533 +43 0.00895025 +44 0.00885031 +45 0.0107127 +46 0.00855909 +47 0.00934234 +48 0.0100792 +49 0.0103586 +50 0.0081799 +51 0.00952556 +52 0.0104142 +53 0.0111808 +54 0.0115229 +55 0.0116895 +56 0.0101384 +57 0.0104727 +58 0.0117689 +59 0.0150443 +60 0.0128877 +61 0.0178024 +62 0.0215446 +63 0.016072 +64 0.0139744 +65 0.01676 +66 0.0153436 +67 0.0118056 +68 0.0083902 +69 0.0108595 +70 0.0118519 +71 0.0107095 +72 0.0100633 +73 0.0127556 +74 0.00794769 +75 0.0119556 +76 0.0146121 +77 0.0112273 +78 0.0101306 +79 0.0111788 +80 0.0128524 +81 0.0191867 +82 0.0192524 +83 0.0171933 +84 0.0119861 +85 0.0176111 +86 0.0241222 +87 0.0224167 +88 0.0201778 +89 0.00863333 +90 0.0208833 +91 0.0171949 +92 0.0282278 +93 0.022975 +94 0.0094 +95 0.0108111 +96 0.0197267 +97 0.0200444 +98 0.0239333 +99 0.01085 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/Molecules.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/Molecules.txt Tue Nov 22 08:15:16 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-r 76dc74f824ba -r 20d1b0e5838f Orig/contactMapIso.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/contactMapIso.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 0.386 +102 0.433 +104 0.475 +105 0.443 +107 0.426 +114 0.412 +117 0.457 +120 0.515 +122 0.551 +123 0.537 +124 0.462 +125 0.63 +130 0.589 +133 0.629 +136 0.621 +137 0.653 +138 0.615 +140 0.682 +141 0.623 +144 0.649 +146 0.344 +147 0.65 +149 0.541 +150 0.627 +151 0.631 +156 0.733 +157 0.723 +158 0.739 +159 0.715 +162 0.729 +164 0.701 +169 0.769 +181 0.789 +185 0.919 +186 0.917 +187 0.915 +190 0.953 +191 0.954 +192 0.952 +193 0.965 +197 1.009 +198 1.055 +199 1.026 +214 0.975 +220 1.093 +221 1.07 +222 1.107 +224 1.115 +225 1.156 +226 1.162 +228 1.178 +237 1.23 +252 1.299 +264 1.401 +265 1.343 +271 1.404 +274 1.538 +275 1.623 +276 1.503 +277 1.471 +280 1.601 +286 1.612 +302 1.696 +327 1.783 +330 1.761 +334 1.964 +336 1.839 +359 2.097 +360 2.163 +385 2.117 +387 2.149 +388 2.147 +393 2.207 +395 2.236 +414 2.352 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0.01419 +37 0.01445 +38 0.01471 +39 0.01562 +40 0.01423 +41 0.01704 +42 0.01971 +43 0.01602 +44 0.02022 +45 0.01618 +46 0.02056 +47 0.01739 +48 0.01871 +49 0.01775 +50 0.01764 +51 0.01992 +52 0.01969 +53 0.01888 +54 0.01948 +55 0.02089 +56 0.01696 +57 0.01885 +58 0.02133 +59 0.02238 +60 0.0212 +61 0.02596 +62 0.02761 +63 0.02743 +64 0.02153 +65 0.02993 +66 0.03349 +67 0.0314 +68 0.0351 +69 0.0246 +70 0.02679 +71 0.03377 +72 0.0278 +73 0.02835 +74 0.02895 +75 0.02715 +76 0.03037 +77 0.03238 +78 0.03211 +79 0.03462 +80 0.03723 +81 0.03793 +82 0.03227 +83 0.03539 +84 0.03406 +85 0.03441 +86 0.04042 +87 0.03769 +88 0.03697 +89 0.04212 +91 0.03941 +93 0.04149 +94 0.04669 +95 0.04263 +96 0.04419 +97 0.04054 +98 0.04179 +99 0.04812 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/proteinIso.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/proteinIso.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,234 @@ +535 1.069 +554 1.143 +571 1.299 +614 1.331 +634 1.439 +636 1.477 +639 1.451 +730 1.87 +731 1.903 +734 2.025 +735 1.897 +737 1.938 +755 1.982 +760 2.175 +788 2.207 +794 2.373 +795 2.183 +802 2.206 +816 2.268 +819 2.419 +831 2.409 +833 2.432 +838 2.376 +858 2.511 +861 2.592 +874 2.581 +877 2.585 +879 2.609 +886 2.851 +899 2.731 +957 3.039 +982 3.273 +995 3.267 +1019 3.489 +1022 3.462 +1046 3.622 +1049 3.629 +1060 3.837 +1073 3.861 +1076 3.853 +1097 4.068 +1140 4.22 +1156 4.367 +1183 4.528 +1192 4.608 +1202 5.595 +1227 4.938 +1271 5.279 +1300 5.488 +1304 5.437 +1312 5.72 +1356 5.976 +1494 7.134 +1502 8.15 +1559 7.822 +1568 7.824 +1576 8.457 +1605 8.123 +1608 8.801 +1625 8.968 +1632 9.29 +1635 9.113 +1648 9.986 +1674 8.959 +1684 9.119 +1732 9.504 +1744 10.306 +1750 9.97 +1754 11.228 +1764 10.573 +1775 10.007 +1826 10.534 +1844 11.526 +1886 11.179 +1930 11.844 +1938 11.87 +1948 11.886 +1995 12.518 +2000 13.394 +2020 12.788 +2044 13.162 +2086 14.446 +2115 14.052 +2123 14.273 +2126 14.565 +2138 14.486 +2168 14.635 +2190 14.84 +2222 15.311 +2250 15.622 +2298 16.504 +2314 16.704 +2316 16.47 +2325 18.018 +2327 16.582 +2329 16.898 +2344 17.201 +2348 16.955 +2355 17.22 +2379 17.399 +2383 17.421 +2413 17.892 +2423 17.98 +2464 18.903 +2466 18.911 +2512 19.285 +2546 19.636 +2553 20.608 +2593 20.407 +2604 20.587 +2605 22.925 +2607 22.02 +2610 22.058 +2611 20.714 +2612 22.399 +2613 20.741 +2616 20.734 +2620 20.816 +2622 20.836 +2625 23.002 +2648 21.074 +2651 21.203 +2716 22.383 +2733 22.638 +2775 23.708 +2812 24.236 +2838 24.318 +2857 24.726 +2872 24.967 +2887 25.232 +2916 25.613 +2920 25.956 +2928 26.125 +2949 26.15 +2966 26.491 +2977 28.792 +2991 26.936 +3104 29.15 +3130 29.567 +3131 29.599 +3134 29.537 +3135 29.58 +3172 32.619 +3182 30.799 +3209 30.968 +3243 31.605 +3255 31.917 +3310 32.849 +3366 37.254 +3429 35.516 +3459 35.948 +3469 36.118 +3483 36.311 +3497 37.724 +3502 36.693 +3505 36.826 +3515 37.302 +3517 37 +3522 37.049 +3523 37.207 +3526 37.279 +3529 37.504 +3560 37.933 +3563 37.97 +3566 38.104 +3574 41.274 +3582 38.911 +3584 38.693 +3618 42.161 +3619 39.329 +3623 42.255 +3703 41.236 +3711 41.072 +3819 43.676 +3829 43.854 +3839 44.288 +3870 44.748 +3906 45.735 +4025 48.558 +4083 50.176 +4156 52.911 +4194 52.261 +4220 52.864 +4244 57.808 +4283 58.769 +4353 56.757 +4382 56.877 +4389 57.042 +4421 57.984 +4446 59.382 +4521 60.846 +4531 60.868 +4532 60.751 +4568 62.485 +4573 66.916 +4582 62.751 +4604 67.785 +4629 64.027 +4710 66.067 +4713 66.413 +4797 68.111 +4879 70.489 +4970 72.896 +5059 75.988 +5126 77.531 +5241 80.788 +5254 81.485 +5271 82.209 +5318 89.984 +5346 84.387 +5455 87.669 +5492 88.66 +5608 92.515 +5735 97.315 +5775 98.216 +5777 98.269 +5795 99.811 +5845 100.942 +5859 101.083 +5877 101.587 +5945 105.329 +5974 104.948 +5976 104.704 +5983 105.662 +6261 115.383 +6606 128.002 +6713 132.525 +6778 145.161 +6883 138.36 +7349 158.714 +7453 163.23 +9114 242.24 +9174 244.585 +10081 295.288 diff -r 76dc74f824ba -r 20d1b0e5838f Orig/proteinsIso.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Orig/proteinsIso.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,234 @@ +535 0.1026 +554 0.1098 +571 0.1199 +614 0.131 +634 0.1407 +636 0.1439 +639 0.1433 +730 0.1818 +731 0.1823 +734 0.1857 +735 0.1834 +737 0.1883 +755 0.1935 +760 0.1947 +788 0.2109 +794 0.214 +795 0.2146 +802 0.2168 +816 0.225 +819 0.227 +831 0.2305 +833 0.2311 +838 0.2345 +858 0.2436 +861 0.2476 +874 0.2516 +877 0.2528 +879 0.257 +886 0.2598 +899 0.2651 +957 0.2984 +982 0.316 +995 0.3251 +1019 0.3412 +1022 0.3391 +1046 0.3555 +1049 0.3567 +1060 0.3628 +1073 0.3769 +1076 0.3735 +1097 0.3869 +1140 0.4171 +1156 0.4276 +1183 0.4452 +1192 0.4535 +1202 0.4626 +1227 0.4815 +1271 0.5108 +1300 0.5313 +1304 0.5402 +1312 0.5451 +1356 0.5765 +1494 0.7052 +1502 0.7017 +1559 0.7573 +1568 0.7675 +1576 0.7678 +1605 0.7975 +1608 0.8003 +1625 0.8217 +1632 0.8223 +1635 0.8315 +1648 0.8444 +1674 0.8698 +1684 0.8829 +1732 0.9302 +1744 0.9393 +1750 0.9475 +1754 0.9523 +1764 0.9621 +1775 0.9685 +1826 1.0346 +1844 1.0558 +1886 1.0843 +1930 1.1393 +1938 1.1485 +1948 1.1616 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16.1769 +9114 24.0921 +9174 24.3965 +10081 29.5504 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PP/ContactMaps.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PP/ContactMaps.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 3.5e-06 +102 7.5e-06 +104 4e-06 +105 4.5e-06 +107 1.15e-05 +114 4e-06 +117 4e-06 +120 0.000131 +122 1.7e-05 +123 4e-06 +124 5.5e-06 +125 4e-06 +130 5e-06 +133 7e-05 +136 4e-06 +137 7.9e-05 +138 1.65e-05 +140 4e-06 +141 4.5e-06 +144 4e-06 +146 5.5e-06 +147 1.5e-05 +149 4.5e-06 +150 5.5e-06 +151 5.7e-05 +156 4.5e-06 +157 7.95e-05 +158 6.3e-05 +159 4e-06 +162 4.5e-06 +164 4.65e-05 +169 7e-06 +181 2.55e-05 +185 4e-05 +186 5.4e-05 +187 8.1e-05 +190 6e-06 +191 8.5e-06 +192 8.5e-06 +193 8e-06 +197 5e-06 +198 5e-06 +199 8e-06 +214 2.95e-05 +220 1.55e-05 +221 1.5e-05 +222 1.55e-05 +224 1.55e-05 +225 1.6e-05 +226 1.75e-05 +228 4.45e-05 +237 5e-06 +252 1.35e-05 +264 5.9e-05 +265 2.95e-05 +271 6e-06 +274 1.25e-05 +275 8.5e-06 +276 0.000102 +277 9.5e-06 +280 2.15e-05 +286 2.95e-05 +302 1.2e-05 +327 4.85e-05 +330 7e-06 +334 1.3e-05 +336 3.45e-05 +359 3.75e-05 +360 9.5e-06 +385 2.15e-05 +387 2.1e-05 +388 4.25e-05 +393 8e-06 +395 1.3e-05 +414 5.5e-05 +428 3.45e-05 +457 1.6e-05 +484 2.2e-05 +501 0.0001035 +556 2.05e-05 +558 1.8e-05 +564 0.0001795 +585 1e-05 +599 1.05e-05 +600 2.35e-05 +611 6e-05 +617 7.65e-05 +694 4.2e-05 +741 5.1e-05 +773 1.25e-05 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/ContactMaps.128.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/ContactMaps.128.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,78 @@ +130 0.21792 +133 0.206453 +136 0.18227 +137 0.19744 +138 0.194798 +140 0.183628 +141 0.197397 +144 0.18094 +146 0.18127 +147 0.18145 +149 0.18299 +150 0.19709 +151 0.182827 +156 0.18175 +157 0.18332 +158 0.201365 +159 0.18257 +162 0.18674 +164 0.18161 +169 0.18196 +181 0.183455 +185 0.18381 +186 0.18457 +187 0.185 +190 0.184773 +191 0.18527 +192 0.18578 +193 0.18477 +197 0.18346 +198 0.19277 +199 0.185593 +214 0.18428 +220 0.184415 +221 0.184715 +222 0.184155 +224 0.18542 +225 0.18469 +226 0.19775 +228 0.18681 +237 0.18285 +252 0.18522 +264 0.183765 +265 0.18731 +271 0.18376 +274 0.1883 +275 0.18192 +276 0.19777 +277 0.18465 +280 0.18727 +286 0.18944 +302 0.18793 +327 0.18918 +330 0.1847 +334 0.23633 +336 0.18691 +359 0.18908 +360 0.18771 +385 0.188424 +387 0.18786 +388 0.188245 +393 0.187311 +395 0.1865 +414 0.19499 +428 0.19705 +457 0.188996 +484 0.19269 +501 0.190735 +556 0.18734 +558 0.18791 +564 0.19292 +585 0.18656 +599 0.18901 +600 0.18782 +611 0.194 +617 0.19671 +694 0.19161 +741 0.19442 +773 0.18798 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/ContactMaps.16.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/ContactMaps.16.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 0.019345 +102 0.0150133 +104 0.0227257 +105 0.01121 +107 0.01894 +114 0.01077 +117 0.00981 +120 0.015715 +122 0.0131767 +123 0.01938 +124 0.01095 +125 0.01409 +130 0.0141 +133 0.0190733 +136 0.01496 +137 0.0161 +138 0.0170433 +140 0.01501 +141 0.012775 +144 0.01467 +146 0.00958 +147 0.01204 +149 0.01141 +150 0.01426 +151 0.0202267 +156 0.01272 +157 0.01638 +158 0.01535 +159 0.01484 +162 0.01691 +164 0.01168 +169 0.00949 +181 0.014425 +185 0.01558 +186 0.01418 +187 0.02293 +190 0.0151533 +191 0.01364 +192 0.01509 +193 0.01429 +197 0.01203 +198 0.01302 +199 0.0142067 +214 0.01951 +220 0.02259 +221 0.02243 +222 0.023105 +224 0.02043 +225 0.0222 +226 0.02513 +228 0.03011 +237 0.01415 +252 0.0301 +264 0.034265 +265 0.02139 +271 0.03121 +274 0.01806 +275 0.01844 +276 0.0323 +277 0.03082 +280 0.03764 +286 0.03835 +302 0.07492 +327 0.02595 +330 0.03104 +334 0.03348 +336 0.02795 +359 0.03069 +360 0.03952 +385 0.036552 +387 0.038095 +388 0.03684 +393 0.03682 +395 0.03583 +414 0.03295 +428 0.03886 +457 0.075984 +484 0.04126 +501 0.048595 +556 0.04653 +558 0.04114 +564 0.05777 +585 0.04359 +599 0.04116 +600 0.05301 +611 0.07247 +617 0.09244 +694 0.0529 +741 0.04632 +773 0.03695 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/ContactMaps.256.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/ContactMaps.256.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,37 @@ +264 0.59937 +265 0.47191 +271 0.45243 +274 0.54542 +275 0.62296 +276 0.54024 +277 0.54541 +280 0.45277 +286 0.45466 +302 0.49129 +327 0.45317 +330 0.45594 +334 0.48659 +336 0.45676 +359 0.46156 +360 0.46787 +385 0.459854 +387 0.45049 +388 0.46847 +393 0.4521 +395 0.45054 +414 0.47651 +428 0.46057 +457 0.473196 +484 0.45915 +501 0.462805 +556 0.46011 +558 0.45866 +564 0.4773 +585 0.46391 +599 0.46784 +600 0.46408 +611 0.47136 +617 0.49935 +694 0.48924 +741 0.46741 +773 0.45467 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/ContactMaps.32.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/ContactMaps.32.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 0.03069 +102 0.02584 +104 0.0261714 +105 0.021855 +107 0.03249 +114 0.02201 +117 0.02163 +120 0.0219 +122 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0.03184 +611 0.04138 +617 0.04892 +694 0.03757 +741 0.0387 +773 0.03241 diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/ContactMaps.64.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/ContactMaps.64.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,90 @@ +99 0.062625 +102 0.0682767 +104 0.07076 +105 0.06959 +107 0.07619 +114 0.06267 +117 0.06185 +120 0.070745 +122 0.0654967 +123 0.06279 +124 0.06319 +125 0.06227 +130 0.06383 +133 0.0635533 +136 0.06295 +137 0.0644833 +138 0.0673367 +140 0.06757 +141 0.0648283 +144 0.062 +146 0.06235 +147 0.06422 +149 0.06357 +150 0.06665 +151 0.06307 +156 0.06283 +157 0.065105 +158 0.071395 +159 0.06262 +162 0.0647878 +164 0.06265 +169 0.06325 +181 0.063735 +185 0.06434 +186 0.06447 +187 0.06389 +190 0.0682533 +191 0.06632 +192 0.06668 +193 0.06848 +197 0.06189 +198 0.06466 +199 0.0685467 +214 0.06483 +220 0.06617 +221 0.064295 +222 0.066105 +224 0.06561 +225 0.06577 +226 0.06735 +228 0.06575 +237 0.06392 +252 0.06848 +264 0.065955 +265 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@@ -0,0 +1,90 @@ +99 3.55861e-05 +102 3.48204e-05 +104 3.82857e-05 +105 3.27361e-05 +107 4.155e-05 +114 2.985e-05 +117 2.91611e-05 +120 3.40694e-05 +122 3.21074e-05 +123 3.34333e-05 +124 3.04222e-05 +125 3.40778e-05 +130 7.72042e-05 +133 7.44806e-05 +136 6.73958e-05 +137 7.2275e-05 +138 7.38313e-05 +140 6.9265e-05 +141 7.06243e-05 +144 6.62625e-05 +146 6.535e-05 +147 6.59417e-05 +149 6.74542e-05 +150 6.75e-05 +151 6.84764e-05 +156 6.6225e-05 +157 6.81729e-05 +158 7.44188e-05 +159 6.7125e-05 +162 6.9319e-05 +164 6.64e-05 +169 6.52042e-05 +181 6.78417e-05 +185 6.90083e-05 +186 6.86083e-05 +187 7.00875e-05 +190 6.96403e-05 +191 6.99542e-05 +192 7.03875e-05 +193 6.995e-05 +197 6.64583e-05 +198 6.95e-05 +199 7.16069e-05 +214 7.11792e-05 +220 7.20187e-05 +221 7.10063e-05 +222 7.22021e-05 +224 7.13417e-05 +225 7.18417e-05 +226 7.27333e-05 +228 7.45125e-05 +237 6.83292e-05 +252 7.39167e-05 +264 0.00016664 +265 0.00013917 +271 0.000139285 +274 0.000156853 +275 0.000175157 +276 0.000159083 +277 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diff -r 76dc74f824ba -r 20d1b0e5838f VF2PPLabel/moleculeIso.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/VF2PPLabel/moleculeIso.