COIN-OR::LEMON - Graph Library

source: glpk-cmake/src/amd/amd_2.c @ 1:c445c931472f

Last change on this file since 1:c445c931472f was 1:c445c931472f, checked in by Alpar Juttner <alpar@…>, 10 years ago

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1/* ========================================================================= */
2/* === AMD_2 =============================================================== */
3/* ========================================================================= */
4
5/* ------------------------------------------------------------------------- */
6/* AMD, Copyright (c) Timothy A. Davis,                                      */
7/* Patrick R. Amestoy, and Iain S. Duff.  See ../README.txt for License.     */
8/* email: davis at cise.ufl.edu    CISE Department, Univ. of Florida.        */
9/* web: http://www.cise.ufl.edu/research/sparse/amd                          */
10/* ------------------------------------------------------------------------- */
11
12/* AMD_2:  performs the AMD ordering on a symmetric sparse matrix A, followed
13 * by a postordering (via depth-first search) of the assembly tree using the
14 * AMD_postorder routine.
15 */
16
17#include "amd_internal.h"
18
19/* ========================================================================= */
20/* === clear_flag ========================================================== */
21/* ========================================================================= */
22
23static Int clear_flag (Int wflg, Int wbig, Int W [ ], Int n)
24{
25    Int x ;
26    if (wflg < 2 || wflg >= wbig)
27    {
28        for (x = 0 ; x < n ; x++)
29        {
30            if (W [x] != 0) W [x] = 1 ;
31        }
32        wflg = 2 ;
33    }
34    /*  at this point, W [0..n-1] < wflg holds */
35    return (wflg) ;
36}
37
38
39/* ========================================================================= */
40/* === AMD_2 =============================================================== */
41/* ========================================================================= */
42
43GLOBAL void AMD_2
44(
45    Int n,              /* A is n-by-n, where n > 0 */
46    Int Pe [ ],         /* Pe [0..n-1]: index in Iw of row i on input */
47    Int Iw [ ],         /* workspace of size iwlen. Iw [0..pfree-1]
48                         * holds the matrix on input */
49    Int Len [ ],        /* Len [0..n-1]: length for row/column i on input */
50    Int iwlen,          /* length of Iw. iwlen >= pfree + n */
51    Int pfree,          /* Iw [pfree ... iwlen-1] is empty on input */
52
53    /* 7 size-n workspaces, not defined on input: */
54    Int Nv [ ],         /* the size of each supernode on output */
55    Int Next [ ],       /* the output inverse permutation */
56    Int Last [ ],       /* the output permutation */
57    Int Head [ ],
58    Int Elen [ ],       /* the size columns of L for each supernode */
59    Int Degree [ ],
60    Int W [ ],
61
62    /* control parameters and output statistics */
63    double Control [ ], /* array of size AMD_CONTROL */
64    double Info [ ]     /* array of size AMD_INFO */
65)
66{
67
68/*
69 * Given a representation of the nonzero pattern of a symmetric matrix, A,
70 * (excluding the diagonal) perform an approximate minimum (UMFPACK/MA38-style)
71 * degree ordering to compute a pivot order such that the introduction of
72 * nonzeros (fill-in) in the Cholesky factors A = LL' is kept low.  At each
73 * step, the pivot selected is the one with the minimum UMFAPACK/MA38-style
74 * upper-bound on the external degree.  This routine can optionally perform
75 * aggresive absorption (as done by MC47B in the Harwell Subroutine
76 * Library).
77 *
78 * The approximate degree algorithm implemented here is the symmetric analog of
79 * the degree update algorithm in MA38 and UMFPACK (the Unsymmetric-pattern
80 * MultiFrontal PACKage, both by Davis and Duff).  The routine is based on the
81 * MA27 minimum degree ordering algorithm by Iain Duff and John Reid.
82 *
83 * This routine is a translation of the original AMDBAR and MC47B routines,
84 * in Fortran, with the following modifications:
85 *
86 * (1) dense rows/columns are removed prior to ordering the matrix, and placed
87 *      last in the output order.  The presence of a dense row/column can
88 *      increase the ordering time by up to O(n^2), unless they are removed
89 *      prior to ordering.
90 *
91 * (2) the minimum degree ordering is followed by a postordering (depth-first
92 *      search) of the assembly tree.  Note that mass elimination (discussed
93 *      below) combined with the approximate degree update can lead to the mass
94 *      elimination of nodes with lower exact degree than the current pivot
95 *      element.  No additional fill-in is caused in the representation of the
96 *      Schur complement.  The mass-eliminated nodes merge with the current
97 *      pivot element.  They are ordered prior to the current pivot element.
98 *      Because they can have lower exact degree than the current element, the
99 *      merger of two or more of these nodes in the current pivot element can
100 *      lead to a single element that is not a "fundamental supernode".  The
101 *      diagonal block can have zeros in it.  Thus, the assembly tree used here
102 *      is not guaranteed to be the precise supernodal elemination tree (with
103 *      "funadmental" supernodes), and the postordering performed by this
104 *      routine is not guaranteed to be a precise postordering of the
105 *      elimination tree.
106 *
107 * (3) input parameters are added, to control aggressive absorption and the
108 *      detection of "dense" rows/columns of A.
109 *
110 * (4) additional statistical information is returned, such as the number of
111 *      nonzeros in L, and the flop counts for subsequent LDL' and LU
112 *      factorizations.  These are slight upper bounds, because of the mass
113 *      elimination issue discussed above.
114 *
115 * (5) additional routines are added to interface this routine to MATLAB
116 *      to provide a simple C-callable user-interface, to check inputs for
117 *      errors, compute the symmetry of the pattern of A and the number of
118 *      nonzeros in each row/column of A+A', to compute the pattern of A+A',
119 *      to perform the assembly tree postordering, and to provide debugging
120 *      ouput.  Many of these functions are also provided by the Fortran
121 *      Harwell Subroutine Library routine MC47A.
122 *
123 * (6) both int and UF_long versions are provided.  In the descriptions below
124 *      and integer is and int or UF_long depending on which version is
125 *      being used.
126
127 **********************************************************************
128 ***** CAUTION:  ARGUMENTS ARE NOT CHECKED FOR ERRORS ON INPUT.  ******
129 **********************************************************************
130 ** If you want error checking, a more versatile input format, and a **
131 ** simpler user interface, use amd_order or amd_l_order instead.    **
132 ** This routine is not meant to be user-callable.                   **
133 **********************************************************************
134
135 * ----------------------------------------------------------------------------
136 * References:
137 * ----------------------------------------------------------------------------
138 *
139 *  [1] Timothy A. Davis and Iain Duff, "An unsymmetric-pattern multifrontal
140 *      method for sparse LU factorization", SIAM J. Matrix Analysis and
141 *      Applications, vol. 18, no. 1, pp. 140-158.  Discusses UMFPACK / MA38,
142 *      which first introduced the approximate minimum degree used by this
143 *      routine.
144 *
145 *  [2] Patrick Amestoy, Timothy A. Davis, and Iain S. Duff, "An approximate
146 *      minimum degree ordering algorithm," SIAM J. Matrix Analysis and
147 *      Applications, vol. 17, no. 4, pp. 886-905, 1996.  Discusses AMDBAR and
148 *      MC47B, which are the Fortran versions of this routine.
149 *
150 *  [3] Alan George and Joseph Liu, "The evolution of the minimum degree
151 *      ordering algorithm," SIAM Review, vol. 31, no. 1, pp. 1-19, 1989.
152 *      We list below the features mentioned in that paper that this code
153 *      includes:
154 *
155 *      mass elimination:
156 *          Yes.  MA27 relied on supervariable detection for mass elimination.
157 *
158 *      indistinguishable nodes:
159 *          Yes (we call these "supervariables").  This was also in the MA27
160 *          code - although we modified the method of detecting them (the
161 *          previous hash was the true degree, which we no longer keep track
162 *          of).  A supervariable is a set of rows with identical nonzero
163 *          pattern.  All variables in a supervariable are eliminated together.
164 *          Each supervariable has as its numerical name that of one of its
165 *          variables (its principal variable).
166 *
167 *      quotient graph representation:
168 *          Yes.  We use the term "element" for the cliques formed during
169 *          elimination.  This was also in the MA27 code.  The algorithm can
170 *          operate in place, but it will work more efficiently if given some
171 *          "elbow room."
172 *
173 *      element absorption:
174 *          Yes.  This was also in the MA27 code.
175 *
176 *      external degree:
177 *          Yes.  The MA27 code was based on the true degree.
178 *
179 *      incomplete degree update and multiple elimination:
180 *          No.  This was not in MA27, either.  Our method of degree update
181 *          within MC47B is element-based, not variable-based.  It is thus
182 *          not well-suited for use with incomplete degree update or multiple
183 *          elimination.
184 *
185 * Authors, and Copyright (C) 2004 by:
186 * Timothy A. Davis, Patrick Amestoy, Iain S. Duff, John K. Reid.
187 *
188 * Acknowledgements: This work (and the UMFPACK package) was supported by the
189 * National Science Foundation (ASC-9111263, DMS-9223088, and CCR-0203270).
190 * The UMFPACK/MA38 approximate degree update algorithm, the unsymmetric analog
191 * which forms the basis of AMD, was developed while Tim Davis was supported by
192 * CERFACS (Toulouse, France) in a post-doctoral position.  This C version, and
193 * the etree postorder, were written while Tim Davis was on sabbatical at
194 * Stanford University and Lawrence Berkeley National Laboratory.
195
196 * ----------------------------------------------------------------------------
197 * INPUT ARGUMENTS (unaltered):
198 * ----------------------------------------------------------------------------
199
200 * n:  The matrix order.  Restriction:  n >= 1.