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,91 @@ +8 0.007 +10 0.007 +11 0.01 +12 0.01 +13 0.007 +14 0.01 +15 0.012 +16 0.009 +17 0.016 +18 0.009 +19 0.008 +20 0.013 +21 0.014 +22 0.013 +23 0.014 +24 0.013 +25 0.023 +26 0.016 +27 0.015 +28 0.018 +29 0.02 +30 0.017 +31 0.02 +32 0.021 +33 0.027 +34 0.023 +35 0.021 +36 0.026 +37 0.025 +38 0.028 +39 0.026 +40 0.026 +41 0.03 +42 0.031 +43 0.024 +44 0.029 +45 0.027 +46 0.034 +47 0.028 +48 0.031 +49 0.031 +50 0.027 +51 0.034 +52 0.042 +53 0.031 +54 0.032 +55 0.04 +56 0.026 +57 0.03 +58 0.039 +59 0.037 +60 0.036 +61 0.038 +62 0.044 +63 0.048 +64 0.035 +65 0.057 +66 0.051 +67 0.055 +68 0.055 +69 0.039 +70 0.038 +71 0.083 +72 0.039 +73 0.042 +74 0.037 +75 0.035 +76 0.047 +77 0.043 +78 0.046 +79 0.041 +80 0.048 +81 0.049 +82 0.05 +83 0.048 +84 0.048 +85 0.048 +86 0.062 +87 0.069 +88 0.055 +89 0.056 +90 0.087 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This is not technically correct. +%I did something largely similar with Walter Lewin's video lectures, +%(which are wonderful), posted online through MIT OCW. Since a +%'proper' way of citing it wasn't built in, I made one up. If +%you're doing a bibliography where correct formatting is very +%important, you'd do best not to follow my example. Otherwise, +%I think it's good enough. + +@Article{CordellaVentoSymbolRecognition, + author = "L.P. Cordella, M. Vento", + title = "Symbol recognition in documents: a collection of techniques?", + journal = "International Journal on Document Analysis and Recognition Volume 3, Issue 2, pp 73-88", + year = 2000, + month = "December" +} + +@Article{ProteinDataBank, + author = "Protein Data Bank", + title = "http://www.rcsb.org/pdb", + journal = "", + year = 2015, + month = "June" +} + +@Article{LEMON, + author = "LEMON: Library for Efficient Modeling and Optimization in Networks", + title = "https://lemon.cs.elte.hu", + journal = "", + year = 2015, + month = "March" +} + +@Article{QUANTUMBIO, + author = "QuantumBio Inc.", + title = "http://www.quantumbioinc.com/" +} + +@Article{Content, + author = "Vento, Mario and Jiang, Xiaoyi and Foggia, Pasquale", + title = "International contest on pattern search in biological databases, http://biograph2014.unisa.it ", + journal = "", + year = 2015, + month = "June" +} + +@Article{VF, + author = "L. P. Cordella, P. Foggia, C. Sansone and M. Vento", + title = "Performance Evaluation of the VF Graph Matching Algorithm", + journal = "Proc. of the 10th ICIAP, IEEE Computer Society Press, pp. 1172-1177", + year = 1999, + month = "" +} + +@Article{UllmannBit, + author = "Julian R. Ullmann", + title = "Bit-vector algorithms for binary constraint satisfaction and subgraph isomorphism", + journal = "Journal of Experimental Algorithmics (JEA),Volume 15, Article No. 1.6, ACM New York, NY, USA ", + year = 2010, + month = "" +} + +@Article{Ullmann, + author = "J. R. Ullmann", + title = "An Algorithm for Subgraph Isomorphism", + journal = "Journal of the ACM (JACM), Volume 23 Issue 1, Pages 31-42 ", + year = 1976, + month = "January" +} + +@Article{SubgraphNPC, + author = "Cook, S. A.", + title = "The complexity of theorem-proving procedures", + journal = "Proc. 3rd ACM Symposium on Theory of Computing, pp. 151-158", + year = 1971, + month = "" +} + +@Article{VF2, + author = "Cordella, L.P. and Foggia, P. and Sansone, C. and Vento, M.", + title = "A (sub)graph isomorphism algorithm for matching large graphs ", + journal = "Journal of the ACM (JACM) JACM Homepage archive Volume 23 Issue 1, Pages 31-42 ", + year = 2004, + month = "" +} + +@Article{VF2Plus, + author = "Carletti Vincenzo and Pasquale Foggia and Mario Vento", + title = "VF2 Plus: An Improved Version of VF2 For Biological Graphs", + journal = "Conference: Graph-Based Representations in Pattern Recognition, At Beijing ", + year = 2015, + month = "May" +} + +@Article{RI, + author = "Vincenzo Bonnici and Rosalba Giugno and Alfredo Pulvirenti and Dennis Shasha and Alfredo Ferro", + title = "A subgraph isomorphism algorithm and its application to biochemical data", + journal = "BMC Bioinformatics. 2013; 14(Suppl 7): S13. ", + year = 2013, + month = "April" +} + +@Article{Lad, + author = "Brendan D. McKay", + title = "AllDifferent-based filtering for subgraph isomorphism", + journal = "Artificial Intelligence 174,850-864", + year = 2010, + month = "" +} +@Article{Nauty, + author = "Christine Solnon", + title = "Practical Graph Isomorphism", + journal = "Congressus Numerantium, vol.30, pp. 45-87", + year = 1981, + month = "" +} + +@Article{JianzhuangYongFaceIdentification, + author = "Jianzhuang Liu and Yong Tsui Lee", + title = "A Graph-Based Method for Face Identification from a Single 2D Line Drawing", + journal = "IEEE Transactions on Pattern Analysis and Machine Intelligence - Graph Algorithms and Computer Vision archive Volume 23 Issue 10", + year = 2001, + month = "October" +} + +@Article{HorstBunkeApplications, + author = "Horst Bunke", + title = "Graph Matching: Theoretical Foundations, Algorithms, and Applications", + journal = "In International Conference on Vision Interface, pp. 82-84", + year = 2000, + month = "May" +} + +@Article{AlexandruApplications, + author = "Alexandru T. Balaban", + title = "Applications of graph theory in chemistry", + journal = "J. Chem. Inf. Comput. Sci., 1985,25(3),pp 334-343", + year = 1985, + month = "March" +} + +@Article{PlanarGraphIso, + author = "J. E. Hopcroft and J. K. Wong", + title = "Linear time algorithm for isomorphism of planar graphs", + journal = "Proceeding STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing, Pages 172-184 ", + year = 1974, + month = "April" +} + +@Article{BondedDegGraphIso, + author = "Eugene M. Luks", + title = "Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time", + journal = "Journal of Computer and System Sciences, Volume 25, Issue 1, Pages 42-65", + year = 1982, + month = "August" +} + +@Article{IntervalGraphIso, + author = "George S. Lue and Kellogg S. Booth", + title = "A Linear Time Algorithm for Deciding Interval Graph Isomorphism", + journal = "Journal of the ACM (JACM), Volume 26, Issue 2, Pages 183-195 ", + year = 1979, + month = "April" +} + +@Article{PermGraphIso, + author = "Charles J. Colbourn", + title = "On testing isomorphism of permutation graphs", + journal = "Networks, Volume 11, Issue 1, Pages 13-21 ", + year = 1981, + month = "March" +} diff -r 76dc74f824ba -r 20d1b0e5838f damecco.tex --- a/damecco.tex Fri Nov 18 16:45:03 2016 +0100 +++ b/damecco.tex Tue Nov 22 08:15:16 2016 +0100 @@ -47,6 +47,34 @@ %% for the whole article with \linenumbers. %% \usepackage{lineno} +\usepackage{amsmath} +%% \usepackage[pdftex]{graphicx} + +\usepackage{pgfplots} +\pgfplotsset{width=9cm} +\pgfplotsset{compat=1.8} + +\usepackage{caption} +\usepackage{subcaption} + +\usepackage{algorithm} +\usepackage{algpseudocode} +\usepackage{tikz} + +\usepackage{amsthm,amssymb} +\renewcommand{\qedsymbol}{\rule{0.7em}{0.7em}} + +\newtheorem{theorem}{Theorem}[subsection] +\newtheorem{corollary}{Corollary}[theorem] +\newtheorem{claim}[theorem]{Claim} + +\newtheorem{definition}{Definition}[subsection] +\newtheorem{notation}{Notation}[subsection] +\newtheorem{example}{Example}[subsection] +\usetikzlibrary{decorations.markings} +\let\oldproofname=\proofname +%% \renewcommand{\proofname}{\rm\bf{Proof:}} + \journal{Discrete Applied Mathematics} \begin{document} @@ -147,8 +175,1201 @@ %% \linenumbers %% main text -\section{} -\label{} +\section{Introduction} +\label{sec:intro} + +In the last decades, combinatorial structures, and especially graphs have been considered with ever increasing interest, and applied to the solution of several new and revised questions. +The expressiveness, the simplicity and the studiedness of graphs make them practical for modelling and appear constantly in several seemingly independent fields. +Bioinformatics and chemistry are amongst the most relevant and most important fields. + +Complex biological systems arise from the interaction and cooperation of plenty of molecular components. Getting acquainted with such systems at the molecular level has primary importance, since protein-protein interaction, DNA-protein interaction, metabolic interaction, transcription factor binding, neuronal networks, and hormone signaling networks can be understood only this way. + +For instance, a molecular structure can be considered as a graph, whose nodes correspond to atoms and whose edges to chemical bonds. The secondary structure of a protein can also be represented as a graph, where nodes are associated with aminoacids and the edges with hydrogen bonds. The nodes are often whole molecular components and the edges represent some relationships among them. +The similarity and dissimilarity of objects corresponding to nodes are incorporated to the model by \emph{node labels}. +Many other chemical and biological structures can easily be modeled in a similar way. Understanding such networks basically requires finding specific subgraphs, which can not avoid the application of graph matching algorithms. + +Finally, let some of the other real-world fields related to some variants of graph matching be briefly mentioned: pattern recognition and machine vision \cite{HorstBunkeApplications}, symbol recognition \cite{CordellaVentoSymbolRecognition}, face identification \cite{JianzhuangYongFaceIdentification}. +\\ + +Subgraph and induced subgraph matching problems are known to be NP-Complete\cite{SubgraphNPC}, while the graph isomorphism problem is one of the few problems in NP neither known to be in P nor NP-Complete. Although polynomial time isomorphism algorithms are known for various graph classes, like trees and planar graphs\cite{PlanarGraphIso}, bounded valence graphs\cite{BondedDegGraphIso}, interval graphs\cite{IntervalGraphIso} or permutation graphs\cite{PermGraphIso}. + +In the following, some algorithms based on other approaches are summarized, which do not need any restrictions on the graphs. However, an overall polynomial behaviour is not expectable from such an alternative, it may often have good performance, even on a graph class for which polynomial algorithm is known. Note that this summary containing only exact matching algorithms is far not complete, neither does it cover all the recent algorithms. + +The first practically usable approach was due to \textbf{Ullmann}\cite{Ullmann} which is a commonly used depth-first search based algorithm with a complex heuristic for reducing the number of visited states. A major problem is its $\Theta(n^3)$ space complexity, which makes it impractical in the case of big sparse graphs. + +In a recent paper, \textbf{Ullmann}\cite{UllmannBit} presents an improved version of this algorithm based on a bit-vector solution for the binary Constraint Satisfaction Problem. + +The \textbf{Nauty} algorithm\cite{Nauty} transforms the two graphs to a canonical form before starting to check for the isomorphism. It has been considered as one of the fastest graph isomorphism algorithms, although graph categories were shown in which it takes exponentially many steps. This algorithm handles only the graph isomorphism problem. + +The \textbf{LAD} algorithm\cite{Lad} uses a depth-first search strategy and formulates the matching as a Constraint Satisfaction Problem to prune the search tree. The constraints are that the mapping has to be injective and edge-preserving, hence it is possible to handle new matching types as well. + +The \textbf{RI} algorithm\cite{RI} and its variations are based on a state space representation. After reordering the nodes of the graphs, it uses some fast executable heuristic checks without using any complex pruning rules. It seems to run really efficiently on graphs coming from biology, and won the International Contest on Pattern Search in Biological Databases\cite{Content}. + +The currently most commonly used algorithm is the \textbf{VF2}\cite{VF2}, the improved version of VF\cite{VF}, which was designed for solving pattern matching and computer vision problems, and has been one of the best overall algorithms for more than a decade. Although, it can't be up to new specialized algorithms, it is still widely used due to its simplicity and space efficiency. VF2 uses a state space representation and checks some conditions in each state to prune the search tree. + +Our first graph matching algorithm was the first version of VF2 which recognizes the significance of the node ordering, more opportunities to increase the cutting efficiency and reduce its computational complexity. This project was initiated and sponsored by QuantumBio Inc.\cite{QUANTUMBIO} and the implementation --- along with a source code --- has been published as a part of LEMON\cite{LEMON} open source graph library. + +This thesis introduces \textbf{VF2++}, a new further improved algorithm for the graph and (induced)subgraph isomorphism problem, which uses efficient cutting rules and determines a node order in which VF2 runs significantly faster on practical inputs. + +Meanwhile, another variant called \textbf{VF2 Plus}\cite{VF2Plus} has been published. It is considered to be as efficient as the RI algorithm and has a strictly better behavior on large graphs. The main idea of VF2 Plus is to precompute a heuristic node order of the small graph, in which the VF2 works more efficiently.\newline +\newline + +\section{Problem Statement} +This section provides a detailed description of the problems to be solved. +\subsection{Definitions} + +Throughout the paper $G_{small}=(V_{small}, E_{small})$ and $G_{large}=(V_{large}, E_{large})$ denote two undirected graphs. +\begin{definition}\label{sec:ismorphic} +$G_{small}$ and $G_{large}$ are \textbf{isomorphic} if $\exists M: V_{small} \longrightarrow V_{large}$ bijection, for which the following is true: +\begin{center} +$\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (M(u),M(v))\in{E_{large}}$ +\end{center} +\end{definition} +For the sake of simplicity in this paper subgraphs and induced subgraphs are defined in a more general way than usual: +\begin{definition} +$G_{small}$ is a \textbf{subgraph} of $G_{large}$ if $\exists I: V_{small}\longrightarrow V_{large}$ injection, for which the following is true: +\begin{center} +$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Rightarrow (I(u),I(v))\in E_{large}$ +\end{center} +\end{definition} + +\begin{definition} +$G_{small}$ is an \textbf{induced subgraph} of $G_{large}$ if $\exists I: V_{small}\longrightarrow V_{large}$ injection, for which the following is true: +\begin{center} +$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (I(u),I(v))\in E_{large}$ +\end{center} +\end{definition} + +\begin{definition} +$lab: (V_{small}\cup V_{large}) \longrightarrow K$ is a \textbf{node label function}, where K is an arbitrary set. The elements in K are the \textbf{node labels}. Two nodes, u and v are said to be \textbf{equivalent}, if $lab(u)=lab(v)$. +\end{definition} + +When node labels are also given, the matched nodes must have the same labels. +For example, the node labeled isomorphism is phrased by +\begin{definition} +$G_{small}$ and $G_{large}$ are \textbf{isomorphic by the node label function lab} if $\exists M: V_{small} \longrightarrow V_{large}$ bijection, for which the following is true: +\begin{center} +$(\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (M(u),M(v))\in{E_{large}})$ + and $(\forall u\in{V_{small}} : lab(u)=lab(M(u)))$ +\end{center} +\end{definition} + +The other two definitions can be extended in the same way. + +Note that edge label function can be defined similarly to node label function, and all the definitions can be extended with