201 *
202 * iwlen:  The size of the Iw array.  On input, the matrix is stored in
203 *      Iw [0..pfree-1].  However, Iw [0..iwlen-1] should be slightly larger
204 *      than what is required to hold the matrix, at least iwlen >= pfree + n.
205 *      Otherwise, excessive compressions will take place.  The recommended
206 *      value of iwlen is 1.2 * pfree + n, which is the value used in the
207 *      user-callable interface to this routine (amd_order.c).  The algorithm
208 *      will not run at all if iwlen < pfree.  Restriction: iwlen >= pfree + n.
209 *      Note that this is slightly more restrictive than the actual minimum
210 *      (iwlen >= pfree), but AMD_2 will be very slow with no elbow room.
211 *      Thus, this routine enforces a bare minimum elbow room of size n.
212 *
213 * pfree: On input the tail end of the array, Iw [pfree..iwlen-1], is empty,
214 *      and the matrix is stored in Iw [0..pfree-1].  During execution,
215 *      additional data is placed in Iw, and pfree is modified so that
216 *      Iw [pfree..iwlen-1] is always the unused part of Iw.
217 *
218 * Control:  A double array of size AMD_CONTROL containing input parameters
219 *      that affect how the ordering is computed.  If NULL, then default
220 *      settings are used.
221 *
222 *      Control [AMD_DENSE] is used to determine whether or not a given input
223 *      row is "dense".  A row is "dense" if the number of entries in the row
224 *      exceeds Control [AMD_DENSE] times sqrt (n), except that rows with 16 or
225 *      fewer entries are never considered "dense".  To turn off the detection
226 *      of dense rows, set Control [AMD_DENSE] to a negative number, or to a
227 *      number larger than sqrt (n).  The default value of Control [AMD_DENSE]
228 *      is AMD_DEFAULT_DENSE, which is defined in amd.h as 10.
229 *
230 *      Control [AMD_AGGRESSIVE] is used to determine whether or not aggressive
231 *      absorption is to be performed.  If nonzero, then aggressive absorption
232 *      is performed (this is the default).
233
234 * ----------------------------------------------------------------------------
235 * INPUT/OUPUT ARGUMENTS:
236 * ----------------------------------------------------------------------------
237 *
238 * Pe:  An integer array of size n.  On input, Pe [i] is the index in Iw of
239 *      the start of row i.  Pe [i] is ignored if row i has no off-diagonal
240 *      entries.  Thus Pe [i] must be in the range 0 to pfree-1 for non-empty
241 *      rows.
242 *
243 *      During execution, it is used for both supervariables and elements:
244 *
245 *      Principal supervariable i:  index into Iw of the description of
246 *          supervariable i.  A supervariable represents one or more rows of
247 *          the matrix with identical nonzero pattern.  In this case,
248 *          Pe [i] >= 0.
249 *
250 *      Non-principal supervariable i:  if i has been absorbed into another
251 *          supervariable j, then Pe [i] = FLIP (j), where FLIP (j) is defined
252 *          as (-(j)-2).  Row j has the same pattern as row i.  Note that j
253 *          might later be absorbed into another supervariable j2, in which
254 *          case Pe [i] is still FLIP (j), and Pe [j] = FLIP (j2) which is
255 *          < EMPTY, where EMPTY is defined as (-1) in amd_internal.h.
256 *
257 *      Unabsorbed element e:  the index into Iw of the description of element
258 *          e, if e has not yet been absorbed by a subsequent element.  Element
259 *          e is created when the supervariable of the same name is selected as
260 *          the pivot.  In this case, Pe [i] >= 0.
261 *
262 *      Absorbed element e:  if element e is absorbed into element e2, then
263 *          Pe [e] = FLIP (e2).  This occurs when the pattern of e (which we
264 *          refer to as Le) is found to be a subset of the pattern of e2 (that
265 *          is, Le2).  In this case, Pe [i] < EMPTY.  If element e is "null"
266 *          (it has no nonzeros outside its pivot block), then Pe [e] = EMPTY,
267 *          and e is the root of an assembly subtree (or the whole tree if
268 *          there is just one such root).
269 *
270 *      Dense variable i:  if i is "dense", then Pe [i] = EMPTY.
271 *
272 *      On output, Pe holds the assembly tree/forest, which implicitly
273 *      represents a pivot order with identical fill-in as the actual order
274 *      (via a depth-first search of the tree), as follows.  If Nv [i] > 0,
275 *      then i represents a node in the assembly tree, and the parent of i is
276 *      Pe [i], or EMPTY if i is a root.  If Nv [i] = 0, then (i, Pe [i])
277 *      represents an edge in a subtree, the root of which is a node in the
278 *      assembly tree.  Note that i refers to a row/column in the original
279 *      matrix, not the permuted matrix.
280 *
281 * Info:  A double array of size AMD_INFO.  If present, (that is, not NULL),
282 *      then statistics about the ordering are returned in the Info array.
283 *      See amd.h for a description.
284
285 * ----------------------------------------------------------------------------
286 * INPUT/MODIFIED (undefined on output):
287 * ----------------------------------------------------------------------------
288 *
289 * Len:  An integer array of size n.  On input, Len [i] holds the number of
290 *      entries in row i of the matrix, excluding the diagonal.  The contents
291 *      of Len are undefined on output.
292 *
293 * Iw:  An integer array of size iwlen.  On input, Iw [0..pfree-1] holds the
294 *      description of each row i in the matrix.  The matrix must be symmetric,
295 *      and both upper and lower triangular parts must be present.  The
296 *      diagonal must not be present.  Row i is held as follows:
297 *
298 *          Len [i]:  the length of the row i data structure in the Iw array.
299 *          Iw [Pe [i] ... Pe [i] + Len [i] - 1]:
300 *              the list of column indices for nonzeros in row i (simple
301 *              supervariables), excluding the diagonal.  All supervariables
302 *              start with one row/column each (supervariable i is just row i).
303 *              If Len [i] is zero on input, then Pe [i] is ignored on input.
304 *
305 *          Note that the rows need not be in any particular order, and there
306 *          may be empty space between the rows.
307 *
308 *      During execution, the supervariable i experiences fill-in.  This is
309 *      represented by placing in i a list of the elements that cause fill-in
310 *      in supervariable i:
311 *
312 *          Len [i]:  the length of supervariable i in the Iw array.
313 *          Iw [Pe [i] ... Pe [i] + Elen [i] - 1]:
314 *              the list of elements that contain i.  This list is kept short
315 *              by removing absorbed elements.
316 *          Iw [Pe [i] + Elen [i] ... Pe [i] + Len [i] - 1]:
317 *              the list of supervariables in i.  This list is kept short by
318 *              removing nonprincipal variables, and any entry j that is also
319 *              contained in at least one of the elements (j in Le) in the list
320 *              for i (e in row i).
321 *
322 *      When supervariable i is selected as pivot, we create an element e of
323 *      the same name (e=i):
324 *
325 *          Len [e]:  the length of element e in the Iw array.
326 *          Iw [Pe [e] ... Pe [e] + Len [e] - 1]:
327 *              the list of supervariables in element e.
328 *
329 *      An element represents the fill-in that occurs when supervariable i is
330 *      selected as pivot (which represents the selection of row i and all
331 *      non-principal variables whose principal variable is i).  We use the
332 *      term Le to denote the set of all supervariables in element e.  Absorbed
333 *      supervariables and elements are pruned from these lists when
334 *      computationally convenient.
335 *
336 *  CAUTION:  THE INPUT MATRIX IS OVERWRITTEN DURING COMPUTATION.
337 *  The contents of Iw are undefined on output.
338
339 * ----------------------------------------------------------------------------
340 * OUTPUT (need not be set on input):
341 * ----------------------------------------------------------------------------
342 *
343 * Nv:  An integer array of size n.  During execution, ABS (Nv [i]) is equal to
344 *      the number of rows that are represented by the principal supervariable
345 *      i.  If i is a nonprincipal or dense variable, then Nv [i] = 0.
346 *      Initially, Nv [i] = 1 for all i.  Nv [i] < 0 signifies that i is a
347 *      principal variable in the pattern Lme of the current pivot element me.
348 *      After element me is constructed, Nv [i] is set back to a positive
349 *      value.
350 *
351 *      On output, Nv [i] holds the number of pivots represented by super
352 *      row/column i of the original matrix, or Nv [i] = 0 for non-principal
353 *      rows/columns.  Note that i refers to a row/column in the original
354 *      matrix, not the permuted matrix.
355 *
356 * Elen:  An integer array of size n.  See the description of Iw above.  At the
357 *      start of execution, Elen [i] is set to zero for all rows i.  During
358 *      execution, Elen [i] is the number of elements in the list for
359 *      supervariable i.  When e becomes an element, Elen [e] = FLIP (esize) is
360 *      set, where esize is the size of the element (the number of pivots, plus
361 *      the number of nonpivotal entries).  Thus Elen [e] < EMPTY.
362 *      Elen (i) = EMPTY set when variable i becomes nonprincipal.
363 *
364 *      For variables, Elen (i) >= EMPTY holds until just before the
365 *      postordering and permutation vectors are computed.  For elements,
366 *      Elen [e] < EMPTY holds.
367 *
368 *      On output, Elen [i] is the degree of the row/column in the Cholesky
369 *      factorization of the permuted matrix, corresponding to the original row
370 *      i, if i is a super row/column.  It is equal to EMPTY if i is
371 *      non-principal.  Note that i refers to a row/column in the original
372 *      matrix, not the permuted matrix.