additional conditions, but it is out of the scope of this work. + +The equivalence of two nodes is usually defined by another relation, $\\R\subseteq (V_{small}\cup V_{large})^2$. This overlaps with the definition given above if R is an equivalence relation, which does not mean restriction in biological and chemical applications. + +\subsection{Common problems}\label{sec:CommProb} + +The focus of this paper is on two extensively studied topics, the subgraph isomorphism and its variations. However, the following problems also appear in many applications. + +The \textbf{subgraph matching problem} is the following: is $G_{small}$ isomorphic to any subgraph of $G_{large}$ by a given node label? + +The \textbf{induced subgraph matching problem} asks the same about the existence of an induced subgraph. + +The \textbf{graph isomorphism problem} can be defined as induced subgraph matching problem where the sizes of the two graphs are equal. + +In addition to existence, it may be needed to show such a subgraph, or it may be necessary to list all of them. + +It should be noted that some authors misleadingly refer to the term \emph{subgraph isomorphism problem} as an \emph{induced subgraph isomorphism problem}. + +The following sections give the descriptions of VF2, VF2++, VF2 Plus and a particular comparison. + +\section{The VF2 Algorithm} +This algorithm is the basis of both the VF2++ and the VF2 Plus. +VF2 is able to handle all the variations mentioned in \textbf{Section \ref{sec:CommProb})}. +Although it can also handle directed graphs, for the sake of simplicity, only the undirected case will be discussed. + + +\subsection{Common notations} +\indent +Assume $G_{small}$ is searched in $G_{large}$. +The following definitions and notations will be used throughout the whole paper. +\begin{definition} +A set $M\subseteq V_{small}\times V_{large}$ is called \textbf{mapping}, if no node of $V_{small}$ or of $V_{large}$ appears in more than one pair in M. +That is, M uniquely associates some of the nodes in $V_{small}$ with some nodes of $V_{large}$ and vice versa. +\end{definition} + +\begin{definition} +Mapping M \textbf{covers} a node v, if there exists a pair in M, which contains v. +\end{definition} + +\begin{definition} +A mapping $M$ is $\mathbf{whole\ mapping}$, if $M$ covers all the nodes in $V_{small}$. +\end{definition} + +\begin{notation} +Let $\mathbf{M_{small}(s)} := \{u\in V_{small} : \exists v\in V_{large}: (u,v)\in M(s)\}$ and $\mathbf{M_{large}(s)} := \{v\in V_{large} : \exists u\in V_{small}: (u,v)\in M(s)\}$. +\end{notation} + +\begin{notation} +Let $\mathbf{Pair(M,v)}$ be the pair of $v$ in $M$, if such a node exist, otherwise $\mathbf{Pair(M,v)}$ is undefined. For a mapping $M$ and $v\in V_{small}\cup V_{large}$. +\end{notation} + +Note that if $\mathbf{Pair(M,v)}$ exists, then it is unique + +The definitions of the isomorphism types can be rephrased on the existence of a special whole mapping $M$, since it represents a bijection. For example +\begin{center} +$M\subseteq V_{small}\times V_{large}$ represents an induced subgraph isomorphism $\Leftrightarrow$ $M$ is whole mapping and $\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow (Pair(M,u),Pair(M,v))\in E_{large}$. +\end{center} + +\begin{definition} +A set of whole mappings is called \textbf{problem type}. +\end{definition} +Throughout the paper, $\mathbf{PT}$ denotes a generic problem type which can be substituted by any problem type. + +A whole mapping $W\mathbf{\ is\ of\ type\ PT}$, if $W\in PT$. Using this notations, VF2 searches a whole mapping $W$ of type $PT$. + +For example the problem type of graph isomorphism problem is the following. +A whole mapping $W$ is in $\mathbf{ISO}$, iff the bijection represented by $W$ satisfies \textbf{Definition \ref{sec:ismorphic})}. +The subgraph- and induced subgraph matching problems can be formalized in a similar way. Let their problem types be denoted as $\mathbf{SUB}$ and $\mathbf{IND}$. + +\begin{definition} +\label{expPT} +$PT$ is an \textbf{expanding problem type} if $\ \forall\ W\in PT:\ \forall u_1,u_2\in V_{small}:\ (u_1,u_2)\in E_{small}\Rightarrow (Pair(W,u_1),Pair(W,u_2))\in E_{large}$, that is each edge of $G_{small}$ has to be mapped to an edge of $G_{large}$ for each mapping in $PT$. +\end{definition} + +Note that $ISO$, $SUB$ and $IND$ are expanding problem types. + +This paper deals with the three problem types mentioned above only, but +the following generic definitions make it possible to handle other types as well. +Although it may be challenging to find a proper consistency function and an efficient +cutting function. + +\begin{definition} +Let M be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a \textbf{consistency function by } $\mathbf{PT}$, if the following holds. If there exists whole mapping $W$ of type PT for which $M\subseteq W$, then $Cons_{PT}(M)$ is true. +\end{definition} + +\begin{definition} +Let M be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a \textbf{cutting function by } $\mathbf{PT}$, if the following holds. $\mathbf{Cut_{PT}(M)}$ is false if $M$ can be extended to a whole mapping W of type PT. +\end{definition} + +\begin{definition} +$M$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$, if $Cons_{PT}(M)$ is true. +\end{definition} + +$Cons_{PT}$ and $Cut_{PT}$ will often be used in the following form. +\begin{notation} +Let $\mathbf{Cons_{PT}(p, M)}:=Cons_{PT}(M\cup\{p\})$ and $\mathbf{Cut_{PT}(p, M)}:=Cut_{PT}(M\cup\{p\})$, where $p\in{V_{small}\!\times\!V_{large}}$ and $M\cup\{p\}$ is mapping. +\end{notation} + +$Cons_{PT}$ will be used to check the consistency of the already covered nodes, while $Cut_{PT}$ is for looking ahead to recognize if no whole consistent mapping can contain the current mapping. + +\subsection{Overview of the algorithm} +VF2 uses a state space representation of mappings, $Cons_{PT}$ for excluding inconsistency with the problem type and $Cut_{PT}$ for pruning the search tree. +Each state $s$ of the matching process can be associated with a mapping $M(s)$. + +\textbf{Algorithm~\ref{alg:VF2Pseu})} is a high level description of the VF2 matching algorithm. + + +\begin{algorithm} +\algtext*{EndIf}%ne nyomtasson end if-et +\algtext*{EndFor}%ne nyomtasson .. +\algtext*{EndProcedure}%ne nyomtasson .. +\caption{\hspace{0.5cm}$A\ high\ level\ description\ of\ VF2$}\label{alg:VF2Pseu} +\begin{algorithmic}[1] + +\Procedure{VF2}{State $s$, ProblemType $PT$} + \If{$M(s$) covers $V_{small}$} + \State Output($M(s)$) + \Else + + \State Compute the set $P(s)$ of the pairs candidate for inclusion in $M(s)$ + \ForAll{$p\in{P(s)}$} + \If{Cons$_{PT}$($p, M(s)$) $\wedge$ $\neg$Cut$_{PT}$($p, M(s)$)} + \State Compute the nascent state $\tilde{s}$ by adding $p$ to $M(s)$ + \State \textbf{call} VF2($\tilde{s}$, $PT$) + \EndIf + \EndFor + \EndIf +\EndProcedure +\end{algorithmic} +\end{algorithm} + + +The initial state $s_0$ is associated with $M(s_0)=\emptyset$, i.e. it starts with an empty mapping. + +For each state $s$, the algorithm computes $P(s)$, the set of candidate node pairs for adding to the current state $s$. + +For each pair $p$ in $P(s)$, $Cons_{PT}(p,M(s))$ and $Cut_{PT}(p,M(s))$ are evaluated. If $Cons_{PT}(p,M(s))$ is true and $Cut_{PT}(p,M(s))$ is false, the successor state $\tilde{s}=s\cup \{p\}$ is computed, and the whole process is recursively applied to $\tilde{s}$. Otherwise, $\tilde{s}$ is not consistent by $PT$ or it can be proved that $s$ can not be extended to a whole mapping. + +In order to make sure of the correctness see +\begin{claim} +Through consistent mappings, only consistent whole mappings can be reached, and all of the whole mappings are reachable through consistent mappings. +\end{claim} + +Note that a state may be reached in many different ways, since the order of insertions into M does not influence the nascent mapping. In fact, the number of different ways which lead to the same state can be exponentially large. If $G_{small}$ and $G_{large}$ are circles with n nodes and n different node labels, there exists exactly one graph isomorphism between them, but it will be reached in $n!$ different ways. + +However, one may observe + +\begin{claim} +\label{claim:claimTotOrd} +Let $\prec$ an arbitrary total ordering relation on $V_{small}$. +If the algorithm ignores each $p=(u,v) \in P(s)$, for which +\begin{center} +$\exists (\hat{u},\hat{v})\in P(s): \hat{u} \prec u$, +\end{center} +then no state can be reached more than ones and each state associated with a whole mapping remains reachable. +\end{claim} + +Note that the cornerstone of the improvements to VF2 is a proper choice of a total ordering. + +\subsection{The candidate set P(s)} +\label{candidateComputingVF2} +$P(s)$ is the set of the candidate pairs for inclusion in $M(s)$. +Suppose that $PT$ is an expanding problem type, see \textbf{Definition~\ref{expPT})}. + +\begin{notation} +Let $\mathbf{T_{small}(s)}:=\{u \in V_{small} : u$ is not covered by $M(s)\wedge\exists \tilde{u}\in{V_{small}: (u,\tilde{u})\in E_{small}} \wedge \tilde{u}$ is covered by $M(s)\}$, and \\ $\mathbf{T_{large}(s)}\!:=\!\{v \in\!V_{large}\!:\!v$ is not covered by $M(s)\wedge\!\exists\tilde{v}\!\in\!{V_{large}\!:\!(v,\tilde{v})\in\!E_{large}} \wedge \tilde{v}$ is covered by $M(s)\}$ +\end{notation} + +The set $P(s)$ includes the pairs of uncovered neighbours of covered nodes and if there is not such a node pair, all the pairs containing two uncovered nodes are added. Formally, let +\[ + P(s)\!=\! + \begin{cases} + T_{small}(s)\times T_{large}(s)&\hspace{-0.15cm}\text{if } T_{small}(s)\!\neq\!\emptyset\!\wedge\!T_{large}(s)\!\neq \emptyset,\\ + (V_{small}\!\setminus\!M_{small}(s))\!\times\!(V_{large}\!\setminus\!M_{large}(s)) &\hspace{-0.15cm}otherwise. + \end{cases} +\] + +\subsection{Consistency} +This section defines the consistency functions for the different problem types mentioned in \textbf{Section \ref{sec:CommProb})}. +\begin{notation} +Let $\mathbf{\Gamma_{small} (u)}:=\{\tilde{u}\in V_{small} : (u,\tilde{u})\in E_{small}\}$\\ +Let $\mathbf{\Gamma_{large} (v)}:=\{\tilde{v}\in V_{large} : (v,\tilde{v})\in E_{large}\}$ +\end{notation} +Suppose $p=(u,v)$, where $u\in V_{small}$ and $v\in V_{large}$, +$s$ is a state of the matching procedure, +$M(s)$ is consistent mapping by $PT$ and $lab(u)=lab(v)$. +$Cons_{PT}(p,M(s))$ checks whether including pair $p$ into $M(s)$ leads to a consistent mapping by $PT$. + +\subsubsection{Induced subgraph isomorphism} +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $IND$ $\Leftrightarrow (\forall \tilde{u}\in M_{small}: (u,\tilde{u})\in E_{small} \Leftrightarrow (v,Pair(M(s),\tilde{u}))\in E_{large})$.\newline +The following formulation gives an efficient way of calculating $Cons_{IND}$. +\begin{claim} +$Cons_{IND}((u,v),M(s)):=(\forall \tilde{v}\in \Gamma_{large}(v) \ \cap\ M_{large}(s):\\(Pair(M(s),\tilde{v}),u)\in E_{small})\wedge +(\forall \tilde{u}\in \Gamma_{small}(u) \ \cap\ M_{small}(s):(v,Pair(M(s),\tilde{u}))\in E_{large})$ is a consistency function in the case of $IND$. +\end{claim} + +\subsubsection{Graph isomorphism} +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $ISO$ $\Leftrightarrow$ $M(s)\cup \{(u,v)\}$ is a consistent mapping by $IND$. +\begin{claim} +$Cons_{ISO}((u,v),M(s))$ is a consistency function by $ISO$ if and only if it is a consistency function by $IND$. +\end{claim} +\subsubsection{Subgraph isomorphism} +$M(s)\cup \{(u,v)\}$ is a consistent mapping by $SUB$ $\Leftrightarrow (\forall \tilde{u}\in M_{small}:\\(u,\tilde{u})\in E_{small} \Rightarrow (v,Pair(M(s),\tilde{u}))\in E_{large})$. +\newline +The following formulation gives an efficient way of calculating $Cons_{SUB}$. +\begin{claim} +$Cons_{SUB}((u,v),M(s)):= +(\forall \tilde{u}\in \Gamma_{small}(u) \ \cap\ M_{small}(s):\\(v,Pair(M(s),\tilde{u}))\in E_{large})$ is a consistency function by $SUB$. +\end{claim} + +\subsection{Cutting rules} +$Cut_{PT}(p,M(s))$ is defined by a collection of efficiently verifiable conditions. The requirement is that $Cut_{PT}(p,M(s))$ can be true only if it is impossible to extended $M(s)\cup \{p\}$ to a whole mapping. +\begin{notation} + +Let $\mathbf{\tilde{T}_{small}}(s):=(V_{small}\backslash M_{small}(s))\backslash T_{small}(s)$, and \\ $\mathbf{\tilde{T}_{large}}(s):=(V_{large}\backslash M_{large}(s))\backslash T_{large}(s)$. +\end{notation} +\subsubsection{Induced subgraph isomorphism} +\begin{claim} +$Cut_{IND}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is a cutting function by $IND$. +\end{claim} +\subsubsection{Graph isomorphism} +Note that the cutting function of induced subgraph isomorphism defined above is a cutting function by $ISO$, too, however it is less efficient than the following while their computational complexity is the same. +\begin{claim} +$Cut_{ISO}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| \neq |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| \neq |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is a cutting function by $ISO$. +\end{claim} + +\subsubsection{Subgraph isomorphism} +\begin{claim} +$Cut_{SUB}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)|$ is a cutting function by $SUB$. +\end{claim} +Note that there is a significant difference between induced and non-induced subgraph isomorphism: + +\begin{claim} +\label{claimSUB} +$Cut_{SUB}'((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| < |\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}(u)\cap \tilde{T}_{small}(s)|$ is \textbf{not} a cutting function by $SUB$. +\end{claim} +\begin{proof}$ $\\ +\vspace*{-0.5cm} + +\begin{figure} +\begin{center} +\begin{tikzpicture} + [scale=.8,auto=left,every node/.style={circle,fill=black!15}] + \node[rectangle,fill=black!15] at (4,6) {$G_{small}$}; + \node (u4) at (2.5,10) {$u_4$}; + \node (u3) at (5.5,10) {$u_3$}; + \node (u1) at (2.5,7) {$u_1$}; + \node (u2) at (5.5,7) {$u_2$}; + + \node[rectangle,fill=black!30] at (13.5,6) {$G_{large}$}; + \node[fill=black!30] (v4) at (12,10) {$v_4$}; + \node[fill=black!30] (v3) at (15,10) {$v_3$}; + \node[fill=black!30] (v1) at (12,7) {$v_1$}; + \node[fill=black!30] (v2) at (15,7) {$v_2$}; + + + \foreach \from/\to in {u1/u2,u2/u3,u3/u4,u4/u1} + \draw (\from) -- (\to); + \foreach \from/\to in {v1/v2,v2/v3,v3/v4,v4/v1,v1/v3} + \draw (\from) -- (\to); +% \draw[dashed] (\from) -- (\to); +\end{tikzpicture} +\caption{Graphs for the proof of \textbf{Claim \ref{claimSUB}}} \label{fig:proofSUB} +\end{center} +\end{figure} +Let the two graphs of \textbf{Figure \ref{fig:proofSUB})} be the input graphs. +Suppose the total ordering relation is $u_1 \prec u_2 \prec u_3 \prec u_4$,$M(s)\!=\{(u_1,v_1)\}$, and VF2 tries to add $(u_2,v_2)\in P(s)$.\newline +$Cons_{SUB}((u_2,v_2),M(s))=true$, so $M\cup \{(u_2,v_2)\}$ is consistent by $SUB$. The cutting function $Cut_{SUB}((u_2,v_2),M(s))$ is false, so it does not let cut the tree.