373 *
374 *      Note that the contents of Elen on output differ from the Fortran
375 *      version (Elen holds the inverse permutation in the Fortran version,
376 *      which is instead returned in the Next array in this C version,
377 *      described below).
378 *
379 * Last: In a degree list, Last [i] is the supervariable preceding i, or EMPTY
380 *      if i is the head of the list.  In a hash bucket, Last [i] is the hash
381 *      key for i.
382 *
383 *      Last [Head [hash]] is also used as the head of a hash bucket if
384 *      Head [hash] contains a degree list (see the description of Head,
385 *      below).
386 *
387 *      On output, Last [0..n-1] holds the permutation.  That is, if
388 *      i = Last [k], then row i is the kth pivot row (where k ranges from 0 to
389 *      n-1).  Row Last [k] of A is the kth row in the permuted matrix, PAP'.
390 *
391 * Next: Next [i] is the supervariable following i in a link list, or EMPTY if
392 *      i is the last in the list.  Used for two kinds of lists:  degree lists
393 *      and hash buckets (a supervariable can be in only one kind of list at a
394 *      time).
395 *
396 *      On output Next [0..n-1] holds the inverse permutation.  That is, if
397 *      k = Next [i], then row i is the kth pivot row. Row i of A appears as
398 *      the (Next[i])-th row in the permuted matrix, PAP'.
399 *
400 *      Note that the contents of Next on output differ from the Fortran
401 *      version (Next is undefined on output in the Fortran version).
402
403 * ----------------------------------------------------------------------------
404 * LOCAL WORKSPACE (not input or output - used only during execution):
405 * ----------------------------------------------------------------------------
406 *
407 * Degree:  An integer array of size n.  If i is a supervariable, then
408 *      Degree [i] holds the current approximation of the external degree of
409 *      row i (an upper bound).  The external degree is the number of nonzeros
410 *      in row i, minus ABS (Nv [i]), the diagonal part.  The bound is equal to
411 *      the exact external degree if Elen [i] is less than or equal to two.
412 *
413 *      We also use the term "external degree" for elements e to refer to
414 *      |Le \ Lme|.  If e is an element, then Degree [e] is |Le|, which is the
415 *      degree of the off-diagonal part of the element e (not including the
416 *      diagonal part).
417 *
418 * Head:   An integer array of size n.  Head is used for degree lists.
419 *      Head [deg] is the first supervariable in a degree list.  All
420 *      supervariables i in a degree list Head [deg] have the same approximate
421 *      degree, namely, deg = Degree [i].  If the list Head [deg] is empty then
422 *      Head [deg] = EMPTY.
423 *
424 *      During supervariable detection Head [hash] also serves as a pointer to
425 *      a hash bucket.  If Head [hash] >= 0, there is a degree list of degree
426 *      hash.  The hash bucket head pointer is Last [Head [hash]].  If
427 *      Head [hash] = EMPTY, then the degree list and hash bucket are both
428 *      empty.  If Head [hash] < EMPTY, then the degree list is empty, and
429 *      FLIP (Head [hash]) is the head of the hash bucket.  After supervariable
430 *      detection is complete, all hash buckets are empty, and the
431 *      (Last [Head [hash]] = EMPTY) condition is restored for the non-empty
432 *      degree lists.
433 *
434 * W:  An integer array of size n.  The flag array W determines the status of
435 *      elements and variables, and the external degree of elements.
436 *
437 *      for elements:
438 *          if W [e] = 0, then the element e is absorbed.
439 *          if W [e] >= wflg, then W [e] - wflg is the size of the set
440 *              |Le \ Lme|, in terms of nonzeros (the sum of ABS (Nv [i]) for
441 *              each principal variable i that is both in the pattern of
442 *              element e and NOT in the pattern of the current pivot element,
443 *              me).
444 *          if wflg > W [e] > 0, then e is not absorbed and has not yet been
445 *              seen in the scan of the element lists in the computation of
446 *              |Le\Lme| in Scan 1 below.
447 *
448 *      for variables:
449 *          during supervariable detection, if W [j] != wflg then j is
450 *          not in the pattern of variable i.
451 *
452 *      The W array is initialized by setting W [i] = 1 for all i, and by
453 *      setting wflg = 2.  It is reinitialized if wflg becomes too large (to
454 *      ensure that wflg+n does not cause integer overflow).
455
456 * ----------------------------------------------------------------------------
457 * LOCAL INTEGERS:
458 * ----------------------------------------------------------------------------
459 */
460
461    Int deg, degme, dext, lemax, e, elenme, eln, i, ilast, inext, j,
462        jlast, jnext, k, knt1, knt2, knt3, lenj, ln, me, mindeg, nel, nleft,
463        nvi, nvj, nvpiv, slenme, wbig, we, wflg, wnvi, ok, ndense, ncmpa,
464        dense, aggressive ;
465
466    unsigned Int hash ;     /* unsigned, so that hash % n is well defined.*/
467
468/*
469 * deg:         the degree of a variable or element
470 * degme:       size, |Lme|, of the current element, me (= Degree [me])
471 * dext:        external degree, |Le \ Lme|, of some element e
472 * lemax:       largest |Le| seen so far (called dmax in Fortran version)
473 * e:           an element
474 * elenme:      the length, Elen [me], of element list of pivotal variable
475 * eln:         the length, Elen [...], of an element list
476 * hash:        the computed value of the hash function
477 * i:           a supervariable
478 * ilast:       the entry in a link list preceding i
479 * inext:       the entry in a link list following i
480 * j:           a supervariable
481 * jlast:       the entry in a link list preceding j
482 * jnext:       the entry in a link list, or path, following j
483 * k:           the pivot order of an element or variable
484 * knt1:        loop counter used during element construction
485 * knt2:        loop counter used during element construction
486 * knt3:        loop counter used during compression
487 * lenj:        Len [j]
488 * ln:          length of a supervariable list
489 * me:          current supervariable being eliminated, and the current
490 *                  element created by eliminating that supervariable
491 * mindeg:      current minimum degree
492 * nel:         number of pivots selected so far
493 * nleft:       n - nel, the number of nonpivotal rows/columns remaining
494 * nvi:         the number of variables in a supervariable i (= Nv [i])
495 * nvj:         the number of variables in a supervariable j (= Nv [j])
496 * nvpiv:       number of pivots in current element
497 * slenme:      number of variables in variable list of pivotal variable
498 * wbig:        = INT_MAX - n for the int version, UF_long_max - n for the
499 *                  UF_long version.  wflg is not allowed to be >= wbig.
500 * we:          W [e]
501 * wflg:        used for flagging the W array.  See description of Iw.
502 * wnvi:        wflg - Nv [i]
503 * x:           either a supervariable or an element
504 *
505 * ok:          true if supervariable j can be absorbed into i
506 * ndense:      number of "dense" rows/columns
507 * dense:       rows/columns with initial degree > dense are considered "dense"
508 * aggressive:  true if aggressive absorption is being performed
509 * ncmpa:       number of garbage collections
510
511 * ----------------------------------------------------------------------------
512 * LOCAL DOUBLES, used for statistical output only (except for alpha):
513 * ----------------------------------------------------------------------------
514 */
515
516    double f, r, ndiv, s, nms_lu, nms_ldl, dmax, alpha, lnz, lnzme ;
517
518/*
519 * f:           nvpiv
520 * r:           degme + nvpiv
521 * ndiv:        number of divisions for LU or LDL' factorizations
522 * s:           number of multiply-subtract pairs for LU factorization, for the
523 *                  current element me
524 * nms_lu       number of multiply-subtract pairs for LU factorization
525 * nms_ldl      number of multiply-subtract pairs for LDL' factorization
526 * dmax:        the largest number of entries in any column of L, including the
527 *                  diagonal
528 * alpha:       "dense" degree ratio
529 * lnz:         the number of nonzeros in L (excluding the diagonal)
530 * lnzme:       the number of nonzeros in L (excl. the diagonal) for the
531 *                  current element me
532
533 * ----------------------------------------------------------------------------
534 * LOCAL "POINTERS" (indices into the Iw array)
535 * ----------------------------------------------------------------------------
536*/
537
538    Int p, p1, p2, p3, p4, pdst, pend, pj, pme, pme1, pme2, pn, psrc ;
539
540/*
541 * Any parameter (Pe [...] or pfree) or local variable starting with "p" (for
542 * Pointer) is an index into Iw, and all indices into Iw use variables starting
543 * with "p."  The only exception to this rule is the iwlen input argument.
544 *
545 * p:           pointer into lots of things
546 * p1:          Pe [i] for some variable i (start of element list)
547 * p2:          Pe [i] + Elen [i] -  1 for some variable i
548 * p3:          index of first supervariable in clean list
549 * p4:         
550 * pdst:        destination pointer, for compression
551 * pend:        end of memory to compress
552 * pj:          pointer into an element or variable
553 * pme:         pointer into the current element (pme1...pme2)
554 * pme1:        the current element, me, is stored in Iw [pme1...pme2]
555 * pme2:        the end of the current element
556 * pn:          pointer into a "clean" variable, also used to compress
557 * psrc:        source pointer, for compression
558*/
559
560/* ========================================================================= */
561/*  INITIALIZATIONS */
562/* ========================================================================= */
563
564    /* Note that this restriction on iwlen is slightly more restrictive than
565     * what is actually required in AMD_2.  AMD_2 can operate with no elbow
566     * room at all, but it will be slow.  For better performance, at least
567     * size-n elbow room is enforced. */
568    ASSERT (iwlen >= pfree + n) ;
569    ASSERT (n > 0) ;
570
571    /* initialize output statistics */
572    lnz = 0 ;
573    ndiv = 0 ;
574    nms_lu = 0 ;
575    nms_ldl = 0 ;
576    dmax = 1 ;
577    me = EMPTY ;
578
579    mindeg = 0 ;
580    ncmpa = 0 ;
581    nel = 0 ;
582    lemax = 0 ;
583
584    /* get control parameters */
585    if (Control != (double *) NULL)
586    {
587        alpha = Control [AMD_DENSE] ;
588        aggressive = (Control [AMD_AGGRESSIVE] != 0) ;
589    }
590    else
591    {
592        alpha = AMD_DEFAULT_DENSE ;
593        aggressive = AMD_DEFAULT_AGGRESSIVE ;
594    }
595    /* Note: if alpha is NaN, this is undefined: */
596    if (alpha < 0)
597    {
598        /* only remove completely dense rows/columns */
599        dense = n-2 ;
600    }
601    else
602    {
603        dense = alpha * sqrt ((double) n) ;
604    }
605    dense = MAX (16, dense) ;
606    dense = MIN (n,  dense) ;
607    AMD_DEBUG1 (("\n\nAMD (debug), alpha %g, aggr. "ID"\n",
608        alpha, aggressive)) ;
609
610    for (i = 0 ; i < n ; i++)
611    {
612        Last [i] = EMPTY ;
613        Head [i] = EMPTY ;
614        Next [i] = EMPTY ;
615        /* if separate Hhead array is used for hash buckets: *
616        Hhead [i] = EMPTY ;
617        */
618        Nv [i] = 1 ;
619        W [i] = 1 ;
620        Elen [i] = 0 ;
621        Degree [i] = Len [i] ;
622    }
623
624#ifndef NDEBUG
625    AMD_DEBUG1 (("\n======Nel "ID" initial\n", nel)) ;
626    AMD_dump (n, Pe, Iw, Len, iwlen, pfree, Nv, Next, Last,
627                Head, Elen, Degree, W, -1) ;
628#endif
629
630    /* initialize wflg */
631    wbig = Int_MAX - n ;
632    wflg = clear_flag (0, wbig, W, n) ;
633
634    /* --------------------------------------------------------------------- */
635    /* initialize degree lists and eliminate dense and empty rows */
636    /* --------------------------------------------------------------------- */
637
638    ndense = 0 ;
639
640    for (i = 0 ; i < n ; i++)
641    {
642        deg = Degree [i] ;
643        ASSERT (deg >= 0 && deg < n) ;
644        if (deg == 0)
645        {
646
647            /* -------------------------------------------------------------
648             * we have a variable that can be eliminated at once because
649             * there is no off-diagonal non-zero in its row.  Note that
650             * Nv [i] = 1 for an empty variable i.  It is treated just
651             * the same as an eliminated element i.
652             * ------------------------------------------------------------- */
653
654            Elen [i] = FLIP (1) ;
655            nel++ ;
656            Pe [i] = EMPTY ;
657            W [i] = 0 ;
658
659        }
660        else if (deg > dense)
661        {
662
663            /* -------------------------------------------------------------
664             * Dense variables are not treated as elements, but as unordered,
665             * non-principal variables that have no parent.  They do not take
666             * part in the postorder, since Nv [i] = 0.  Note that the Fortran
667             * version does not have this option.
668             * ------------------------------------------------------------- */
669
670            AMD_DEBUG1 (("Dense node "ID" degree "ID"\n", i, deg)) ;
671            ndense++ ;
672            Nv [i] = 0 ;                /* do not postorder this node */
673            Elen [i] = EMPTY ;
674            nel++ ;
675            Pe [i] = EMPTY ;
676
677        }
678        else
679        {
680
681            /* -------------------------------------------------------------
682             * place i in the degree list corresponding to its degree
683             * ------------------------------------------------------------- */
684
685            inext = Head [deg] ;
686            ASSERT (inext >= EMPTY && inext < n) ;
687            if (inext != EMPTY) Last [inext] = i ;
688            Next [i] = inext ;
689            Head [deg] = i ;
690
691        }
692    }
693
694/* ========================================================================= */
695/* WHILE (selecting pivots) DO */
696/* ========================================================================= */
697
698    while (nel < n)
699    {
700
701#ifndef NDEBUG
702        AMD_DEBUG1 (("\n======Nel "ID"\n", nel)) ;
703        if (AMD_debug >= 2)
704        {
705            AMD_dump (n, Pe, Iw, Len, iwlen, pfree, Nv, Next,
706                    Last, Head, Elen, Degree, W, nel) ;
707        }
708#endif
709
710/* ========================================================================= */
711/* GET PIVOT OF MINIMUM DEGREE */
712/* ========================================================================= */
713
714        /* ----------------------------------------------------------------- */
715        /* find next supervariable for elimination */
716        /* ----------------------------------------------------------------- */
717
718        ASSERT (mindeg >= 0 && mindeg < n) ;
719        for (deg = mindeg ; deg < n ; deg++)
720        {
721            me = Head [deg] ;
722            if (me != EMPTY) break ;
723        }
724        mindeg = deg ;
725        ASSERT (me >= 0 && me < n) ;
726        AMD_DEBUG1 (("=================me: "ID"\n", me)) ;
727
728        /* ----------------------------------------------------------------- */
729        /* remove chosen variable from link list */
730        /* ----------------------------------------------------------------- */
731
732        inext = Next [me] ;
733        ASSERT (inext >= EMPTY && inext < n) ;
734        if (inext != EMPTY) Last [inext] = EMPTY ;
735        Head [deg] = inext ;
736
737        /* ----------------------------------------------------------------- */
738        /* me represents the elimination of pivots nel to nel+Nv[me]-1. */
739        /* place me itself as the first in this set. */
740        /* ----------------------------------------------------------------- */
741
742        elenme = Elen [me] ;
743        nvpiv = Nv [me] ;
744        ASSERT (nvpiv > 0) ;
745        nel += nvpiv ;
746
747/* ========================================================================= */
748/* CONSTRUCT NEW ELEMENT */
749/* ========================================================================= */
750
751        /* -----------------------------------------------------------------
752         * At this point, me is the pivotal supervariable.  It will be
753         * converted into the current element.  Scan list of the pivotal
754         * supervariable, me, setting tree pointers and constructing new list
755         * of supervariables for the new element, me.  p is a pointer to the
756         * current position in the old list.
757         * ----------------------------------------------------------------- */
758
759        /* flag the variable "me" as being in Lme by negating Nv [me] */
760        Nv [me] = -nvpiv ;
761        degme = 0 ;
762        ASSERT (Pe [me] >= 0 && Pe [me] < iwlen) ;
763
764        if (elenme == 0)
765        {
766
767            /* ------------------------------------------------------------- */
768            /* construct the new element in place */
769            /* ------------------------------------------------------------- */
770
771            pme1 = Pe [me] ;
772            pme2 = pme1 - 1 ;
773
774            for (p = pme1 ; p <= pme1 + Len [me] - 1 ; p++)
775            {
776                i = Iw [p] ;
777                ASSERT (i >= 0 && i < n && Nv [i] >= 0) ;
778                nvi = Nv [i] ;
779                if (nvi > 0)
780                {
781
782                    /* ----------------------------------------------------- */
783                    /* i is a principal variable not yet placed in Lme. */
784                    /* store i in new list */
785                    /* ----------------------------------------------------- */
786
787                    /* flag i as being in Lme by negating Nv [i] */
788                    degme += nvi ;
789                    Nv [i] = -nvi ;
790                    Iw [++pme2] = i ;
791
792                    /* ----------------------------------------------------- */
793                    /* remove variable i from degree list. */
794                    /* ----------------------------------------------------- */
795
796                    ilast = Last [i] ;
797                    inext = Next [i] ;
798                    ASSERT (ilast >= EMPTY && ilast < n) ;
799                    ASSERT (inext >= EMPTY && inext < n) ;
800                    if (inext != EMPTY) Last [inext] = ilast ;
801                    if (ilast != EMPTY)
802                    {
803                        Next [ilast] = inext ;
804                    }
805                    else
806                    {
807                        /* i is at the head of the degree list */
808                        ASSERT (Degree [i] >= 0 && Degree [i] < n) ;
809                        Head [Degree [i]] = inext ;
810                    }
811                }
812            }
813        }
814        else
815        {
816
817            /* ------------------------------------------------------------- */
818            /* construct the new element in empty space, Iw [pfree ...] */
819            /* ------------------------------------------------------------- */
820
821            p = Pe [me] ;
822            pme1 = pfree ;
823            slenme = Len [me] - elenme ;
824
825            for (knt1 = 1 ; knt1 <= elenme + 1 ; knt1++)
826            {
827
828                if (knt1 > elenme)
829                {
830                    /* search the supervariables in me. */
831                    e = me ;
832                    pj = p ;
833                    ln = slenme ;
834                    AMD_DEBUG2 (("Search sv: "ID" "ID" "ID"\n", me,pj,ln)) ;
835                }
836                else
837                {
838                    /* search the elements in me. */
839                    e = Iw [p++] ;
840                    ASSERT (e >= 0 && e < n) ;
841                    pj = Pe [e] ;
842                    ln = Len [e] ;
843                    AMD_DEBUG2 (("Search element e "ID" in me "ID"\n", e,me)) ;
844                    ASSERT (Elen [e] < EMPTY && W [e] > 0 && pj >= 0) ;
845                }
846                ASSERT (ln >= 0 && (ln == 0 || (pj >= 0 && pj < iwlen))) ;
847
848                /* ---------------------------------------------------------
849                 * search for different supervariables and add them to the
850                 * new list, compressing when necessary. this loop is
851                 * executed once for each element in the list and once for
852                 * all the supervariables in the list.