\newline +On the other hand $Cut_{SUB}'((u_2,v_2),M(s))$ is true, since\\$0=|\Gamma_{large}(v_2)\cap \tilde{T}_{large}(s)|<|\Gamma_{small}(u_2)\cap \tilde{T}_{small}(s)|=1$ is true, but still the tree can not be pruned, because otherwise the $\{(u_1,v_1)(u_2,v_2)(u_3,v_3)(u_4,v_4)\}$ mapping can not be found. +\end{proof} + +\newpage +\section{The VF2++ Algorithm} +Although any total ordering relation makes the search space of VF2 a tree, its +choice turns out to dramatically influence the number of visited states. The goal is to determine an efficient one as quickly as possible. + +The main reason for VF2++' superiority over VF2 is twofold. Firstly, taking into account the structure and the node labeling of the graph, VF2++ determines a state order in which most of the unfruitful branches of the search space can be pruned immediately. Secondly, introducing more efficient --- nevertheless still easier to compute --- cutting rules reduces the chance of going astray even further. + +In addition to the usual subgraph isomorphism, specialized versions for induced subgraph isomorphism and for graph isomorphism have been designed. VF2++ has gained a runtime improvement of one order of magnitude respecting induced subgraph isomorphism and a better asymptotical behaviour in the case of graph isomorphism problem. + +Note that a weaker version of the cutting rules and the more efficient candidate +set calculating were described in \cite{VF2Plus}, too. + +It should be noted that all the methods described in this section are extendable to handle directed graphs and edge labels as well. + +The basic ideas and the detailed description of VF2++ are provided in the following. + +\subsection{Preparations} +\begin{claim} +\label{claim:claimCoverFromLeft} +The total ordering relation uniquely determines a node order, in which the nodes of $V_{small}$ will be covered by VF2. From the point of view of the matching procedure, this means, that always the same node of $G_{small}$ will be covered on the d-th level. +\end{claim} +\begin{proof} +In order to make the search space a tree, the pairs in $\{(u,v)\in P(s) : \exists \hat{u} : \hat{u}\prec u\}$ are excluded from $P(s)$. +\newline +Let $\tilde{P}(s):=P(s)\backslash \{(u,v)\in P(s) : \exists \hat{u} : \hat{u}\prec u\}$ +\newline +The relation $\prec$ is a total ordering, so $\exists!\ \tilde{u} : \forall\ (u,v)\in \tilde{P}(s): u=\tilde u$. Since a pair form $\tilde{P}(s)$ is chosen for including into $M(s)$, it is obvious, that only $\tilde{u}$ can be covered in $V_{small}$. Actually, $\tilde{u}$ is the smallest element in $T_{small}(s)$ (or in $V_{small}\backslash M_{small}(s)$, if $T_{small}(s)$ were empty), and $T_{small}(s)$ depends only on the covered nodes of $G_{small}$. +\newline +Simple induction on $d$ shows that the set of covered nodes of $G_{small}$ is unique, if $d$ is given, so $\tilde{u}$ is unique if $d$ is given. +\end{proof} + +\begin{definition} +An order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ of $V_{small}$ is \textbf{matching order}, if exists $\prec$ total ordering relation, s.t. the VF2 with $\prec$ on the d-th level finds pair for $u_{\sigma(d)}$ for all $d\in\{1,..,|V_{small}|\}$. +\end{definition} + +\begin{claim}\label{claim:MOclaim} +A total ordering is matching order, iff the nodes of every component form an interval in the node sequence, and every node connects to a previous node in its component except the first node of the component. The order of the components is arbitrary. +\\Formally spoken, an order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ of $V_{small}$ is matching order $\Leftrightarrow$ $\forall G'_{small}=(V'_{small},E'_{small})\ component\ of\ G_{small}: \forall i: (\exists j : j depth \Rightarrow M[i]= INVALID$. $M[depth]$ changes while the state is being processed, but the property is held before both stepping back to a predecessor state and exploring a successor state. + +The necessary part of the candidate set is easily maintainable or computable by following \textbf{Section~\ref{candidateComputingVF2})}. A much faster method has been designed for biological- and sparse graphs, see the next section for details. + +\subsubsection{Calculating the candidates for a node} +Being aware of \textbf{Claim~\ref{claim:claimCoverFromLeft})}, the task is not to maintain the candidate set, but to generate the candidate nodes in $G_{large}$ for a given node $u\in V_{small}$. +In case of an expanding problem type and $M$ mapping, if a node $v\in V_{large}$ is a potential pair of $u\in V_{small}$, then $\forall u'\in V_{small} : (u,u')\in E_{small}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in E_{large}$. That is, each covered neighbour of $u$ has to be mapped to a covered neighbour of $v$. + +Having said that, an algorithm running in $\Theta(deg)$ time is describable if there exists a covered node in the component containing $u$. In this case choose a covered neighbour $u'$ of $u$ arbitrarily --- such a node exists based on \textbf{Claim~\ref{claim:MOclaim})}. With all the candidates of $u$ being among the uncovered neighbours of $Pair(M,u')$, there are solely $deg(Pair(M,u'))$ nodes to check. + +An easy trick is to choose an $u'$, for which $|\{uncovered\ neighbours\ $ $of\ Pair(M,u')\}|$ is the smallest possible. + +Note that if $u$ is the first node of its component, then all the uncovered nodes of $G_{large}$ are candidates, so giving a sublinear method is impossible. + + +\subsubsection{Determining the node order} +This section describes how the node order preprocessing method of VF2++ can efficiently be implemented. + +For using lookup tables, the node labels are associated with the numbers $\{0,1,..,|K|-1\}$, where $K$ is the set of the labels. It enables $F_\mathcal{M}$ to be stored in an array, for which $F_\mathcal{M}[i]=F_\mathcal{M}(i)$ where $i=0,1,..,|K|-1$. At first, $\mathcal{M}=\emptyset$, so $F_\mathcal{M}[i]$ is the number of nodes in $V_{small}$ having label i, which is easy to compute in $\Theta(|V_{small}|)$ steps. + +$\mathcal{M}\subseteq V_{small}$ can be represented as an array of size $|V_{small}|$. + +The BFS tree is computed by using a FIFO data structure which is usually implemented as a linked list, but one can avoid it by using the array $\mathcal{M}$ itself. $\mathcal{M}$ contains all the nodes seen before, a pointer shows where the first node of the FIFO is, and another one shows where the next explored node has to be inserted. So the nodes of each level of the BFS tree can be processed by \textbf{Algorithm \ref{alg:VF2PPProcess1})} and \textbf{\ref{alg:VF2PPProcess2})} in place by swapping nodes. + +After a node $u$ gets to the next place of the node order, $F_\mathcal{M}[lab[u]]$ has to be decreased by one, because there is one less covered node in $V_{large}$ with label $lab(u)$, that is why min selection sort is preferred which gives the elements from left to right in descending order, see \textbf{Algorithm \ref{alg:VF2PPProcess1})}. + +Note that using a $\Theta(n^2)$ sort absolutely does not slow down the procedure on biological (and on sparse) graphs, since they have few nodes on a level. If a level had a large number of nodes, \textbf{Algorithm \ref{alg:VF2PPProcess2})} would seem to be a better choice with a $\Theta(nlog(n))$ or Bucket sort, but it may reduce the efficiency of the matching procedure, since $F_\mathcal{M}(i)$ can not be immediately refreshed, so it is unable to provide up-to-date label information. + +Note that the \textit{while loop} of \textbf{Algorithm \ref{alg:VF2PPPseu})} takes one iteration per graph component and the graphs in biology are mostly connected. +\subsubsection{Cutting rules} +In \textbf{Section \ref{VF2PPCuttingRules})}, the cutting rules were described using the sets $T_{small}$, $T_{large}$, $\tilde T_{small}$ and $\tilde T_{large}$, which are dependent on the all-time mapping (i.e. on the all-time state). The aim is to check the labeled cutting rules of VF2++ in $\Theta(deg)$ time. + +Firstly, suppose that these four sets are given in such a way, that checking whether a node is in a certain set takes constant time, e.g. they are given by their 0-1 characteristic vectors. Let $L$ be an initially zero integer lookup table of size $|K|$. After incrementing $L[lab(u')]$ for all $u'\in \Gamma_{small}(u) \cap T_{small}(s)$ and decrementing $L[lab(v')]$ for all $v'\in\Gamma_{large} (v) \cap T_{large}(s)$, the first part of the cutting rules is checkable in $\Theta(deg)$ time by considering the proper signs of $L$. Setting $L$ to zero takes $\Theta(deg)$ time again, which makes it possible to use the same table through the whole algorithm. +The second part of the cutting rules can be verified using the same method with $\tilde T_{small}$ and $\tilde T_{large}$ instead of $T_{small}$ and $T_{large}$. Thus, the overall complexity is $\Theta(deg)$. + +An other integer lookup table storing the number of covered neighbours of each node in $G_{large}$ gives all the information about the sets $T_{large}$ and $\tilde T_{large}$, which is maintainable in $\Theta(deg)$ time when a pair is added or substracted by incrementing or decrementing the proper indices. A further improvement is that the values of $L[lab(u')]$ in case of checking $u$ is dependent only on $u$, i.e. on the size of the mapping, so for each $u\in V_{small}$ an array of pairs (label, number of such labels) can be stored to skip the maintaining operations. Note that these arrays are at most of size $deg$. Skipping this trick, the number of covered neighbours has to be stored for each node of $G_{small}$ as well to get the sets $T_{small}$ and $\tilde T_{small}$. + +Using similar tricks, the consistency function can be evaluated in $\Theta(deg)$ steps, as well. + +\section{The VF2 Plus Algorithm} +The VF2 Plus algorithm is a recently improved version of VF2. It was compared with the state of the art algorithms in \cite{VF2Plus} and has proven itself to be competitive with RI, the best algorithm on biological graphs. +\\ +A short summary of VF2 Plus follows, which uses the notation and the conventions of the original paper. + +\subsection{Ordering procedure} +VF2 Plus uses a sorting procedure that prefers nodes in $V_{small}$ with the lowest probability to find a pair in $V_{small}$ and the highest number of connections with the nodes already sorted by the algorithm. + +\begin{definition} +$(u,v)$ is a \textbf{feasible pair}, if $lab(u)=lab(v)$ and $deg(u)\leq deg(v)$, where $u\in{V_{small}}$ and $ v\in{V_{large}}$. +\end{definition} +$P_{lab}(L):=$ a priori probability to find a node with label $L$ in $V_{large}$ +\newline +$P_{deg}(d):=$ a priori probability to find a node with degree $d$ in $V_{large}$ +\newline +$P(u):=P_{lab}(L)*\bigcup_{d'>d}P_{deg}(d')$\\ +$M$ is the set of already sorted nodes, $T$ is the set of nodes candidate to be selected, and $degreeM$ of a node is the number of its neighbours in $M$. +\begin{algorithm} +\algtext*{EndIf}%ne nyomtasson end if-et +\algtext*{EndFor}%ne nyomtasson .. +\algtext*{EndProcedure}%ne nyomtasson .. +\algtext*{EndWhile} +\caption{}\label{alg:VF2PlusPseu} +\begin{algorithmic}[1] +\Procedure{VF2 Plus order}{} + \State Select the node with the lowest $P$. + \If {more nodes share the same $P$} + \State select the one with maximum degree + \EndIf + \If {more nodes share the same $P$ and have the max degree} + \State select the first + \EndIf + \State Put the selected node in the set $M$. \label{alg:putIn} + \State Put all its unsorted neighbours in the set $T$. + \If {$M\neq V_{small}$} + \State From set $T$ select the node with maximum $degreeM$. + \If {more nodes have maximum $degreeM$} + \State Select the one with the lowest $P$ + \EndIf + \If {more nodes have maximum $degreeM$ and $P$} + \State Select the first. + \EndIf + \State \textbf{goto \ref{alg:putIn}.} + \EndIf +\EndProcedure +\end{algorithmic} +\end{algorithm} + +Using these notations, \textbf{Algorithm~\ref{alg:VF2PlusPseu})} provides the description of the sorting procedure. + +Note that $P(u)$ is not the exact probability of finding a consistent pair for $u$ by choosing a node of $V_{large}$ randomly, since $P_{lab}$ and $P_{deg}$ are not independent, though calculating the real probability would take quadratic time, which may be reduced by using fittingly lookup tables. + +\newpage +\section{Experimental results} +This section compares the performance of VF2++ and VF2 Plus. Both algorithms have run faster with orders of magnitude than VF2, thus its inclusion was not reasonable. +\subsection{Biological graphs} +The tests have been executed on a recent biological dataset created for the International Contest on Pattern Search in Biological Databases\cite{Content}, which has been constructed of Molecule, Protein and Contact Map graphs extracted from the Protein Data Bank\cite{ProteinDataBank}. + +The molecule dataset contains small graphs with less than 100 nodes and an average degree of less than 3. The protein dataset contains graphs having 500-10 000 nodes and an average degree of 4, while the contact map dataset contains graphs with 150-800 nodes and an average degree of 20. +\\ + +In the following, the induced subgraph isomorphism and the graph isomorphism will be examined. + +\subsubsection{Induced subgraph isomorphism} +This dataset contains a set of graph pairs, and \textbf{all} the induced subgraph ismorphisms have to be found between them. \textbf{Figure \ref{fig:INDProt}), \ref{fig:INDContact}),} and \textbf{ \ref{fig:INDMolecule})} show the solution time of the problems in the problem set. + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} + \begin{axis}[title=Proteins IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid + =major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] + %\addplot+[only marks] table {proteinsOrig.txt}; + \addplot[mark=*,mark size=1.2pt,color=blue] table {Orig/Proteins.256.txt}; + \addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/Proteins.256.txt}; + \end{axis} + \end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{Both the algorithms have linear behaviour on protein graphs. VF2++ is more than 10 times faster than VF2 Plus.} \label{fig:INDProt} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title=Contact Maps IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {Orig/ContactMaps.128.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/ContactMaps.128.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{On Contact Maps, VF2++ runs in near constant time, while VF2 Plus has a near linear behaviour.} \label{fig:INDContact} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title=Molecules IND,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {Orig/Molecules.32.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/Molecules.32.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{In the case of Molecules, the algorithms seem to have a similar behaviour, but VF2++ is almost two times faster even on such small graphs.