853                 * --------------------------------------------------------- */
854
855                for (knt2 = 1 ; knt2 <= ln ; knt2++)
856                {
857                    i = Iw [pj++] ;
858                    ASSERT (i >= 0 && i < n && (i == me || Elen [i] >= EMPTY));
859                    nvi = Nv [i] ;
860                    AMD_DEBUG2 ((": "ID" "ID" "ID" "ID"\n",
861                                i, Elen [i], Nv [i], wflg)) ;
862
863                    if (nvi > 0)
864                    {
865
866                        /* ------------------------------------------------- */
867                        /* compress Iw, if necessary */
868                        /* ------------------------------------------------- */
869
870                        if (pfree >= iwlen)
871                        {
872
873                            AMD_DEBUG1 (("GARBAGE COLLECTION\n")) ;
874
875                            /* prepare for compressing Iw by adjusting pointers
876                             * and lengths so that the lists being searched in
877                             * the inner and outer loops contain only the
878                             * remaining entries. */
879
880                            Pe [me] = p ;
881                            Len [me] -= knt1 ;
882                            /* check if nothing left of supervariable me */
883                            if (Len [me] == 0) Pe [me] = EMPTY ;
884                            Pe [e] = pj ;
885                            Len [e] = ln - knt2 ;
886                            /* nothing left of element e */
887                            if (Len [e] == 0) Pe [e] = EMPTY ;
888
889                            ncmpa++ ;   /* one more garbage collection */
890
891                            /* store first entry of each object in Pe */
892                            /* FLIP the first entry in each object */
893                            for (j = 0 ; j < n ; j++)
894                            {
895                                pn = Pe [j] ;
896                                if (pn >= 0)
897                                {
898                                    ASSERT (pn >= 0 && pn < iwlen) ;
899                                    Pe [j] = Iw [pn] ;
900                                    Iw [pn] = FLIP (j) ;
901                                }
902                            }
903
904                            /* psrc/pdst point to source/destination */
905                            psrc = 0 ;
906                            pdst = 0 ;
907                            pend = pme1 - 1 ;
908
909                            while (psrc <= pend)
910                            {
911                                /* search for next FLIP'd entry */
912                                j = FLIP (Iw [psrc++]) ;
913                                if (j >= 0)
914                                {
915                                    AMD_DEBUG2 (("Got object j: "ID"\n", j)) ;
916                                    Iw [pdst] = Pe [j] ;
917                                    Pe [j] = pdst++ ;
918                                    lenj = Len [j] ;
919                                    /* copy from source to destination */
920                                    for (knt3 = 0 ; knt3 <= lenj - 2 ; knt3++)
921                                    {
922                                        Iw [pdst++] = Iw [psrc++] ;
923                                    }
924                                }
925                            }
926
927                            /* move the new partially-constructed element */
928                            p1 = pdst ;
929                            for (psrc = pme1 ; psrc <= pfree-1 ; psrc++)
930                            {
931                                Iw [pdst++] = Iw [psrc] ;
932                            }
933                            pme1 = p1 ;
934                            pfree = pdst ;
935                            pj = Pe [e] ;
936                            p = Pe [me] ;
937
938                        }
939
940                        /* ------------------------------------------------- */
941                        /* i is a principal variable not yet placed in Lme */
942                        /* store i in new list */
943                        /* ------------------------------------------------- */
944
945                        /* flag i as being in Lme by negating Nv [i] */
946                        degme += nvi ;
947                        Nv [i] = -nvi ;
948                        Iw [pfree++] = i ;
949                        AMD_DEBUG2 (("     s: "ID"     nv "ID"\n", i, Nv [i]));
950
951                        /* ------------------------------------------------- */
952                        /* remove variable i from degree link list */
953                        /* ------------------------------------------------- */
954
955                        ilast = Last [i] ;
956                        inext = Next [i] ;
957                        ASSERT (ilast >= EMPTY && ilast < n) ;
958                        ASSERT (inext >= EMPTY && inext < n) ;
959                        if (inext != EMPTY) Last [inext] = ilast ;
960                        if (ilast != EMPTY)
961                        {
962                            Next [ilast] = inext ;
963                        }
964                        else
965                        {
966                            /* i is at the head of the degree list */
967                            ASSERT (Degree [i] >= 0 && Degree [i] < n) ;
968                            Head [Degree [i]] = inext ;
969                        }
970                    }
971                }
972
973                if (e != me)
974                {
975                    /* set tree pointer and flag to indicate element e is
976                     * absorbed into new element me (the parent of e is me) */
977                    AMD_DEBUG1 ((" Element "ID" => "ID"\n", e, me)) ;
978                    Pe [e] = FLIP (me) ;
979                    W [e] = 0 ;
980                }
981            }
982
983            pme2 = pfree - 1 ;
984        }
985
986        /* ----------------------------------------------------------------- */
987        /* me has now been converted into an element in Iw [pme1..pme2] */
988        /* ----------------------------------------------------------------- */
989
990        /* degme holds the external degree of new element */
991        Degree [me] = degme ;
992        Pe [me] = pme1 ;
993        Len [me] = pme2 - pme1 + 1 ;
994        ASSERT (Pe [me] >= 0 && Pe [me] < iwlen) ;
995
996        Elen [me] = FLIP (nvpiv + degme) ;
997        /* FLIP (Elen (me)) is now the degree of pivot (including
998         * diagonal part). */
999
1000#ifndef NDEBUG
1001        AMD_DEBUG2 (("New element structure: length= "ID"\n", pme2-pme1+1)) ;
1002        for (pme = pme1 ; pme <= pme2 ; pme++) AMD_DEBUG3 ((" "ID"", Iw[pme]));
1003        AMD_DEBUG3 (("\n")) ;
1004#endif
1005
1006        /* ----------------------------------------------------------------- */
1007        /* make sure that wflg is not too large. */
1008        /* ----------------------------------------------------------------- */
1009
1010        /* With the current value of wflg, wflg+n must not cause integer
1011         * overflow */
1012
1013        wflg = clear_flag (wflg, wbig, W, n) ;
1014
1015/* ========================================================================= */
1016/* COMPUTE (W [e] - wflg) = |Le\Lme| FOR ALL ELEMENTS */
1017/* ========================================================================= */
1018
1019        /* -----------------------------------------------------------------
1020         * Scan 1:  compute the external degrees of previous elements with
1021         * respect to the current element.  That is:
1022         *       (W [e] - wflg) = |Le \ Lme|
1023         * for each element e that appears in any supervariable in Lme.  The
1024         * notation Le refers to the pattern (list of supervariables) of a
1025         * previous element e, where e is not yet absorbed, stored in
1026         * Iw [Pe [e] + 1 ... Pe [e] + Len [e]].  The notation Lme
1027         * refers to the pattern of the current element (stored in
1028         * Iw [pme1..pme2]).   If aggressive absorption is enabled, and
1029         * (W [e] - wflg) becomes zero, then the element e will be absorbed
1030         * in Scan 2.
1031         * ----------------------------------------------------------------- */
1032
1033        AMD_DEBUG2 (("me: ")) ;
1034        for (pme = pme1 ; pme <= pme2 ; pme++)
1035        {
1036            i = Iw [pme] ;
1037            ASSERT (i >= 0 && i < n) ;
1038            eln = Elen [i] ;
1039            AMD_DEBUG3 ((""ID" Elen "ID": \n", i, eln)) ;
1040            if (eln > 0)
1041            {
1042                /* note that Nv [i] has been negated to denote i in Lme: */
1043                nvi = -Nv [i] ;
1044                ASSERT (nvi > 0 && Pe [i] >= 0 && Pe [i] < iwlen) ;
1045                wnvi = wflg - nvi ;
1046                for (p = Pe [i] ; p <= Pe [i] + eln - 1 ; p++)
1047                {
1048                    e = Iw [p] ;
1049                    ASSERT (e >= 0 && e < n) ;
1050                    we = W [e] ;
1051                    AMD_DEBUG4 (("    e "ID" we "ID" ", e, we)) ;
1052                    if (we >= wflg)
1053                    {
1054                        /* unabsorbed element e has been seen in this loop */
1055                        AMD_DEBUG4 (("    unabsorbed, first time seen")) ;
1056                        we -= nvi ;
1057                    }
1058                    else if (we != 0)
1059                    {
1060                        /* e is an unabsorbed element */
1061                        /* this is the first we have seen e in all of Scan 1 */
1062                        AMD_DEBUG4 (("    unabsorbed")) ;
1063                        we = Degree [e] + wnvi ;
1064                    }
1065                    AMD_DEBUG4 (("\n")) ;
1066                    W [e] = we ;
1067                }
1068            }
1069        }
1070        AMD_DEBUG2 (("\n")) ;
1071
1072/* ========================================================================= */
1073/* DEGREE UPDATE AND ELEMENT ABSORPTION */
1074/* ========================================================================= */
1075
1076        /* -----------------------------------------------------------------
1077         * Scan 2:  for each i in Lme, sum up the degree of Lme (which is
1078         * degme), plus the sum of the external degrees of each Le for the
1079         * elements e appearing within i, plus the supervariables in i.
1080         * Place i in hash list.