} \label{fig:INDMolecule} +\end{figure} + + +\subsubsection{Graph ismorphism} +In this experiment, the nodes of each graph in the database have been shuffled and an isomorphism between the shuffled and the original graph has been searched. For runtime results, see \textbf{Figure \ref{fig:ISOProt}), \ref{fig:ISOContact}),} and \textbf{\ref{fig:ISOMolecule})}. +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title=Proteins ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {Orig/proteinsIso.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/proteinsIso.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{On protein graphs, VF2 Plus has a super linear time complexity, while VF2++ runs in near constant time. The difference is about two order of magnitude on large graphs.}\label{fig:ISOProt} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title=Contact Maps ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {Orig/contactMapsIso.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/contactMapsIso.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{The results are closer to each other on Contact Maps, but VF2++ still performs consistently better.}\label{fig:ISOContact} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title=Molecules ISO,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {Orig/moleculesIso.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {VF2PPLabel/moleculesIso.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{In the case of Molecules, there is not such a significant difference, but VF2++ seems to be faster as the number of nodes increases.}\label{fig:ISOMolecule} +\end{figure} + + +\subsection{Random graphs} +This section compares VF2++ with VF2 Plus on random graphs of a large size. The node labels are uniformly distributed. +Let $\delta$ denote the average degree. +For the parameters of problems solved in the experiments, please see the top of each chart. +\subsubsection{Graph isomorphism} +To evaluate the efficiency of the algorithms in the case of graph isomorphism, connected graphs of less than 20 000 nodes have been considered. Generating a random graph and shuffling its nodes, an isomorphism had to be found. \textbf{Figure \ref{fig:randISO5}), \ref{fig:randISO10}), \ref{fig:randISO15}), \ref{fig:randISO35}), \ref{fig:randISO45}),} and \textbf{\ref{fig:randISO100}) } show the runtime results on graph sets of various density. + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 5$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso5_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso5_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO5} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 10$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso10_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso10_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO10} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 15$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso15_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso15_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO15} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 35$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso35_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso35_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO35} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 45$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso45_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso45_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO45} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random ISO, $\delta = 100$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/iso/vf2pIso100_1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/iso/vf2ppIso100_1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{}\label{fig:randISO100} +\end{figure} + + +Considering the graph isomorphism problem, VF2++ consistently outperforms its rival especially on sparse graphs. The reason for the slightly super linear behaviour of VF2++ on denser graphs is the larger number of nodes in the BFS tree constructed in \textbf{Algorithm \ref{alg:VF2PPPseu})}. + +\subsubsection{Induced subgraph isomorphism} +This section provides a comparison of VF2++ and VF2 Plus in the case of induced subgraph isomorphism. In addition to the size of the large graph, that of the small graph dramatically influences the hardness of a given problem too, so the overall picture is provided by examining small graphs of various size. + +For each chart, a number $0<\rho< 1$ has been fixed and the following has been executed 150 times. Generating a large graph $G_{large}$, choose 10 of its induced subgraphs having $\rho\ |V_{large}|$ nodes, and for all the 10 subgraphs find a mapping by using both the graph matching algorithms. +The $\delta = 5, 10, 35$ and $\rho = 0.05, 0.1, 0.3, 0.6, 0.8, 0.95$ cases have been examined (see \textbf{Figure \ref{fig:randIND5}), \ref{fig:randIND10})} and \textbf{\ref{fig:randIND35})}), and for each $\delta$, a cumulative chart is given as well, which excludes $\rho = 0.05$ and $0.1$ for the sake of perspicuity (see \textbf{Figure \ref{fig:randIND5Sum}), \ref{fig:randIND10Sum})} and \textbf{\ref{fig:randIND35Sum})}). + + + + + +\begin{figure}[H] +\vspace*{-0.8cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.05.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.05.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\hspace{1cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.3.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.6.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\begin{subfigure}[b]{0.55\textwidth} + +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.8.txt}; +\end{axis} +\end{tikzpicture} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 5$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd5_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{subfigure} +\vspace*{-0.8cm} +\caption{IND on graphs having an average degree of 5.}\label{fig:randIND5} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Rand IND Summary, $\delta = 5$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.3.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.6.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.8.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd5_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd5_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{Cummulative chart for $\delta=5$.}\label{fig:randIND5Sum} +\end{figure} + + + +\begin{figure}[H] +\vspace*{-0.8cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.05.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.05.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\hspace{1cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.3.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.6.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\begin{subfigure}[b]{0.55\textwidth} + +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.8.txt}; +\end{axis} +\end{tikzpicture} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 10$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd10_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{subfigure} +\vspace*{-0.8cm} +\caption{IND on graphs having an average degree of 10.}\label{fig:randIND10} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Rand IND Summary, $\delta = 10$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.3.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.6.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.8.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd10_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd10_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{Cummulative chart for $\delta=10$.}\label{fig:randIND10Sum} +\end{figure} + + + +\begin{figure}[H] +\vspace*{-0.8cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.05$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.05.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.05.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.1$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.1.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.1.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\hspace{1cm} +\begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.3$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.3.txt}; +\end{axis} +\end{tikzpicture} +\end{center} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.6$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.6.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\end{subfigure} +\begin{subfigure}[b]{0.55\textwidth} + +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.8$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \space}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.8.txt}; +\end{axis} +\end{tikzpicture} + \end{subfigure} + \begin{subfigure}[b]{0.55\textwidth} +\begin{tikzpicture} +\begin{axis}[title={Random IND, $\delta = 35$, $\rho = 0.95$},width=7.2cm,height=6cm,xlabel={target size},ylabel={time (ms)},ylabel near ticks,legend entries={VF2 Plus,VF2++},grid +=major,mark size=1.2pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot table {randGraph/ind/vf2pInd35_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{subfigure} +\vspace*{-0.8cm} +\caption{IND on graphs having an average degree of 35.}\label{fig:randIND35} +\end{figure} + +\begin{figure}[H] +\begin{center} +\begin{tikzpicture} +\begin{axis}[title={Rand IND Summary, $\delta = 35$, $\rho = 0.3, 0.6, 0.8, 0.95$},height=17cm,width=16cm,xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},line width=0.8pt,grid +=major,mark size=1pt, legend style={at={(0,1)},anchor=north west},scaled x ticks = false,x tick label style={/pgf/number format/1000 sep = \thinspace}] +%\addplot+[only marks] table {proteinsOrig.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.3.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.3.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.6.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.6.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.8.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.8.txt}; +\addplot[mark=*,mark size=1.5pt,color=blue] table {randGraph/ind/vf2pInd35_0.95.txt}; +\addplot[mark=triangle*,mark size=1.8pt,color=red] table {randGraph/ind/vf2ppInd35_0.95.txt}; +\end{axis} +\end{tikzpicture} +\end{center} +\vspace*{-0.8cm} +\caption{Cummulative chart for $\delta=35$.}\label{fig:randIND35Sum} +\end{figure} + +Based on these experiments, VF2++ is faster than VF2 Plus and able to handle really large graphs in milliseconds. Note that when $IND$ was considered and the small graphs had proportionally few nodes ($\rho = 0.05$, or $\rho = 0.1$), then VF2 Plus produced some inefficient node orders(e.g. see the $\delta=10$ case on \textbf{Figure \ref{fig:randIND10})}). If these examples had been excluded, the charts would have seemed to be similar to the other ones. +Unsurprisingly, as denser graphs are considered, both VF2++ and VF2 Plus slow slightly down, but remain practically usable even on graphs having 10 000 nodes. + + + + +\newpage +\section{Conclusion} +In this thesis, after providing a short summary of the recent algorithms, a new graph matching algorithm based on VF2, called VF2++, has been presented and analyzed from a practical viewpoint. + +Recognizing the importance of the node order and determining an efficient one, VF2++ is able to match graphs of thousands of nodes in near practically linear time including preprocessing. In addition to the proper order, VF2++ uses more efficient consistency and cutting rules which are easy to compute and make the algorithm able to prune most of the unfruitful branches without going astray. + +In order to show the efficiency of the new method, it has been compared to VF2 Plus, which is the best concurrent algorithm based on \cite{VF2Plus}. + +The experiments show that VF2++ consistently outperforms VF2 Plus on biological graphs. It seems to be asymptotically faster on protein and on contact map graphs in the case of induced subgraph isomorphism, while in the case of graph isomorphism, it has definitely better asymptotic behaviour on protein graphs. + +Regarding random sparse graphs, not only has VF2++ proved itself to be faster than VF2 Plus, but it has a practically linear behaviour both in the case of induced subgraph- and graph isomorphism, as well. + + %% The Appendices part is started with the command \appendix; %% appendix sections are then done as normal sections @@ -160,20 +1381,21 @@ %% If you have bibdatabase file and want bibtex to generate the %% bibitems, please use %% -%% \bibliographystyle{elsarticle-num} -%% \bibliography{} +\bibliographystyle{elsarticle-num} +\bibliography{bibliography} %% else use the following coding to input the bibitems directly in the %% TeX file. -\begin{thebibliography}{00} +%% \begin{thebibliography}{00} -%% \bibitem{label} -%% Text of bibliographic item +%% %% \bibitem{label} +%% %% Text of bibliographic item -\bibitem{} +%% \bibitem{} -\end{thebibliography} +%% \end{thebibliography} + \end{document} \endinput %% diff -r 76dc74f824ba -r 20d1b0e5838f proteins.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/proteins.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,6 @@ +1 10 +2 11 +3 15 +4 7 +1 7 +1000 6 diff -r 76dc74f824ba -r 20d1b0e5838f proteinsBfs.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/proteinsBfs.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,234 @@ +535 0.0013412 +554 0.0007026 +571 0.0015756 +614 0.0021506 +634 0.0007692 +636 0.0009666 +639 0.0007728 +730 0.0007754 +731 0.0010338 +734 0.000867 +735 0.0007172 +737 0.0017526 +755 0.0007422 +760 0.0007674 +788 0.0050546 +794 0.003566 +795 0.000748 +802 0.0007864 +816 0.001123 +819 0.0008854 +831 0.0007446 +833 0.0008486 +838 0.0011482 +858 0.0007336 +861 0.0009128 +874 0.000735 +877 0.0007424 +879 0.0011854 +886 0.0009686 +899 0.0012188 +957 0.000763 +982 0.0008034 +995 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b/randGraph/ind/vf2pInd10_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +354 0.0506667 +446 0.0713333 +631 0.122 +668 0.128667 +842 0.192667 +1296 0.379333 +1327 0.402333 +1495 0.486667 +1713 0.614333 +2027 0.811667 +2106 0.911667 +2901 1.551 +2967 1.64133 +3169 1.91167 +3279 2.42867 +3320 1.978 +3510 2.36667 +4337 3.20533 +4535 3.66033 +4841 4.295 +5020 4.30967 +5111 4.54233 +5163 4.843 +5359 5.03067 +5542 5.158 +5547 5.323 +5562 5.51767 +5631 7.10833 +5877 5.91167 +6101 10.3863 +6295 6.685 +6584 7.42 +6594 13.237 +6788 22.985 +6849 10.606 +7173 26.576 +7279 10.1943 +7361 8.77533 +7384 11.583 +7461 10.739 +7467 16.974 +8091 11.3097 +8451 11.6927 +8827 481.983 +9031 14.1033 +9134 262.166 +9427 793.866 +9639 14.7117 +9720 37.7563 +9943 45.4893 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd10_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd10_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,48 @@ +70 0.0116667 +377 0.0986667 +390 0.108 +788 0.311 +1086 0.513333 +1922 1.40767 +2057 1.694 +2071 1.62867 +2169 1.81033 +2406 2.13267 +2958 3.30467 +3323 5.543 +3631 4.47733 +4224 6.646 +4380 7.38133 +4427 6.581 +4663 7.362 +4942 10.6473 +5118 8.73467 +5183 8.83933 +5204 11.3987 +5368 13.5937 +5704 11.7243 +5826 14.3633 +5859 11.0753 +5925 11.8847 +5948 12.146 +5961 13.188 +6052 12.8867 +6472 13.8913 +6871 17.5173 +7015 17.576 +7039 261.191 +7221 18.2443 +7238 17.9493 +7776 22.4403 +7806 23.5107 +8478 23.7227 +8679 26.0337 +8966 32.3923 +9251 46.147 +9336 364.076 +9499 47.7787 +9583 31.54 +9585 33.146 +9649 312.417 +9816 2676 +9932 86.0923 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd10_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd10_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +326 0.229 +399 0.312 +476 0.405667 +1069 1.47167 +1389 2.27967 +1415 2.38233 +1654 3.13433 +1909 4.10367 +1961 4.20967 +2638 7.205 +3005 9.56467 +3045 9.63067 +3144 9.97967 +3229 10.535 +3275 11.0397 +3803 14.7707 +4459 19.1907 +4816 22.2127 +4988 24.2533 +5097 24.7153 +5424 27.775 +5465 28.1747 +5634 29.9557 +5645 30.4397 +6022 34.0627 +6094 34.7597 +6159 35.3867 +6291 36.908 +6361 37.735 +6638 40.9127 +6677 41.3867 +6838 43.3643 +7043 45.8147 +7531 53.295 +7566 53.4237 +7623 53.3137 +7647 53.7857 +7822 56.64 +7994 58.8747 +8003 58.516 +8475 65.5097 +8996 73.5733 +9118 75.6233 +9256 78.6417 +9702 88.3083 +9750 86.066 +9753 87.0243 +9787 86.666 +9848 87.6113 +9849 87.6647 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd10_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd10_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +379 0.659 +431 0.756333 +665 1.45333 +718 1.63233 +826 2.04067 +1251 4.01467 +1279 4.19167 +1443 5.15233 +1500 5.49767 +2193 10.6097 +2231 10.9357 +2280 11.4147 +2632 14.7793 +2687 15.294 +2907 18.394 +3053 19.4757 +3091 19.9403 +3437 24.123 +3587 26.1283 +3976 31.5933 +4702 43.1907 +4751 43.979 +5349 54.99 +5350 55.042 +5353 55.103 +5722 64.9803 +5764 63.4153 +5844 64.99 +5946 67.22 +5969 68.138 +5993 68.2493 +6013 68.6733 +6323 76.4853 +6883 88.9913 +6917 89.7673 +7218 97.8563 +7383 101.738 +7663 108.983 +8038 120.23 +8119 122.024 +8133 122.049 +8165 123.306 +8204 124.93 +8694 138.633 +8805 142.476 +9262 157.005 +9334 159.177 +9569 167.333 +9811 176.083 +9876 178.658 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd10_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd10_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +22 0.031 +155 0.288 +176 0.334333 +911 3.326 +1065 4.422 +1107 4.559 +1526 7.77567 +1623 8.72367 +1782 10.411 +1863 10.9807 +1911 11.4317 +2165 14.2203 +2192 14.5207 +2217 14.8703 +2247 15.8033 +3050 26.4777 +3167 28.3113 +3398 32.1737 +3627 36.211 +4092 45.6143 +4240 50.091 +4346 50.883 +4615 56.5223 +4673 58.2247 +4708 58.7723 +4739 59.827 +4818 61.338 +5058 67.4893 +5222 73.828 +5414 76.2007 +5868 92.0477 +5946 90.9583 +6061 97.862 +6263 101.006 +6611 111.016 +6699 114.082 +6799 118.224 +6915 121.078 +7100 127.908 +7136 128.96 +7218 131.624 +7369 136.738 +7736 150.767 +8219 170.233 +8377 174.837 +8482 178.895 +8632 184.569 +8886 195.908 +9351 215.669 +9771 234.352 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd10_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd10_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,39 @@ +550 1.89167 +628 2.30633 +660 2.44667 +1192 6.26733 +1663 11.008 +1749 12.0033 +2082 16.0437 +2355 20.0497 +2375 20.4247 +2946 31.0957 +3499 40.91 +3567 42.3037 +3774 47.1067 +4189 57.4717 +4205 57.1887 +4214 57.5237 +4414 62.6257 +4430 65.425 +4481 66.262 +4500 64.8567 +4861 75.0313 +5162 83.857 +5650 99.221 +5724 104.668 +5795 107.559 +5944 109.123 +6006 114.083 +6057 113.29 +6462 127.495 +6536 130.378 +6668 135.983 +6996 149.48 +7594 173.137 +7936 188.337 +8716 229 +9142 250.944 +9164 248.217 +9435 263.048 +9465 264.229 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,29 @@ +216 0.03 +407 0.0723333 +538 0.105667 +927 0.258 +1009 0.279667 +1277 0.448333 +1668 0.648 +2391 1.245 +3229 2.398 +3754 3.21067 +3953 2.97767 +4254 3.59933 +4525 3.90133 +4780 4.31133 +5281 5.78067 +5861 7.20833 +6317 7.559 +6434 7.90267 +6766 7.867 +6881 8.46033 +7047 8.52233 +7316 11.2903 +7396 10.8353 +8696 25.6483 +9051 186.357 +9075 51.8077 +9377 41.2747 +9573 19.3877 +9687 27.1643 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +344 0.105333 +1434 1.012 +2070 1.78833 +2175 2.12467 +2480 2.36267 +2685 2.756 +3150 3.74 +3611 4.70533 +3814 5.17033 +4085 6.02267 +4089 5.83567 +4565 7.13067 +4742 7.711 +4778 8.84367 +5395 11.3127 +5587 17.3593 +5598 11.304 +5741 11.2983 +6033 12.098 +6357 15.332 +6401 14.5977 +6581 17.253 +6593 14.869 +6869 16.6577 +7079 19.5183 +7280 17.8077 +7882 24.0283 +8847 26.2117 +9039 30.609 +9728 33.457 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.2.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.2.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +32 0.0143333 +471 0.318333 +1247 1.378 +2258 3.965 +2333 4.39633 +2339 4.09667 +2344 4.3 +2642 5.072 +2756 5.529 +3207 7.45267 +3593 8.88933 +4212 12.124 +5038 16.622 +5085 16.935 +5543 19.945 +5801 23.4663 +5861 23.88 +6042 23.8307 +6385 26.9897 +6959 31.8693 +7205 32.7863 +8042 40.4443 +8173 41.864 +9063 51.899 +9074 52.9547 +9736 58.821 +9770 59.7663 +9805 59.3963 +9833 64.1883 +9914 63.187 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +262 0.207667 +598 0.655 +694 0.811333 +722 0.862 +752 0.915667 +1116 1.74333 +1224 2.021 +1293 2.24967 +1314 2.264 +1836 4.03533 +3127 10.3897 +3213 10.9287 +3390 12.0497 +3613 13.491 +4832 23.1793 +5319 27.7917 +5330 28.0533 +5379 28.2763 +5394 28.4367 +5456 29.08 +6354 38.7267 +6579 41.3643 +6889 45.599 +7096 47.646 +8039 60.601 +8436 66.923 +8686 71.225 +9377 81.8663 +9455 82.5453 +9608 85.13 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.4.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.4.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +30 0.023 +315 0.366667 +439 0.567667 +717 1.18 +1550 4.17967 +1859 5.77533 +1867 5.795 +1868 5.83467 +2687 11.057 +2734 11.48 +3357 15.9177 +3709 19.078 +5091 34.3267 +5510 39.729 +5514 40.0913 +5654 42.2553 +5732 44.3647 +6082 49.7163 +6549 57.0103 +6816 61.572 +6920 62.6347 +7859 78.1077 +8363 87.6337 +8757 97.692 +8854 97.8513 +8868 98.074 +9036 105.005 +9515 115.057 +9686 118.096 +9782 119.309 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.5.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.5.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +313 0.5 +542 1.03933 +623 1.258 +1531 5.05733 +1753 6.40633 +1856 7.12267 +1903 7.40533 +2022 8.23633 +2064 8.50633 +3364 20.2557 +3982 27.4747 +4041 29.015 +5037 42.9267 +6089 60.5607 +6443 67.2083 +6638 70.898 +6892 76.147 +7005 79.9043 +7133 81.553 +7764 97.4677 +7826 96.7693 +7839 99.9547 +7916 99.192 +8055 104.416 +8075 103.3 +8403 111.229 +8627 119.201 +8701 118.668 +9207 134.604 +9907 152.308 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +136 0.212333 +1037 3.34567 +1104 3.686 +1198 4.20867 +1204 4.26333 +1400 5.47233 +1533 6.26367 +1669 7.33233 +1723 7.70567 +2230 11.9047 +3701 29.3703 +4063 35.149 +4258 37.7457 +4476 41.4307 +5205 54.5803 +5304 57.3897 +5718 65.1257 +6123 73.9937 +6141 74.5327 +6192 75.7483 +6711 88.3167 +6973 95.5403 +7075 97.3207 +7294 103.253 +7653 113.638 +7869 118.855 +7878 119.142 +8668 142.724 +8726 146.63 +9523 171.031 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.7.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.7.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +115 0.218 +197 0.397667 +364 0.887 +392 1.00033 +914 3.32667 +950 3.571 +1556 7.707 +1561 7.97633 +1702 9.129 +1740 9.286 +1834 10.1287 +1982 11.714 +1998 11.8017 +2023 12.0333 +2092 12.644 +2187 13.815 +2297 14.967 +2314 15.2977 +2399 16.366 +2450 16.9767 +2534 17.7803 +2570 18.3007 +2666 19.3383 +2712 20.1653 +2725 20.0667 +2729 20.199 +2730 20.3637 +2895 22.468 +2920 22.8947 +2929 22.7347 +2997 23.8593 +3117 26.0253 +3149 26.2507 +3182 26.712 +3451 30.7967 +3464 31.0397 +3537 32.0727 +3564 32.3637 +3594 32.905 +3825 37.2367 +3892 38.0317 +3972 39.4953 +4198 44.1937 +4243 44.3837 +4281 45.103 +4327 46.527 +4776 55.3963 +4882 57.6943 +4938 59.1703 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +57 0.116333 +864 3.572 +1355 7.141 +1402 7.623 +1580 9.21167 +2161 15.619 +2295 17.2717 +2368 18.1977 +2585 21.1427 +2720 23.1157 +3126 29.6183 +3151 30.101 +4218 50.8937 +4886 66.1663 +5022 69.6627 +5239 75.1977 +5344 77.9527 +5361 78.2973 +5375 78.8197 +5819 91.158 +5930 94.378 +6273 104.835 +6477 111.401 +6947 126.704 +7598 150.256 +8269 175.992 +8736 195.773 +8868 200.751 +8972 205.386 +9725 239.337 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd15_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd15_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +28 0.0666667 +322 1.093 +449 1.70833 +561 2.337 +848 4.31167 +1519 10.761 +2125 18.4883 +2211 19.9937 +2218 20.0537 +2232 20.2807 +2533 25.162 +2622 26.9303 +2892 31.6663 +3128 36.231 +3156 36.8187 +3379 41.4687 +3513 44.631 +3807 51.283 +4061 57.5013 +4750 76.1557 +4767 76.6803 +4768 76.782 +5893 112.892 +6203 124.273 +8331 214.711 +8433 220.537 +8547 225.248 +8935 245.129 +9613 281.541 +9627 282.218 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +216 0.0483333 +407 0.115333 +538 0.164667 +927 0.400667 +1009 0.462333 +1277 0.626333 +1668 0.952 +2391 2.458 +3229 2.67367 +3754 3.469 +3953 4.03 +4254 6.02067 +4525 4.731 +4780 5.11633 +5281 8.87567 +5861 7.817 +6317 8.872 +6434 9.72067 +6766 18.1203 +6881 10.3367 +7047 10.7203 +7316 11.1883 +7396 12.2917 +8696 18.3437 +8897 59.4683 +9051 16.5487 +9075 18.851 +9377 40.446 +9573 26.39 +9687 22.042 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +344 0.143667 +1434 1.20933 +2070 2.17767 +2175 2.31733 +2480 2.87733 +2685 3.26067 +3150 4.33467 +3611 5.43433 +3814 5.951 +4085 6.683 +4089 6.89333 +4565 8.10867 +4742 8.58733 +4778 8.7 +5395 11.0133 +5587 11.8337 +5598 11.5987 +5741 12.2517 +6033 13.1647 +6357 15.0947 +6401 15.002 +6581 15.561 +6593 15.532 +6869 16.6487 +7079 17.639 +7280 19.2467 +7882 21.8247 +8847 26.6603 +9039 27.7043 +9728 32.275 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +262 0.309333 +598 0.919333 +694 1.15533 +722 1.219 +752 1.30767 +1116 2.30967 +1224 2.66167 +1293 2.898 +1314 2.959 +1836 5.00933 +3127 12.1287 +3213 12.703 +3390 13.9413 +3613 15.6223 +4832 25.7873 +5319 30.644 +5330 30.6967 +5379 31.2433 +5394 31.361 +5456 32.0347 +6354 42.1693 +6579 44.9697 +6889 48.9497 +7096 51.723 +8039 65.0997 +8436 71.3323 +8686 75.3243 +9377 86.862 +9455 88.1637 +9608 90.9447 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +136 0.363333 +1037 4.887 +1104 5.14633 +1198 5.79467 +1204 5.89733 +1400 7.36467 +1533 8.404 +1669 9.62233 +1723 10.1267 +2230 15.155 +3701 34.8173 +4063 42.5703 +4258 43.9077 +4476 47.9777 +5205 62.309 +5304 64.399 +5718 73.5017 +6123 83.217 +6141 83.6287 +6192 85.131 +6711 97.8737 +6973 104.891 +7075 107.821 +7294 114.304 +7653 123.88 +7869 132.351 +7878 130.796 +8668 156.333 +8726 158.004 +9523 185.501 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +248 1.11367 +439 2.172 +443 2.244 +1328 9.922 +1471 11.5383 +2396 24.0727 +2454 25.0777 +2968 33.7757 +2997 34.6437 +3149 37.539 +3328 40.9153 +3472 43.9497 +3850 52.1813 +4055 56.9343 +5013 81.4517 +5059 83.3097 +5663 99.998 +5728 102.454 +6077 115.204 +6448 126.072 +6510 126.617 +6532 128.346 +7140 149.279 +7297 155.374 +7470 162.685 +7628 168.062 +8250 195.35 +8333 199.957 +8834 221.676 +9980 280.124 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd32_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd32_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +28 0.092 +322 1.87367 +449 2.88 +561 3.849 +848 6.687 +1519 15.3643 +2125 24.726 +2211 26.4567 +2218 27.1653 +2232 26.7347 +2533 32.6447 +2622 33.9373 +2892 40.2703 +3128 46.1757 +3156 46.894 +3379 51.711 +3513 55.5957 +3807 61.8537 +4061 70.1517 +4750 89.438 +4767 89.9523 +4768 91.593 +5893 129.177 +6203 141.726 +8331 241.685 +8433 253.332 +8547 252.701 +8935 275.213 +9613 317.321 +9627 310.94 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd35_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd35_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,47 @@ +33 0.004 +684 0.255333 +1057 0.492 +1125 0.534667 +1578 0.919333 +1860 1.234 +1867 1.21633 +1990 1.263 +2036 1.30067 +2595 1.94167 +2607 1.921 +2756 3.03567 +3185 2.718 +3275 2.70667 +3362 3.14733 +3466 3.34367 +3492 3.081 +3564 3.14767 +3656 3.34333 +3834 3.45633 +4345 4.58533 +4605 4.69967 +4740 5.30433 +4921 6.00767 +4941 5.772 +4993 5.941 +5513 7.71433 +5767 7.69133 +6134 9.76067 +6410 9.202 +6534 10.31 +6904 9.506 +7010 11.4237 +7067 11.7407 +7231 11.6627 +7314 13.9493 +7525 12.132 +7538 12.1363 +7660 12.3557 +8422 14.5693 +8696 81.2783 +8885 17.4843 +8916 19.0267 +8919 16.0893 +9337 19.1553 +9878 31.0497 +9999 19.6727 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd35_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd35_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ 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+2638 9.36833 +3005 11.6703 +3045 12.1313 +3144 12.635 +3229 13.213 +3275 13.6237 +3803 17.1753 +4459 23.0647 +4816 26.3773 +4988 27.9623 +5097 28.998 +5424 32.519 +5465 32.8147 +5634 34.8827 +5645 34.648 +6022 39.1953 +6094 40.01 +6159 40.5813 +6291 42.185 +6361 43.3303 +6638 47.9383 +6677 47.2617 +6838 48.9613 +7043 52.919 +7531 58.8037 +7566 59.2117 +7623 59.7657 +7647 60.3467 +7822 62.6957 +7994 65.3453 +8003 65.7403 +8475 73.3503 +8996 85.0193 +9118 83.9353 +9256 87.1167 +9702 93.593 +9750 94.753 +9753 94.6723 +9787 95.2873 +9848 96.5613 +9849 99.8623 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd35_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd35_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +379 1.304 +431 1.55767 +665 2.743 +718 2.95 +826 3.58 +1251 6.52667 +1279 6.73867 +1443 8.25133 +1500 8.65767 +2193 15.2023 +2231 15.5933 +2280 16.1503 +2632 20.2637 +2687 21.059 +2907 23.782 +3053 25.886 +3091 26.434 +3437 31.5983 +3587 34.2333 +3976 39.9213 +4702 53.383 +4751 54.564 +5349 67.4433 +5350 67.262 +5353 67.505 +5722 74.9237 +5764 76.005 +5844 77.7597 +5946 80.0777 +5969 80.8947 +5993 81.0297 +6013 81.995 +6323 89.7763 +6883 104.332 +6917 105.1 +7218 113.653 +7383 120.002 +7663 127.383 +8038 138.808 +8119 140.824 +8133 140.531 +8165 142.339 +8204 144.491 +8694 158.751 +8805 164.991 +9262 180.528 +9334 182.775 +9569 189.227 +9811 198.952 +9876 202.732 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd35_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd35_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +22 0.045 +155 0.700333 +176 0.784333 +911 6.20033 +1065 7.59467 +1107 8.02367 +1526 13.3993 +1623 13.9763 +1782 15.9207 +1863 16.9817 +1911 17.7677 +2165 22.11 +2192 22.0237 +2217 22.17 +2247 22.46 +3050 36.9757 +3167 39.482 +3398 45.8423 +3627 47.983 +4092 60.179 +4240 62.5573 +4346 65.6493 +4615 72.9677 +4673 75.9043 +4708 76.1693 +4739 75.6093 +4818 78.779 +5058 85.1967 +5222 88.906 +5414 95.2437 +5868 109.191 +5946 116.128 +6061 115.374 +6263 122.258 +6611 136.819 +6699 