1081         * ----------------------------------------------------------------- */
1082
1083        for (pme = pme1 ; pme <= pme2 ; pme++)
1084        {
1085            i = Iw [pme] ;
1086            ASSERT (i >= 0 && i < n && Nv [i] < 0 && Elen [i] >= 0) ;
1087            AMD_DEBUG2 (("Updating: i "ID" "ID" "ID"\n", i, Elen[i], Len [i]));
1088            p1 = Pe [i] ;
1089            p2 = p1 + Elen [i] - 1 ;
1090            pn = p1 ;
1091            hash = 0 ;
1092            deg = 0 ;
1093            ASSERT (p1 >= 0 && p1 < iwlen && p2 >= -1 && p2 < iwlen) ;
1094
1095            /* ------------------------------------------------------------- */
1096            /* scan the element list associated with supervariable i */
1097            /* ------------------------------------------------------------- */
1098
1099            /* UMFPACK/MA38-style approximate degree: */
1100            if (aggressive)
1101            {
1102                for (p = p1 ; p <= p2 ; p++)
1103                {
1104                    e = Iw [p] ;
1105                    ASSERT (e >= 0 && e < n) ;
1106                    we = W [e] ;
1107                    if (we != 0)
1108                    {
1109                        /* e is an unabsorbed element */
1110                        /* dext = | Le \ Lme | */
1111                        dext = we - wflg ;
1112                        if (dext > 0)
1113                        {
1114                            deg += dext ;
1115                            Iw [pn++] = e ;
1116                            hash += e ;
1117                            AMD_DEBUG4 ((" e: "ID" hash = "ID"\n",e,hash)) ;
1118                        }
1119                        else
1120                        {
1121                            /* external degree of e is zero, absorb e into me*/
1122                            AMD_DEBUG1 ((" Element "ID" =>"ID" (aggressive)\n",
1123                                e, me)) ;
1124                            ASSERT (dext == 0) ;
1125                            Pe [e] = FLIP (me) ;
1126                            W [e] = 0 ;
1127                        }
1128                    }
1129                }
1130            }
1131            else
1132            {
1133                for (p = p1 ; p <= p2 ; p++)
1134                {
1135                    e = Iw [p] ;
1136                    ASSERT (e >= 0 && e < n) ;
1137                    we = W [e] ;
1138                    if (we != 0)
1139                    {
1140                        /* e is an unabsorbed element */
1141                        dext = we - wflg ;
1142                        ASSERT (dext >= 0) ;
1143                        deg += dext ;
1144                        Iw [pn++] = e ;
1145                        hash += e ;
1146                        AMD_DEBUG4 (("  e: "ID" hash = "ID"\n",e,hash)) ;
1147                    }
1148                }
1149            }
1150
1151            /* count the number of elements in i (including me): */
1152            Elen [i] = pn - p1 + 1 ;
1153
1154            /* ------------------------------------------------------------- */
1155            /* scan the supervariables in the list associated with i */
1156            /* ------------------------------------------------------------- */
1157
1158            /* The bulk of the AMD run time is typically spent in this loop,
1159             * particularly if the matrix has many dense rows that are not
1160             * removed prior to ordering. */
1161            p3 = pn ;
1162            p4 = p1 + Len [i] ;
1163            for (p = p2 + 1 ; p < p4 ; p++)
1164            {
1165                j = Iw [p] ;
1166                ASSERT (j >= 0 && j < n) ;
1167                nvj = Nv [j] ;
1168                if (nvj > 0)
1169                {
1170                    /* j is unabsorbed, and not in Lme. */
1171                    /* add to degree and add to new list */
1172                    deg += nvj ;
1173                    Iw [pn++] = j ;
1174                    hash += j ;
1175                    AMD_DEBUG4 (("  s: "ID" hash "ID" Nv[j]= "ID"\n",
1176                                j, hash, nvj)) ;
1177                }
1178            }
1179
1180            /* ------------------------------------------------------------- */
1181            /* update the degree and check for mass elimination */
1182            /* ------------------------------------------------------------- */
1183
1184            /* with aggressive absorption, deg==0 is identical to the
1185             * Elen [i] == 1 && p3 == pn test, below. */
1186            ASSERT (IMPLIES (aggressive, (deg==0) == (Elen[i]==1 && p3==pn))) ;
1187
1188            if (Elen [i] == 1 && p3 == pn)
1189            {
1190
1191                /* --------------------------------------------------------- */
1192                /* mass elimination */
1193                /* --------------------------------------------------------- */
1194
1195                /* There is nothing left of this node except for an edge to
1196                 * the current pivot element.  Elen [i] is 1, and there are
1197                 * no variables adjacent to node i.  Absorb i into the
1198                 * current pivot element, me.  Note that if there are two or
1199                 * more mass eliminations, fillin due to mass elimination is
1200                 * possible within the nvpiv-by-nvpiv pivot block.  It is this
1201                 * step that causes AMD's analysis to be an upper bound.
1202                 *
1203                 * The reason is that the selected pivot has a lower
1204                 * approximate degree than the true degree of the two mass
1205                 * eliminated nodes.  There is no edge between the two mass
1206                 * eliminated nodes.  They are merged with the current pivot
1207                 * anyway.
1208                 *
1209                 * No fillin occurs in the Schur complement, in any case,
1210                 * and this effect does not decrease the quality of the
1211                 * ordering itself, just the quality of the nonzero and
1212                 * flop count analysis.  It also means that the post-ordering
1213                 * is not an exact elimination tree post-ordering. */
1214
1215                AMD_DEBUG1 (("  MASS i "ID" => parent e "ID"\n", i, me)) ;
1216                Pe [i] = FLIP (me) ;
1217                nvi = -Nv [i] ;
1218                degme -= nvi ;
1219                nvpiv += nvi ;
1220                nel += nvi ;
1221                Nv [i] = 0 ;
1222                Elen [i] = EMPTY ;
1223
1224            }
1225            else
1226            {
1227
1228                /* --------------------------------------------------------- */
1229                /* update the upper-bound degree of i */
1230                /* --------------------------------------------------------- */
1231
1232                /* the following degree does not yet include the size
1233                 * of the current element, which is added later: */
1234
1235                Degree [i] = MIN (Degree [i], deg) ;
1236
1237                /* --------------------------------------------------------- */
1238                /* add me to the list for i */
1239                /* --------------------------------------------------------- */
1240
1241                /* move first supervariable to end of list */
1242                Iw [pn] = Iw [p3] ;
1243                /* move first element to end of element part of list */
1244                Iw [p3] = Iw [p1] ;
1245                /* add new element, me, to front of list. */
1246                Iw [p1] = me ;
1247                /* store the new length of the list in Len [i] */
1248                Len [i] = pn - p1 + 1 ;
1249
1250                /* --------------------------------------------------------- */
1251                /* place in hash bucket.  Save hash key of i in Last [i]. */
1252                /* --------------------------------------------------------- */
1253
1254                /* NOTE: this can fail if hash is negative, because the ANSI C
1255                 * standard does not define a % b when a and/or b are negative.
1256                 * That's why hash is defined as an unsigned Int, to avoid this
1257                 * problem. */
1258                hash = hash % n ;
1259                ASSERT (((Int) hash) >= 0 && ((Int) hash) < n) ;
1260
1261                /* if the Hhead array is not used: */
1262                j = Head [hash] ;
1263                if (j <= EMPTY)
1264                {
1265                    /* degree list is empty, hash head is FLIP (j) */
1266                    Next [i] = FLIP (j) ;
1267                    Head [hash] = FLIP (i) ;
1268                }
1269                else
1270                {
1271                    /* degree list is not empty, use Last [Head [hash]] as
1272                     * hash head. */
1273                    Next [i] = Last [j] ;
1274                    Last [j] = i ;
1275                }
1276
1277                /* if a separate Hhead array is used: *
1278                Next [i] = Hhead [hash] ;
1279                Hhead [hash] = i ;
1280                */
1281
1282                Last [i] = hash ;
1283            }
1284        }
1285
1286        Degree [me] = degme ;
1287
1288        /* ----------------------------------------------------------------- */
1289        /* Clear the counter array, W [...], by incrementing wflg. */
1290        /* ----------------------------------------------------------------- */
1291
1292        /* make sure that wflg+n does not cause integer overflow */
1293        lemax =  MAX (lemax, degme) ;
1294        wflg += lemax ;
1295        wflg = clear_flag (wflg, wbig, W, n) ;
1296        /*  at this point, W [0..n-1] < wflg holds */
1297
1298/* ========================================================================= */
1299/* SUPERVARIABLE DETECTION */
1300/* ========================================================================= */
1301
1302        AMD_DEBUG1 (("Detecting supervariables:\n")) ;
1303        for (pme = pme1 ; pme <= pme2 ; pme++)
1304        {
1305            i = Iw [pme] ;
1306            ASSERT (i >= 0 && i < n) ;
1307            AMD_DEBUG2 (("Consider i "ID" nv "ID"\n", i, Nv [i])) ;
1308            if (Nv [i] < 0)
1309            {
1310                /* i is a principal variable in Lme */
1311
1312                /* ---------------------------------------------------------
1313                 * examine all hash buckets with 2 or more variables.  We do
1314                 * this by examing all unique hash keys for supervariables in
1315                 * the pattern Lme of the current element, me
1316                 * --------------------------------------------------------- */
1317
1318                /* let i = head of hash bucket, and empty the hash bucket */
1319                ASSERT (Last [i] >= 0 && Last [i] < n) ;
1320                hash = Last [i] ;
1321
1322                /* if Hhead array is not used: */
1323                j = Head [hash] ;
1324                if (j == EMPTY)
1325                {
1326                    /* hash bucket and degree list are both empty */
1327                    i = EMPTY ;
1328                }
1329                else if (j < EMPTY)
1330                {
1331                    /* degree list is empty */
1332                    i = FLIP (j) ;
1333                    Head [hash] = EMPTY ;
1334                }
1335                else
1336                {
1337                    /* degree list is not empty, restore Last [j] of head j */
1338                    i = Last [j] ;
1339                    Last [j] = EMPTY ;
1340                }
1341
1342                /* if separate Hhead array is used: *
1343                i = Hhead [hash] ;
1344                Hhead [hash] = EMPTY ;
1345                */
1346
1347                ASSERT (i >= EMPTY && i < n) ;
1348                AMD_DEBUG2 (("----i "ID" hash "ID"\n", i, hash)) ;
1349
1350                while (i != EMPTY && Next [i] != EMPTY)
1351                {
1352
1353                    /* -----------------------------------------------------
1354                     * this bucket has one or more variables following i.
1355                     * scan all of them to see if i can absorb any entries
1356                     * that follow i in hash bucket.  Scatter i into w.
1357                     * ----------------------------------------------------- */
1358
1359                    ln = Len [i] ;
1360                    eln = Elen [i] ;
1361                    ASSERT (ln >= 0 && eln >= 0) ;
1362                    ASSERT (Pe [i] >= 0 && Pe [i] < iwlen) ;
1363                    /* do not flag the first element in the list (me) */
1364                    for (p = Pe [i] + 1 ; p <= Pe [i] + ln - 1 ; p++)
1365                    {
1366                        ASSERT (Iw [p] >= 0 && Iw [p] < n) ;
1367                        W [Iw [p]] = wflg ;
1368                    }
1369
1370                    /* ----------------------------------------------------- */
1371                    /* scan every other entry j following i in bucket */
1372                    /* ----------------------------------------------------- */
1373
1374                    jlast = i ;
1375                    j = Next [i] ;
1376                    ASSERT (j >= EMPTY && j < n) ;
1377
1378                    while (j != EMPTY)
1379                    {
1380                        /* ------------------------------------------------- */
1381                        /* check if j and i have identical nonzero pattern */
1382                        /* ------------------------------------------------- */
1383
1384                        AMD_DEBUG3 (("compare i "ID" and j "ID"\n", i,j)) ;
1385
1386                        /* check if i and j have the same Len and Elen */
1387                        ASSERT (Len [j] >= 0 && Elen [j] >= 0) ;
1388                        ASSERT (Pe [j] >= 0 && Pe [j] < iwlen) ;
1389                        ok = (Len [j] == ln) && (Elen [j] == eln) ;
1390                        /* skip the first element in the list (me) */
1391                        for (p = Pe [j] + 1 ; ok && p <= Pe [j] + ln - 1 ; p++)
1392                        {
1393                            ASSERT (Iw [p] >= 0 && Iw [p] < n) ;
1394                            if (W [Iw [p]] != wflg) ok = 0 ;
1395                        }
1396                        if (ok)
1397                        {
1398                            /* --------------------------------------------- */
1399                            /* found it!  j can be absorbed into i */
1400                            /* --------------------------------------------- */
1401
1402                            AMD_DEBUG1 (("found it! j "ID" => i "ID"\n", j,i));
1403                            Pe [j] = FLIP (i) ;
1404                            /* both Nv [i] and Nv [j] are negated since they */
1405                            /* are in Lme, and the absolute values of each */
1406                            /* are the number of variables in i and j: */
1407                            Nv [i] += Nv [j] ;
1408                            Nv [j] = 0 ;
1409                            Elen [j] = EMPTY ;
1410                            /* delete j from hash bucket */
1411                            ASSERT (j != Next [j]) ;
1412                            j = Next [j] ;
1413                            Next [jlast] = j ;
1414
1415                        }
1416                        else
1417                        {
1418                            /* j cannot be absorbed into i */
1419                            jlast = j ;
1420                            ASSERT (j != Next [j]) ;
1421                            j = Next [j] ;
1422                        }
1423                        ASSERT (j >= EMPTY && j < n) ;
1424                    }
1425
1426                    /* -----------------------------------------------------
1427                     * no more variables can be absorbed into i
1428                     * go to next i in bucket and clear flag array
1429                     * ----------------------------------------------------- */
1430
1431                    wflg++ ;
1432                    i = Next [i] ;
1433                    ASSERT (i >= EMPTY && i < n) ;
1434
1435                }
1436            }
1437        }
1438        AMD_DEBUG2 (("detect done\n")) ;
1439
1440/* ========================================================================= */
1441/* RESTORE DEGREE LISTS AND REMOVE NONPRINCIPAL SUPERVARIABLES FROM ELEMENT */
1442/* ========================================================================= */
1443
1444        p = pme1 ;
1445        nleft = n - nel ;
1446        for (pme = pme1 ; pme <= pme2 ; pme++)
1447        {
1448            i = Iw [pme] ;
1449            ASSERT (i >= 0 && i < n) ;
1450            nvi = -Nv [i] ;
1451            AMD_DEBUG3 (("Restore i "ID" "ID"\n", i, nvi)) ;
1452            if (nvi > 0)
1453            {
1454                /* i is a principal variable in Lme */
1455                /* restore Nv [i] to signify that i is principal */
1456                Nv [i] = nvi ;
1457
1458                /* --------------------------------------------------------- */
1459                /* compute the external degree (add size of current element) */
1460                /* --------------------------------------------------------- */
1461
1462                deg = Degree [i] + degme - nvi ;
1463                deg = MIN (deg, nleft - nvi) ;
1464                ASSERT (IMPLIES (aggressive, deg > 0) && deg >= 0 && deg < n) ;
1465
1466                /* --------------------------------------------------------- */
1467                /* place the supervariable at the head of the degree list */
1468                /* --------------------------------------------------------- */
1469
1470                inext = Head [deg] ;
1471                ASSERT (inext >= EMPTY && inext < n) ;
1472                if (inext != EMPTY) Last [inext] = i ;
1473                Next [i] = inext ;
1474                Last [i] = EMPTY ;
1475                Head [deg] = i ;
1476
1477                /* --------------------------------------------------------- */
1478                /* save the new degree, and find the minimum degree */
1479                /* --------------------------------------------------------- */
1480
1481                mindeg = MIN (mindeg, deg) ;
1482                Degree [i] = deg ;
1483
1484                /* --------------------------------------------------------- */
1485                /* place the supervariable in the element pattern */
1486                /* --------------------------------------------------------- */
1487
1488                Iw [p++] = i ;
1489
1490            }
1491        }
1492        AMD_DEBUG2 (("restore done\n")) ;
1493
1494/* ========================================================================= */
1495/* FINALIZE THE NEW ELEMENT */
1496/* ========================================================================= */
1497
1498        AMD_DEBUG2 (("ME = "ID" DONE\n", me)) ;
1499        Nv [me] = nvpiv ;
1500        /* save the length of the list for the new element me */
1501        Len [me] = p - pme1 ;
1502        if (Len [me] == 0)
1503        {
1504            /* there is nothing left of the current pivot element */
1505            /* it is a root of the assembly tree */
1506            Pe [me] = EMPTY ;
1507            W [me] = 0 ;
1508        }
1509        if (elenme != 0)
1510        {
1511            /* element was not constructed in place: deallocate part of */
1512            /* it since newly nonprincipal variables may have been removed */
1513            pfree = p ;
1514        }
1515
1516        /* The new element has nvpiv pivots and the size of the contribution
1517         * block for a multifrontal method is degme-by-degme, not including
1518         * the "dense" rows/columns.  If the "dense" rows/columns are included,
1519         * the frontal matrix is no larger than
1520         * (degme+ndense)-by-(degme+ndense).
1521         */
1522
1523        if (Info != (double *) NULL)
1524        {
1525            f = nvpiv ;
1526            r = degme + ndense ;
1527            dmax = MAX (dmax, f + r) ;
1528
1529            /* number of nonzeros in L (excluding the diagonal) */
1530            lnzme = f*r + (f-1)*f/2 ;
1531            lnz += lnzme ;
1532
1533            /* number of divide operations for LDL' and for LU */
1534            ndiv += lnzme ;
1535
1536            /* number of multiply-subtract pairs for LU */
1537            s = f*r*r + r*(f-1)*f + (f-1)*f*(2*f-1)/6 ;
1538            nms_lu += s ;
1539
1540            /* number of multiply-subtract pairs for LDL' */
1541            nms_ldl += (s + lnzme)/2 ;
1542        }
1543
1544#ifndef NDEBUG
1545        AMD_DEBUG2 (("finalize done nel "ID" n "ID"\n   ::::\n", nel, n)) ;
1546        for (pme = Pe [me] ; pme <= Pe [me] + Len [me] - 1 ; pme++)
1547        {
1548              AMD_DEBUG3 ((" "ID"", Iw [pme])) ;
1549        }
1550        AMD_DEBUG3 (("\n")) ;
1551#endif
1552
1553    }
1554
1555/* ========================================================================= */
1556/* DONE SELECTING PIVOTS */
1557/* ========================================================================= */
1558
1559    if (Info != (double *) NULL)
1560    {
1561
1562        /* count the work to factorize the ndense-by-ndense submatrix */
1563        f = ndense ;
1564        dmax = MAX (dmax, (double) ndense) ;
1565
1566        /* number of nonzeros in L (excluding the diagonal) */
1567        lnzme = (f-1)*f/2 ;
1568        lnz += lnzme ;
1569
1570        /* number of divide operations for LDL' and for LU */
1571        ndiv += lnzme ;
1572
1573        /* number of multiply-subtract pairs for LU */
1574        s = (f-1)*f*(2*f-1)/6 ;
1575        nms_lu += s ;
1576
1577        /* number of multiply-subtract pairs for LDL' */
1578        nms_ldl += (s + lnzme)/2 ;
1579
1580        /* number of nz's in L (excl. diagonal) */
1581        Info [AMD_LNZ] = lnz ;
1582
1583        /* number of divide ops for LU and LDL' */
1584        Info [AMD_NDIV] = ndiv ;
1585
1586        /* number of multiply-subtract pairs for LDL' */
1587        Info [AMD_NMULTSUBS_LDL] = nms_ldl ;
1588
1589        /* number of multiply-subtract pairs for LU */
1590        Info [AMD_NMULTSUBS_LU] = nms_lu ;
1591
1592        /* number of "dense" rows/columns */
1593        Info [AMD_NDENSE] = ndense ;
1594
1595        /* largest front is dmax-by-dmax */
1596        Info [AMD_DMAX] = dmax ;
1597
1598        /* number of garbage collections in AMD */
1599        Info [AMD_NCMPA] = ncmpa ;
1600
1601        /* successful ordering */
1602        Info [AMD_STATUS] = AMD_OK ;
1603    }
1604
1605/* ========================================================================= */
1606/* POST-ORDERING */
1607/* ========================================================================= */
1608
1609/* -------------------------------------------------------------------------
1610 * Variables at this point:
1611 *
1612 * Pe: holds the elimination tree.  The parent of j is FLIP (Pe [j]),
1613 *      or EMPTY if j is a root.  The tree holds both elements and
1614 *      non-principal (unordered) variables absorbed into them.