138.081 +6799 141.604 +6915 146.624 +7100 152.956 +7136 154.626 +7218 157.368 +7369 167.007 +7736 180.202 +8219 199.971 +8377 210.541 +8482 214.688 +8632 217.674 +8886 234.622 +9351 251.814 +9771 284.116 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd35_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd35_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +20 0.049 +550 3.94 +628 4.72633 +660 5.01467 +1192 11.1367 +1663 18.0227 +1749 20.1853 +1768 19.665 +2082 24.8423 +2355 31.2503 +2375 30.2203 +2617 35.2503 +2946 43.0043 +3499 56.9613 +3567 58.4403 +3774 67.8443 +3923 68.45 +4002 71.0137 +4189 75.521 +4205 79.518 +4214 77.788 +4414 81.9137 +4430 82.603 +4481 84.3963 +4500 86.8007 +4706 92.226 +4861 97.081 +5162 107.331 +5650 126.66 +5724 128.453 +5795 130.164 +5938 138.769 +5944 137.439 +6006 140.665 +6057 147.517 +6462 161.468 +6536 162.08 +6668 170.276 +6996 188.282 +7594 213.816 +7936 230.678 +8716 269.054 +8871 271.437 +9032 283.079 +9142 292.064 +9164 311.828 +9435 308.418 +9465 320.903 +9792 341.184 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd5_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd5_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +354 0.042 +446 0.061 +631 0.104333 +668 0.109667 +842 0.164667 +1296 0.334 +1327 0.354 +1495 0.424333 +1713 0.554667 +2027 0.735333 +2106 0.802667 +2901 1.459 +2967 1.494 +3169 1.79833 +3279 1.942 +3320 1.84067 +3510 2.026 +4337 3.03467 +4535 3.29267 +4841 3.747 +5020 4.00767 +5111 4.14067 +5163 4.54733 +5359 4.85367 +5542 4.838 +5547 4.852 +5562 4.86767 +5631 4.97333 +5877 5.41667 +6101 5.932 +6295 6.19133 +6584 6.75567 +6594 6.82033 +6788 7.783 +6849 7.599 +7173 7.93133 +7279 8.18833 +7361 8.36933 +7384 8.44133 +7461 8.64567 +7467 8.59167 +8091 10.7283 +8451 10.8843 +8827 11.8837 +9031 12.6497 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Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd5_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +326 0.199 +399 0.267333 +476 0.353333 +1069 1.362 +1389 2.15067 +1415 2.15767 +1654 2.86733 +1909 3.84733 +1961 3.961 +2638 6.87267 +3005 8.70933 +3045 8.90867 +3144 9.68167 +3229 9.952 +3275 10.5163 +3803 13.5883 +4459 19.0313 +4816 21.531 +4988 23.239 +5097 24.4823 +5424 28.021 +5465 27.2663 +5634 29.8707 +5645 29.3437 +6022 33 +6094 33.717 +6159 35.262 +6291 36.5397 +6361 37.286 +6638 40.641 +6677 41.6727 +6838 42.07 +7043 45.2177 +7531 52.6313 +7566 53.4523 +7623 52.2727 +7647 52.8467 +7822 71.1413 +7994 59.2187 +8003 59.824 +8475 65.147 +8996 73.607 +9118 76.1177 +9256 81.6237 +9750 87.1797 +9753 91.3717 +9787 91.8217 +9848 86.729 +9849 87.4 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd5_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd5_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +379 0.494667 +431 0.605667 +665 1.189 +718 1.341 +826 1.708 +1251 3.50233 +1279 3.631 +1443 4.50667 +1500 4.84533 +2193 9.664 +2231 10.0093 +2280 10.4237 +2632 13.5977 +2687 14.1333 +2907 16.407 +3053 18.0413 +3091 18.4913 +3437 22.572 +3587 24.5313 +3976 29.7623 +4702 41 +4751 41.851 +5349 52.6303 +5350 52.5897 +5353 52.5687 +5722 59.825 +5764 60.897 +5844 62.9263 +5946 64.5117 +5969 64.9087 +5993 65.7193 +6013 65.8873 +6323 72.5703 +6883 85.8193 +6917 86.475 +7218 93.9543 +7383 98.1227 +7663 105.505 +8038 115.815 +8119 117.976 +8133 118.325 +8165 119.396 +8204 120.515 +8694 134.9 +8805 138.845 +9262 152.516 +9334 156.444 +9569 164.193 +9811 170.963 +9876 172.89 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd5_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd5_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +22 0.0186667 +155 0.201333 +176 0.222667 +911 2.74667 +1065 3.56667 +1107 3.82 +1526 6.75067 +1623 7.55467 +1782 8.96033 +1863 9.66833 +1911 10.1757 +2165 12.923 +2192 13.0273 +2217 13.338 +2247 13.6367 +3050 24.2777 +3167 26.036 +3398 29.746 +3627 33.5413 +4092 42.2407 +4240 45.1543 +4346 48.223 +4615 53.2423 +4673 54.4743 +4708 55.3837 +4739 56.768 +4818 57.6567 +5058 63.4033 +5222 67.2153 +5414 72.2027 +5868 84.217 +5946 86.786 +6061 89.7137 +6263 95.571 +6611 106.224 +6699 108.918 +6799 112.146 +6915 115.882 +7100 122.037 +7136 123.168 +7218 126.608 +7369 131.009 +7736 143.94 +8219 161.959 +8377 167.891 +8482 172.541 +8632 179.712 +8886 188.364 +9351 208.154 +9771 227.267 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd5_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd5_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +20 0.02 +550 1.45267 +628 1.793 +660 1.923 +1192 5.2 +1663 9.43367 +1749 10.3543 +1768 10.5797 +2082 14.1837 +2355 17.9237 +2375 18.6123 +2617 21.9483 +2946 27.5247 +3499 37.583 +3567 38.9343 +3774 43.5577 +3923 46.8577 +4002 48.6863 +4189 52.968 +4205 53.1783 +4214 53.656 +4414 58.3077 +4430 58.6777 +4481 59.972 +4500 60.5043 +4706 66.548 +4861 70.1353 +5162 78.8657 +5650 96.785 +5724 95.6887 +5795 98.4357 +5938 103.126 +5944 103.095 +6006 105.139 +6057 107.132 +6462 121.897 +6536 123.841 +6668 131.597 +6996 141.533 +7594 166.466 +7936 180.745 +8716 216.966 +8871 223.681 +9032 232.128 +9142 237.983 +9164 239.712 +9435 252.498 +9465 255.209 +9792 272.201 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +78 0.00633333 +104 0.00966667 +132 0.012 +142 0.0156667 +173 0.0166667 +196 0.0213333 +218 0.0236667 +276 0.0323333 +293 0.0353333 +296 0.0353333 +366 0.049 +423 0.0603333 +458 0.067 +460 0.0693333 +486 0.073 +561 0.095 +575 0.0953333 +641 0.118333 +792 0.153333 +870 0.178 +922 0.206667 +992 0.236333 +1114 0.267667 +1166 0.292667 +1202 0.311667 +1608 0.505333 +1737 0.573667 +2022 0.802333 +2837 1.41533 +3271 1.84667 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +47 0.006 +85 0.013 +150 0.0256667 +192 0.0356667 +200 0.037 +206 0.038 +273 0.0563333 +302 0.0693333 +387 0.092 +394 0.0936667 +536 0.153333 +790 0.282667 +816 0.311 +851 0.331333 +925 0.374 +1085 0.483333 +1107 0.496667 +1135 0.516667 +1460 0.847 +1488 0.869667 +1954 1.39833 +2054 1.51533 +2170 1.729 +2190 1.67733 +2253 2.786 +2257 1.77167 +2311 1.90167 +2390 2.02967 +2463 2.083 +3366 3.765 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,28 @@ +20 0.00666667 +47 0.0176667 +65 0.0256667 +82 0.0366667 +85 0.0346667 +101 0.0453333 +141 0.0666667 +213 0.116667 +228 0.123333 +272 0.162667 +277 0.172333 +355 0.238333 +491 0.382333 +587 0.505 +654 0.601667 +698 0.673 +716 0.703333 +763 0.779 +851 0.970333 +1032 1.29067 +1095 1.421 +1603 2.78933 +1969 4.05133 +2176 4.86067 +2412 5.86433 +2587 6.67967 +2958 8.63433 +4266 17.2513 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,29 @@ +20 0.0176667 +25 0.0193333 +43 0.0386667 +56 0.0516667 +227 0.266667 +397 0.587333 +448 0.698 +474 0.769 +487 0.795 +509 0.849333 +630 1.18 +727 1.50333 +732 1.52167 +777 1.68267 +826 1.83933 +899 2.108 +1034 2.67733 +1078 2.85433 +1107 3.00767 +1146 3.19433 +1282 3.90833 +1393 4.439 +1410 4.676 +1484 5.00367 +1600 5.71033 +2053 8.91533 +2076 9.08333 +2201 10.1147 +2813 15.9103 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +37 0.0423333 +70 0.0863333 +157 0.238 +199 0.322 +271 0.468 +376 0.770333 +379 0.77 +380 0.774333 +522 1.24 +554 1.35333 +595 1.523 +634 1.652 +718 2.018 +721 2.04067 +730 2.07067 +928 3.07467 +1046 3.753 +1088 3.99333 +1156 4.42567 +1357 5.83733 +1443 6.48867 +1564 7.486 +1579 7.59033 +1780 9.42333 +1860 10.198 +1876 10.3787 +2079 12.4097 +2866 22.3363 +3318 29.3473 +4068 43.058 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2pInd7_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2pInd7_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,28 @@ +20 0.0326667 +59 0.0956667 +80 0.136333 +108 0.180667 +166 0.342667 +176 0.353667 +222 0.465333 +444 1.221 +624 1.98067 +631 2.01 +669 2.198 +757 2.66133 +876 3.37067 +891 3.45033 +1124 5.10967 +1193 5.64333 +1217 5.91667 +1345 6.93567 +1449 7.88367 +1603 9.44067 +1655 9.98667 +2079 15.527 +2129 15.7007 +2567 21.9067 +2716 24.4203 +3103 31.128 +3657 42.242 +4945 74.4017 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +354 0.0213333 +446 0.029 +631 0.044 +668 0.043 +842 0.062 +1296 0.0866667 +1327 0.0963333 +1495 0.105 +1713 0.128 +2027 0.129333 +2106 0.172 +2901 0.207333 +2967 0.235 +3169 0.313333 +3279 1.13867 +3320 0.268333 +3510 0.271333 +4337 0.383 +4535 0.784333 +4841 0.376 +5020 11.99 +5111 0.631333 +5163 0.456 +5359 0.532 +5542 0.409 +5547 0.390667 +5562 1.90533 +5631 1.079 +5877 0.505333 +6101 0.489667 +6295 0.470333 +6584 2.44433 +6594 19.3737 +6788 2.752 +6849 0.527333 +7173 1.077 +7279 2.22867 +7361 0.739667 +7384 2.746 +7461 127.963 +7467 1.13867 +8091 1.29667 +8451 0.656667 +8827 66.1093 +9031 0.911667 +9134 20.6043 +9427 0.798 +9639 1.06033 +9720 5.394 +9943 1.997 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,48 @@ +70 0.008 +377 0.0476667 +390 0.0596667 +788 0.116667 +1086 0.154333 +1922 0.260333 +2057 0.265667 +2071 0.306333 +2169 0.358333 +2406 0.573667 +2958 0.496333 +3323 1.10033 +3631 0.498333 +4224 2.97633 +4380 1.394 +4427 1.05067 +4663 1.098 +4942 1.568 +5118 0.946667 +5183 0.800667 +5204 1.32433 +5368 1.427 +5704 1.175 +5826 1.62467 +5859 1.14 +5925 5.22067 +5948 5.151 +5961 2.336 +6052 7.86367 +6472 4.58433 +6871 5.176 +7015 2.91567 +7039 1.56433 +7221 5.88167 +7238 1.15867 +7776 3.652 +7806 11.2547 +8478 14.108 +8679 9.199 +8966 8.07367 +9251 36.0683 +9336 3.62633 +9499 40.213 +9583 11.5847 +9585 5.372 +9649 499.196 +9816 19.6343 +9932 3.18467 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +326 0.122667 +399 0.158 +476 0.19 +1069 0.440333 +1389 0.578 +1415 0.588 +1654 0.699333 +1909 0.830667 +1961 0.811333 +2638 1.361 +3005 1.39367 +3045 1.407 +3144 1.42767 +3229 1.69733 +3275 1.49733 +3803 1.963 +4459 2.233 +4816 2.28267 +4988 2.47933 +5097 2.67933 +5424 2.91633 +5465 2.99933 +5634 3.07333 +5645 3.224 +6022 2.97333 +6094 3.03967 +6159 3.767 +6291 2.952 +6361 2.99367 +6638 4.44067 +6677 3.07867 +6838 4.14167 +7043 4.32567 +7531 3.77567 +7566 3.97833 +7623 4.11233 +7647 4.552 +7822 4.82467 +7994 4.35267 +8003 4.042 +8475 4.786 +8996 4.78967 +9118 5.09367 +9256 5.06733 +9702 6.66567 +9750 5.32933 +9753 6.09333 +9787 5.00067 +9848 6.21267 +9849 5.135 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +379 0.363667 +431 0.392333 +665 0.625 +718 0.686333 +826 0.797333 +1251 1.23167 +1279 1.29267 +1443 1.50733 +1500 1.52067 +2193 2.234 +2231 2.262 +2280 2.35133 +2632 2.76867 +2687 2.781 +2907 3.16533 +3053 3.29767 +3091 3.36433 +3437 3.69 +3587 3.81933 +3976 4.31133 +4702 5.154 +4751 5.19467 +5349 5.89933 +5350 5.955 +5353 5.91467 +5722 6.85567 +5764 6.48367 +5844 6.50733 +5946 6.64233 +5969 6.803 +5993 6.76567 +6013 6.766 +6323 7.207 +6883 8.03367 +6917 7.98533 +7218 8.47467 +7383 8.57 +7663 8.92767 +8038 9.49367 +8119 9.51667 +8133 10.128 +8165 9.59467 +8204 9.61967 +8694 10.1283 +8805 10.378 +9262 10.8383 +9334 10.9447 +9569 11.28 +9811 11.663 +9876 11.675 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +22 0.0266667 +155 0.21 +176 0.236667 +911 1.297 +1065 1.57533 +1107 1.61733 +1526 2.259 +1623 2.48567 +1782 2.81767 +1863 2.838 +1911 2.89667 +2165 3.31867 +2192 3.348 +2217 3.445 +2247 3.59367 +3050 4.935 +3167 5.151 +3398 5.51833 +3627 5.90267 +4092 6.87 +4240 7.22467 +4346 7.32867 +4615 7.695 +4673 7.90567 +4708 7.988 +4739 8.083 +4818 8.18833 +5058 8.73 +5222 9.14267 +5414 9.27467 +5868 10.5013 +5946 10.2697 +6061 10.8493 +6263 11.3743 +6611 11.4547 +6699 11.75 +6799 12.2597 +6915 12.1893 +7100 12.796 +7136 12.5183 +7218 12.9163 +7369 13.036 +7736 13.905 +8219 14.67 +8377 15.2837 +8482 15.0247 +8632 15.2837 +8886 15.868 +9351 16.7927 +9771 17.4753 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd10_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd10_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,39 @@ +550 0.977333 +628 1.12967 +660 1.16167 +1192 2.21433 +1663 3.18033 +1749 3.43967 +2082 4.02067 +2355 4.75233 +2375 4.75433 +2946 6.37567 +3499 7.39967 +3567 7.56767 +3774 8.10367 +4189 9.01667 +4205 9.05533 +4214 9.09067 +4414 9.57467 +4430 10.0363 +4481 9.95667 +4500 9.74033 +4861 10.8177 +5162 11.4517 +5650 12.598 +5724 13.0937 +5795 13.2703 +5944 13.2853 +6006 13.8167 +6057 13.4997 +6462 14.4263 +6536 14.7693 +6668 15.2183 +6996 15.9877 +7594 17.3567 +7936 18.2223 +8716 20.4137 +9142 21.4613 +9164 21.742 +9435 21.7353 +9465 22.3537 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +216 0.0146667 +407 0.0306667 +538 0.0413333 +927 0.0796667 +1009 0.0803333 +1277 0.0966667 +1668 0.133667 +2391 0.220667 +3229 0.483667 +3754 0.379 +3953 0.362333 +4254 0.448333 +4525 0.44 +4780 0.854 +5281 1.765 +5861 1.72167 +6317 28.055 +6434 8.12967 +6766 0.83 +6881 4.948 +7047 1.42567 +7316 20.8757 +7396 4.272 +8696 45.728 +8897 3.91867 +9051 22.0687 +9075 47.6293 +9377 33.6793 +9573 5.967 +9687 56.9007 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +344 0.0553333 +1434 0.264 +2070 0.424333 +2175 0.442333 +2480 0.467333 +2685 0.545 +3150 0.735333 +3611 0.881 +3814 1.27267 +4085 0.890333 +4089 0.840333 +4565 0.805667 +4742 11.2947 +4778 14.677 +5395 2.54333 +5587 3.559 +5598 1.28433 +5741 17.7933 +6033 1.878 +6357 3.084 +6401 4.80367 +6581 3.23567 +6593 2.65867 +6869 3.31867 +7079 3.54667 +7280 2.801 +7882 9.438 +8847 18.4517 +9039 13.3743 +9728 7.01267 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.2.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.2.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +32 0.012 +471 0.158667 +1247 0.422667 +2258 0.848 +2333 0.870667 +2339 0.860667 +2344 0.877333 +2642 1.05267 +2756 0.966 +3207 1.196 +3593 1.27933 +4212 1.534 +5038 1.90733 +5085 2.155 +5543 2.518 +5801 2.947 +5861 3.33867 +6042 2.218 +6385 2.66867 +6959 2.95133 +7205 2.58767 +8042 4.22967 +8173 3.53733 +9063 5.05867 +9074 5.36933 +9736 4.149 +9770 6.145 +9805 4.74167 +9833 4.91367 +9914 5.165 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +262 0.135333 +598 0.309333 +694 0.366667 +722 0.370333 +752 0.379333 +1116 0.585667 +1224 0.645667 +1293 0.698333 +1314 0.681333 +1836 0.994333 +3127 1.80733 +3213 1.82233 +3390 2.109 +3613 2.00933 +4832 2.88533 +5319 3.144 +5330 3.16367 +5379 3.43933 +5394 3.18233 +5456 3.38367 +6354 4.65767 +6579 3.926 +6889 4.30833 +7096 4.39633 +8039 5.09067 +8436 5.38967 +8686 5.465 +9377 6.039 +9455 6.445 +9608 6.16967 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.4.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.4.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +30 0.0193333 +315 0.226333 +439 0.312333 +717 0.533667 +1550 1.21567 +1859 1.512 +1867 1.473 +1868 1.52367 +2687 2.25567 +2734 2.37 +3357 2.74433 +3709 3.01867 +5091 4.33667 +5510 4.62033 +5514 4.83333 +5654 5.03467 +5732 5.07067 +6082 5.78367 +6549 5.85133 +6816 6.15333 +6920 6.08967 +7859 7.083 +8363 7.626 +8757 7.96433 +8854 7.919 +8868 8.09233 +9036 8.47033 +9515 8.78633 +9686 8.83533 +9782 9.45933 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.5.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.5.