1615 *      Dense variables are non-principal and unordered.
1616 *
1617 * Elen: holds the size of each element, including the diagonal part.
1618 *      FLIP (Elen [e]) > 0 if e is an element.  For unordered
1619 *      variables i, Elen [i] is EMPTY.
1620 *
1621 * Nv: Nv [e] > 0 is the number of pivots represented by the element e.
1622 *      For unordered variables i, Nv [i] is zero.
1623 *
1624 * Contents no longer needed:
1625 *      W, Iw, Len, Degree, Head, Next, Last.
1626 *
1627 * The matrix itself has been destroyed.
1628 *
1629 * n: the size of the matrix.
1630 * No other scalars needed (pfree, iwlen, etc.)
1631 * ------------------------------------------------------------------------- */
1632
1633    /* restore Pe */
1634    for (i = 0 ; i < n ; i++)
1635    {
1636        Pe [i] = FLIP (Pe [i]) ;
1637    }
1638
1639    /* restore Elen, for output information, and for postordering */
1640    for (i = 0 ; i < n ; i++)
1641    {
1642        Elen [i] = FLIP (Elen [i]) ;
1643    }
1644
1645/* Now the parent of j is Pe [j], or EMPTY if j is a root.  Elen [e] > 0
1646 * is the size of element e.  Elen [i] is EMPTY for unordered variable i. */
1647
1648#ifndef NDEBUG
1649    AMD_DEBUG2 (("\nTree:\n")) ;
1650    for (i = 0 ; i < n ; i++)
1651    {
1652        AMD_DEBUG2 ((" "ID" parent: "ID"   ", i, Pe [i])) ;
1653        ASSERT (Pe [i] >= EMPTY && Pe [i] < n) ;
1654        if (Nv [i] > 0)
1655        {
1656            /* this is an element */
1657            e = i ;
1658            AMD_DEBUG2 ((" element, size is "ID"\n", Elen [i])) ;
1659            ASSERT (Elen [e] > 0) ;
1660        }
1661        AMD_DEBUG2 (("\n")) ;
1662    }
1663    AMD_DEBUG2 (("\nelements:\n")) ;
1664    for (e = 0 ; e < n ; e++)
1665    {
1666        if (Nv [e] > 0)
1667        {
1668            AMD_DEBUG3 (("Element e= "ID" size "ID" nv "ID" \n", e,
1669                Elen [e], Nv [e])) ;
1670        }
1671    }
1672    AMD_DEBUG2 (("\nvariables:\n")) ;
1673    for (i = 0 ; i < n ; i++)
1674    {
1675        Int cnt ;
1676        if (Nv [i] == 0)
1677        {
1678            AMD_DEBUG3 (("i unordered: "ID"\n", i)) ;
1679            j = Pe [i] ;
1680            cnt = 0 ;
1681            AMD_DEBUG3 (("  j: "ID"\n", j)) ;
1682            if (j == EMPTY)
1683            {
1684                AMD_DEBUG3 (("  i is a dense variable\n")) ;
1685            }
1686            else
1687            {
1688                ASSERT (j >= 0 && j < n) ;
1689                while (Nv [j] == 0)
1690                {
1691                    AMD_DEBUG3 (("      j : "ID"\n", j)) ;
1692                    j = Pe [j] ;
1693                    AMD_DEBUG3 (("      j:: "ID"\n", j)) ;
1694                    cnt++ ;
1695                    if (cnt > n) break ;
1696                }
1697                e = j ;
1698                AMD_DEBUG3 (("  got to e: "ID"\n", e)) ;
1699            }
1700        }
1701    }
1702#endif
1703
1704/* ========================================================================= */
1705/* compress the paths of the variables */
1706/* ========================================================================= */
1707
1708    for (i = 0 ; i < n ; i++)
1709    {
1710        if (Nv [i] == 0)
1711        {
1712
1713            /* -------------------------------------------------------------
1714             * i is an un-ordered row.  Traverse the tree from i until
1715             * reaching an element, e.  The element, e, was the principal
1716             * supervariable of i and all nodes in the path from i to when e
1717             * was selected as pivot.
1718             * ------------------------------------------------------------- */
1719
1720            AMD_DEBUG1 (("Path compression, i unordered: "ID"\n", i)) ;
1721            j = Pe [i] ;
1722            ASSERT (j >= EMPTY && j < n) ;
1723            AMD_DEBUG3 (("      j: "ID"\n", j)) ;
1724            if (j == EMPTY)
1725            {
1726                /* Skip a dense variable.  It has no parent. */
1727                AMD_DEBUG3 (("      i is a dense variable\n")) ;
1728                continue ;
1729            }
1730
1731            /* while (j is a variable) */
1732            while (Nv [j] == 0)
1733            {
1734                AMD_DEBUG3 (("          j : "ID"\n", j)) ;
1735                j = Pe [j] ;
1736                AMD_DEBUG3 (("          j:: "ID"\n", j)) ;
1737                ASSERT (j >= 0 && j < n) ;
1738            }
1739            /* got to an element e */
1740            e = j ;
1741            AMD_DEBUG3 (("got to e: "ID"\n", e)) ;
1742
1743            /* -------------------------------------------------------------
1744             * traverse the path again from i to e, and compress the path
1745             * (all nodes point to e).  Path compression allows this code to
1746             * compute in O(n) time.
1747             * ------------------------------------------------------------- */
1748
1749            j = i ;
1750            /* while (j is a variable) */
1751            while (Nv [j] == 0)
1752            {
1753                jnext = Pe [j] ;
1754                AMD_DEBUG3 (("j "ID" jnext "ID"\n", j, jnext)) ;
1755                Pe [j] = e ;
1756                j = jnext ;
1757                ASSERT (j >= 0 && j < n) ;
1758            }
1759        }
1760    }
1761
1762/* ========================================================================= */
1763/* postorder the assembly tree */
1764/* ========================================================================= */
1765
1766    AMD_postorder (n, Pe, Nv, Elen,
1767        W,                      /* output order */
1768        Head, Next, Last) ;     /* workspace */
1769
1770/* ========================================================================= */
1771/* compute output permutation and inverse permutation */
1772/* ========================================================================= */
1773
1774    /* W [e] = k means that element e is the kth element in the new
1775     * order.  e is in the range 0 to n-1, and k is in the range 0 to
1776     * the number of elements.  Use Head for inverse order. */
1777
1778    for (k = 0 ; k < n ; k++)
1779    {
1780        Head [k] = EMPTY ;
1781        Next [k] = EMPTY ;
1782    }
1783    for (e = 0 ; e < n ; e++)
1784    {
1785        k = W [e] ;
1786        ASSERT ((k == EMPTY) == (Nv [e] == 0)) ;
1787        if (k != EMPTY)
1788        {
1789            ASSERT (k >= 0 && k < n) ;
1790            Head [k] = e ;
1791        }
1792    }
1793
1794    /* construct output inverse permutation in Next,
1795     * and permutation in Last */
1796    nel = 0 ;
1797    for (k = 0 ; k < n ; k++)
1798    {
1799        e = Head [k] ;
1800        if (e == EMPTY) break ;
1801        ASSERT (e >= 0 && e < n && Nv [e] > 0) ;
1802        Next [e] = nel ;
1803        nel += Nv [e] ;
1804    }
1805    ASSERT (nel == n - ndense) ;
1806
1807    /* order non-principal variables (dense, & those merged into supervar's) */
1808    for (i = 0 ; i < n ; i++)
1809    {
1810        if (Nv [i] == 0)
1811        {
1812            e = Pe [i] ;
1813            ASSERT (e >= EMPTY && e < n) ;
1814            if (e != EMPTY)
1815            {
1816                /* This is an unordered variable that was merged
1817                 * into element e via supernode detection or mass
1818                 * elimination of i when e became the pivot element.
1819                 * Place i in order just before e. */
1820                ASSERT (Next [i] == EMPTY && Nv [e] > 0) ;
1821                Next [i] = Next [e] ;
1822                Next [e]++ ;
1823            }
1824            else
1825            {
1826                /* This is a dense unordered variable, with no parent.
1827                 * Place it last in the output order. */
1828                Next [i] = nel++ ;
1829            }
1830        }
1831    }
1832    ASSERT (nel == n) ;
1833
1834    AMD_DEBUG2 (("\n\nPerm:\n")) ;
1835    for (i = 0 ; i < n ; i++)
1836    {
1837        k = Next [i] ;
1838        ASSERT (k >= 0 && k < n) ;
1839        Last [k] = i ;
1840        AMD_DEBUG2 (("   perm ["ID"] = "ID"\n", k, i)) ;
1841    }
1842}
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