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +313 0.319667 +542 0.548667 +623 0.64 +1531 1.55567 +1753 1.84767 +1856 1.99567 +1903 1.96733 +2022 2.211 +2064 2.209 +3364 3.75433 +3982 4.50333 +4041 4.603 +5037 5.99967 +6089 7.01633 +6443 7.47167 +6638 7.72567 +6892 8.05567 +7005 8.47933 +7133 8.693 +7764 9.36267 +7826 9.21667 +7839 9.55367 +7916 9.45433 +8055 9.772 +8075 9.816 +8403 9.97067 +8627 10.559 +8701 10.5947 +9207 11.37 +9907 12.3923 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +136 0.165667 +1037 1.349 +1104 1.43567 +1198 1.58233 +1204 1.641 +1400 1.924 +1533 2.02433 +1669 2.29033 +1723 2.36067 +2230 3.12033 +3701 5.424 +4063 6.157 +4258 6.25533 +4476 6.64567 +5205 7.83367 +5304 8.06333 +5718 8.58367 +6123 9.443 +6141 9.535 +6192 9.48433 +6711 10.6077 +6973 11.155 +7075 11.1143 +7294 11.6163 +7653 12.2893 +7869 12.6847 +7878 12.678 +8668 13.8913 +8726 14.3277 +9523 15.6823 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.7.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.7.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +115 0.172667 +197 0.288333 +364 0.552 +392 0.621667 +914 1.50733 +950 1.57567 +1556 2.613 +1561 2.69567 +1702 2.94833 +1740 2.94633 +1834 3.12 +1982 3.45367 +1998 3.46433 +2023 3.472 +2092 3.586 +2187 3.82633 +2297 4.02933 +2314 4.08033 +2399 4.28333 +2450 4.43367 +2534 4.54633 +2570 4.59367 +2666 4.739 +2712 4.90267 +2725 4.784 +2729 4.82333 +2730 4.883 +2895 5.25267 +2920 5.23633 +2929 5.23967 +2997 5.37767 +3117 5.81833 +3149 5.81033 +3182 5.79967 +3451 6.29767 +3464 6.463 +3537 6.45167 +3564 6.591 +3594 6.57067 +3825 7.19667 +3892 7.172 +3972 7.33367 +4198 8.09167 +4243 7.99033 +4281 8.07633 +4327 8.302 +4776 9.09833 +4882 9.21733 +4938 9.47633 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.8.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.8.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +57 0.099 +864 1.69167 +1355 2.68633 +1402 2.852 +1580 3.21867 +2161 4.597 +2295 4.885 +2368 5.01667 +2585 5.51133 +2720 5.82433 +3126 6.803 +3151 6.917 +4218 9.72633 +4886 11.0813 +5022 11.658 +5239 12.1167 +5344 12.2433 +5361 12.1497 +5375 12.3153 +5819 13.4813 +5930 13.69 +6273 14.7603 +6477 15.2887 +6947 16.3877 +7598 18.3013 +8269 19.8827 +8736 21.7027 +8868 21.4237 +8972 21.9797 +9725 23.7263 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd15_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd15_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +28 0.058 +322 0.729 +449 1.04533 +561 1.30933 +848 2.12467 +1519 4.12433 +2125 5.89733 +2211 6.30633 +2218 6.33867 +2232 6.37933 +2533 7.33867 +2622 7.69733 +2892 8.50067 +3128 9.12 +3156 9.349 +3379 9.994 +3513 10.518 +3807 11.3437 +4061 12.1577 +4750 14.4103 +4767 14.4057 +4768 14.575 +5893 17.9827 +6203 19.2467 +8331 26.5803 +8433 27.073 +8547 27.4483 +8935 28.6703 +9613 31.179 +9627 31.0557 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd32_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd32_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +216 0.0256667 +407 0.054 +538 0.0753333 +927 0.141667 +1009 0.13 +1277 0.182 +1668 0.246333 +2391 0.79 +3229 0.588667 +3754 1.11433 +3953 1.711 +4254 2.74767 +4525 0.831 +4780 2.55267 +5281 3.972 +5861 1.98633 +6317 2.189 +6434 2.517 +6766 4.74233 +6881 1.70067 +7047 1.75867 +7316 6.84733 +7396 2.67267 +8696 7.369 +8897 64.127 +9051 4.37267 +9075 19.596 +9377 7.534 +9573 26.88 +9687 30.3903 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd32_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd32_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +344 0.0883333 +1434 0.382333 +2070 0.731333 +2175 0.609667 +2480 0.679 +2685 0.820667 +3150 1.18033 +3611 1.29167 +3814 1.39767 +4085 1.19267 +4089 1.387 +4565 1.44533 +4742 1.53067 +4778 1.50533 +5395 2.04567 +5587 1.93833 +5598 2.05633 +5741 1.82233 +6033 2.15367 +6357 1.87467 +6401 3.375 +6581 2.27433 +6593 2.21667 +6869 2.19067 +7079 2.73167 +7280 3.381 +7882 3.30433 +8847 4.94833 +9039 4.95233 +9728 4.448 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd32_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd32_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +262 0.222 +598 0.523667 +694 0.625333 +722 0.641667 +752 0.686667 +1116 1.033 +1224 1.14567 +1293 1.21533 +1314 1.231 +1836 1.76667 +3127 3.10133 +3213 3.18767 +3390 3.39567 +3613 3.69567 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@@ -0,0 +1,30 @@ +248 0.872333 +439 1.55767 +443 1.63633 +1328 5.37267 +1471 6.00033 +2396 10.006 +2454 10.3023 +2968 12.4067 +2997 12.942 +3149 13.509 +3328 14.336 +3472 14.9353 +3850 16.56 +4055 17.5867 +5013 22.144 +5059 23.2843 +5663 25.5683 +5728 26.1993 +6077 28.2707 +6448 29.18 +6510 28.466 +6532 29.3593 +7140 31.576 +7297 32.294 +7470 33.6157 +7628 33.767 +8250 38.49 +8333 39.233 +8834 42.1883 +9980 48.5733 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd32_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd32_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +28 0.0763333 +322 1.45067 +449 2.122 +561 2.68233 +848 4.34533 +1519 8.448 +2125 11.5473 +2211 12.187 +2218 12.574 +2232 12.2737 +2533 14.052 +2622 14.1437 +2892 16.2433 +3128 18.7247 +3156 17.7963 +3379 18.994 +3513 20.174 +3807 20.4843 +4061 22.896 +4750 25.737 +4767 25.796 +4768 26.983 +5893 32.2643 +6203 34.7197 +8331 51.787 +8433 53.5687 +8547 54.17 +8935 55.877 +9613 59.2407 +9627 59.24 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd35_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd35_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,47 @@ +33 0.002 +684 0.103333 +1057 0.156333 +1125 0.226333 +1578 0.288 +1860 0.441667 +1867 0.375 +1990 0.387 +2036 0.582667 +2595 0.420333 +2607 0.611 +2756 0.716667 +3185 0.749333 +3275 0.71 +3362 1.171 +3466 0.787667 +3492 0.979667 +3564 0.615667 +3656 1.07733 +3834 1.38467 +4345 2.50567 +4605 1.11367 +4740 1.585 +4921 1.08533 +4941 1.28733 +4993 1.187 +5513 2.28133 +5767 3.42267 +6134 4.55433 +6410 6.12933 +6534 4.74933 +6904 2.03367 +7010 11.2137 +7067 3.02833 +7231 2.602 +7314 5.80833 +7525 4.636 +7538 5.51833 +7660 3.01067 +8422 8.947 +8696 19.85 +8885 13.444 +8916 8.696 +8919 4.75467 +9337 8.49733 +9878 10.3503 +9999 12.4133 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd35_0.1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd35_0.1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +302 0.0766667 +450 0.119 +697 0.204 +744 0.202333 +805 0.222333 +938 0.259667 +963 0.287667 +1043 0.302 +1240 0.402667 +1797 0.548 +2018 0.637667 +2124 0.641 +2435 0.864333 +2810 0.999333 +3368 1.06533 +3401 1.177 +3531 1.17 +3674 1.36033 +3734 1.14833 +4359 1.41933 +4403 1.459 +4406 1.82333 +4466 1.49833 +4579 1.647 +4685 1.49167 +4940 1.68467 +5007 1.70733 +5312 1.733 +5471 1.767 +5564 2.71533 +5962 2.27367 +6114 2.51267 +6677 2.802 +7019 2.60933 +7128 2.97933 +7468 2.61133 +7488 2.498 +7636 3.679 +7668 4.28667 +7823 2.94333 +8143 4.04233 +8156 3.09033 +8333 3.767 +8572 3.66367 +8961 5.04967 +9009 3.94867 +9443 4.829 +9784 4.55167 +9868 3.76567 +9959 3.98567 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd35_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd35_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,50 @@ +326 0.305333 +399 0.372333 +476 0.450333 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35.4867 +5938 37.4263 +5944 37.0227 +6006 38.4617 +6057 42.5057 +6462 41.0057 +6536 41.8393 +6668 43.6183 +6996 46.7797 +7594 49.0137 +7936 52.2893 +8716 56.329 +8871 55.0203 +9032 58.0967 +9142 58.6687 +9164 66.3373 +9435 58.8493 +9465 67.1867 +9792 69.378 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd5_0.05.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd5_0.05.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +354 0.0166667 +446 0.022 +631 0.033 +668 0.033 +842 0.0453333 +1296 0.065 +1327 0.0686667 +1495 0.073 +1713 0.089 +2027 0.103 +2106 0.111667 +2901 0.144 +2967 0.152333 +3169 0.180333 +3279 0.189 +3320 0.176667 +3510 0.183 +4337 0.236333 +4535 0.235333 +4841 0.256333 +5020 0.272 +5111 0.274333 +5163 0.357333 +5359 0.331333 +5542 0.288333 +5547 0.279333 +5562 0.296667 +5631 0.298 +5877 0.299333 +6101 0.361333 +6295 0.339667 +6584 0.346667 +6594 0.573333 +6788 0.375333 +6849 0.406 +7173 0.384667 +7279 0.385333 +7361 0.722 +7384 0.412667 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20d1b0e5838f randGraph/ind/vf2ppInd5_0.3.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd5_0.3.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,48 @@ +326 0.0933333 +399 0.118333 +476 0.136 +1069 0.345667 +1389 0.458667 +1415 0.462333 +1654 0.489 +1909 0.566333 +1961 0.579 +2638 0.995333 +3005 0.898667 +3045 1.058 +3144 1.033 +3229 1.057 +3275 1.29667 +3803 1.158 +4459 1.47133 +4816 1.68333 +4988 2.487 +5097 2.03067 +5424 2.01667 +5465 2.03633 +5634 1.9 +5645 2.21233 +6022 2.241 +6094 3.47567 +6159 2.819 +6291 2.505 +6361 6.54433 +6638 11.287 +6677 3.48567 +6838 3.951 +7043 3.713 +7531 2.717 +7566 2.86233 +7623 4.40867 +7647 4.656 +7822 4.36233 +7994 3.64 +8003 5.27433 +8475 6.34667 +8996 6.25967 +9118 4.21533 +9750 23.6907 +9753 5.349 +9787 5.91533 +9848 6.192 +9849 10.102 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd5_0.6.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd5_0.6.txt Tue Nov 22 08:15:16 2016 +0100 @@ 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1.70633 +1911 1.76667 +2165 2.38267 +2192 2.04267 +2217 2.12933 +2247 2.09467 +3050 2.965 +3167 3.10467 +3398 3.517 +3627 3.529 +4092 4.08667 +4240 4.17667 +4346 4.381 +4615 4.61533 +4673 4.68533 +4708 4.75833 +4739 4.811 +4818 4.84 +5058 5.27267 +5222 5.36067 +5414 5.62467 +5868 6.06767 +5946 6.277 +6061 6.354 +6263 6.612 +6611 7.035 +6699 7.41967 +6799 7.291 +6915 7.52767 +7100 8.37267 +7136 7.69 +7218 7.85467 +7369 8.376 +7736 8.47367 +8219 8.76167 +8377 8.928 +8482 9.275 +8632 9.286 +8886 9.56033 +9351 10.1563 +9771 10.635 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/ind/vf2ppInd5_0.95.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/ind/vf2ppInd5_0.95.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,49 @@ +20 0.017 +550 0.581333 +628 0.666667 +660 0.68 +1192 1.278 +1663 1.866 +1749 1.997 +1768 2.01533 +2082 2.39367 +2355 2.81433 +2375 2.87133 +2617 3.139 +2946 3.63033 +3499 4.28467 +3567 4.39733 +3774 4.69433 +3923 4.88467 +4002 4.96033 +4189 5.24467 +4205 5.207 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Jan 01 00:00:00 1970 +0000 +++ b/randGraph/iso/vf2pIso.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,149 @@ +127 0.3269 +162 0.432 +240 0.7383 +376 1.2215 +425 1.4418 +491 1.7892 +653 2.6752 +690 2.954 +896 4.3193 +985 4.9454 +1003 5.0853 +1030 5.3009 +1132 6.187 +1328 7.918 +1942 14.7931 +2226 19.1137 +2472 22.4367 +2804 28.1592 +2818 28.1566 +2907 30.4856 +2930 30.6408 +3076 34.0425 +3200 35.2525 +3364 39.1791 +3467 41.12 +3544 42.2586 +3613 44.5656 +3704 45.6906 +3946 51.1882 +4023 53.1333 +4076 55.1514 +4145 56.5863 +4183 57.0518 +4426 64.6284 +4644 71.1045 +4767 73.4162 +5112 81.686 +5204 84.8276 +5243 88.8607 +5263 87.1636 +5384 90.1785 +5408 93.2033 +5490 93.4534 +5653 101.436 +5783 103.567 +5861 108.921 +5873 105.73 +6014 110.53 +6135 118.937 +6340 121.431 +6378 122.979 +6443 125.289 +6530 132.548 +6822 139.731 +7086 151.391 +7180 152.987 +7464 165.421 +7504 171.694 +7509 170.031 +8175 201.68 +8291 201.49 +8478 209.186 +8480 209.196 +8622 220.353 +8626 223.71 +8904 230.857 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01 00:00:00 1970 +0000 +++ b/randGraph/iso/vf2pIso15_1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,150 @@ +176 0.564333 +201 0.680333 +248 0.867 +526 2.36667 +773 4.571 +909 5.37467 +959 5.516 +1023 6.28967 +1048 6.43433 +1186 7.972 +1211 7.93233 +1401 10.1643 +1498 11.476 +1557 12.0227 +1696 15.4357 +1732 14.1733 +1747 14.9837 +2030 18.6953 +2056 19.0063 +2172 20.641 +2345 23.4693 +2551 27.3363 +2813 33.0293 +2915 34.5197 +2979 36.213 +3450 46.0727 +3497 47.4427 +3626 50.8307 +3660 51.666 +3877 57.375 +4246 67.732 +4430 79.4417 +4445 72.8763 +4449 73.146 +4734 82.0657 +4961 88.574 +5009 91.247 +5109 93.131 +5259 98.3897 +5661 112.654 +5722 115.168 +5810 118.769 +5836 118.441 +5904 123.274 +5966 124.405 +5968 124.203 +6410 142.027 +6706 153.247 +6895 167.227 +6995 166.15 +7176 175.387 +7215 196.121 +7270 179.878 +7545 193.7 +7546 193.551 +7988 213.504 +8205 225.142 +8511 238.828 +8620 245.955 +8962 265.485 +8975 269.972 +9238 285.401 +9266 283.003 +9370 288.884 +9491 298.718 +9503 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999.306 +17717 989.295 +17953 1026.75 +18103 1028.22 +18150 1033.33 +18198 1041.62 +18201 1035.62 +18244 1134.24 +18263 1050.58 +18711 1106.11 +18740 1096.77 +19082 1155.23 +19175 1149.44 +19205 1154.88 +19546 1203.89 +19571 1201.74 +19790 1236.07 +19849 1231 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/iso/vf2pIso30_1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/iso/vf2pIso30_1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,78 @@ +431 2.79433 +1427 14.372 +1907 21.9707 +2207 27.3327 +2821 40.378 +2962 43.2867 +3218 49.5623 +3553 58.2343 +3789 64.7903 +4318 81.121 +4735 93.7133 +5023 103.343 +5039 103.272 +5340 115.846 +6476 163.966 +6655 168.633 +6880 178.643 +7063 186.107 +7149 190.327 +7525 209.674 +7532 209.284 +7790 228.302 +7926 231.419 +8163 255.66 +8196 242.812 +8348 250.875 +8694 277.269 +9183 303.451 +9292 306.609 +9327 309.754 +9580 325.144 +9640 326.323 +11211 438.273 +11390 447.947 +11427 452.046 +11469 454.352 +11529 458.106 +11578 470.117 +11731 472.158 +11828 480.679 +11835 483.293 +12073 499.352 +12783 557.565 +13286 596.93 +13550 617.476 +13594 636.049 +13761 642.147 +13907 650.99 +14023 661.275 +14090 673.737 +14400 695.074 +14476 701.143 +14510 719.181 +14761 736.055 +15631 824.767 +15829 836.491 +16433 893.111 +16642 924.982 +16805 937.985 +17136 966.686 +17139 965.419 +17211 980.675 +17231 986.482 +17455 1001.69 +17748 1040.39 +17871 1057.27 +18307 1094.26 +18432 1106.87 +18493 1136.26 +18499 1117.88 +18504 1136.65 +18861 1156.82 +18957 1176.25 +19080 1209.36 +19250 1217.55 +19656 1257.58 +19765 1272.94 +19992 1309.2 diff -r 76dc74f824ba -r 20d1b0e5838f randGraph/iso/vf2pIso35_1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/iso/vf2pIso35_1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,149 @@ +26 0.091 +442 3.31567 +799 7.10933 +912 8.28067 +1000 9.53 +1031 10.031 +1034 10.117 +1126 11.5307 +1185 12.012 +1256 13.194 +1363 15.002 +1461 16.0057 +1550 17.4427 +1569 17.7457 +1602 18.6967 +1678 19.6053 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randGraph/iso/vf2ppIso80_1.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/randGraph/iso/vf2ppIso80_1.txt Tue Nov 22 08:15:16 2016 +0100 @@ -0,0 +1,30 @@ +460 5.97667 +1133 15.4167 +1988 27.5997 +2419 34.2267 +2865 43.3543 +4604 67.9797 +4770 70.3437 +5861 97.6497 +6314 103.801 +7278 123.768 +7672 127.688 +8665 159.771 +9691 171.676 +10150 181.881 +10575 206.934 +11432 230.12 +11832 221.124 +11871 222.772 +13337 270.319 +13880 272.756 +14328 283.048 +14680 295.872 +16107 338.477 +16527 345.87 +17538 393.947 +18692 425.632 +18757 421.26 +19043 417.752 +19207 437.89 +19547 433.38