# HG changeset patch
# User Madarasi Peter
# Date 1479863276 -3600
# Node ID 4989d677d5e34ecdab8fc1bddb5d62705326b328
# Parent  fcb190fd9f56d7bbb04d918c8d9a3a1c29b9f7d8
Bio plots grouped

diff -r fcb190fd9f56 -r 4989d677d5e3 damecco.tex
--- a/damecco.tex	Tue Nov 22 19:30:15 2016 +0100
+++ b/damecco.tex	Wed Nov 23 02:07:56 2016 +0100
@@ -75,6 +75,8 @@
 \let\oldproofname=\proofname
 %% \renewcommand{\proofname}{\rm\bf{Proof:}}
 
+\captionsetup{font=normalsize}
+
 \journal{Discrete Applied Mathematics}
 
 \begin{document}
@@ -927,7 +929,7 @@
 \end{example}
 
 These examples may look artificial, but the same problems also appear
-in real-world examples, even though in a less obvious way.
+in real-world instances, even though in a less obvious way.
 
 \subsection{Total ordering}
 Instead of the total ordering relation, the matching order will be
@@ -1055,7 +1057,7 @@
 
 \subsubsection{Subgraph isomorphism}
 \begin{claim}
-\[LabCut_{IND}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!<\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\] is a cutting function by SUB.
+\[LabCut_{SUB}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!<\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\] is a cutting function by SUB.
 \end{claim}
 
 
@@ -1276,8 +1278,8 @@
 \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
+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
@@ -1289,16 +1291,121 @@
 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. Figure~\ref{fig:INDProt}, \ref{fig:INDContact}, and
-\ref{fig:INDMolecule} show the solution time of the
-problems in the problem set.
+This dataset provides graph pairs, between which all the induced subgraph isomorphisms have to be found. For run time results, please see Figure~\ref{fig:bioIND}.
+
+In an other experiment, the nodes of each graph in the database had been
+shuffled, and an isomorphism between the shuffled and the original
+graph was searched. The solution times are shown on Figure~\ref{fig:bioISO}.
+
+
 
 \begin{figure}[H]
+\vspace*{-2cm}
+\hspace*{-1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
+\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}
+\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}
+\end{subfigure}
+\hspace*{1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
+\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}
+\caption{The results are closer to each other on contact maps, but
+  VF2++ still performs consistently better.}\label{fig:ISOContact}
+\end{figure}
+\end{subfigure}
+
 \begin{center}
-\begin{tikzpicture}
+\vspace*{-0.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
+\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}
+\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}
+\end{subfigure}
+\end{center}
+\vspace*{-0.6cm}
+\caption{\normalsize{Graph isomomorphism on biological graphs}}\label{fig:bioISO}
+\end{figure}
+
+
+\begin{figure}[H]
+\vspace*{-2cm}
+\hspace*{-1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
+\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}
+\caption{In the case of molecules, the algorithms have
+  similar behaviour, but VF2++ is almost two times faster even on such
+  small graphs.} \label{fig:INDMolecule}
+\end{figure}
+\end{subfigure}
+\hspace*{1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
+\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}
+\caption{On contact maps, VF2++ runs in near constant time, while VF2
+  Plus has a near linear behaviour.} \label{fig:INDContact}
+\end{figure}
+\end{subfigure}
+
+\begin{center}
+\vspace*{-0.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
+\begin{figure}[H]
+\begin{tikzpicture}[trim axis left, trim axis right]
   \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
@@ -1308,114 +1415,18 @@
       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{subfigure}
 \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}
+\vspace*{-0.5cm}
+\caption{\normalsize{Graph isomomorphism on biological graphs}}\label{fig:bioIND}
 \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
-Figure~\ref{fig:ISOProt}, \ref{fig:ISOContact}, and
-\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}
@@ -1427,18 +1438,22 @@
 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. Figure \ref{fig:randISO5},
-\ref{fig:randISO10}, \ref{fig:randISO15}, \ref{fig:randISO35},
-\ref{fig:randISO45}, and \ref{fig:randISO100} show the runtime results
+isomorphism had to be found. Figure \ref{fig:randISO} shows the runtime results
 on graph sets of various density.
 
+
+
+
 \begin{figure}[H]
+\vspace*{-1.5cm}
+\hspace*{-1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 5$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 5$},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}]
+  format/1000 sep = \space}]
 %\addplot+[only marks] table {proteinsOrig.txt};
 \addplot table {randGraph/iso/vf2pIso5_1.txt};
 \addplot[mark=triangle*,mark size=1.8pt,color=red] table
@@ -1446,17 +1461,15 @@
 \end{axis}
 \end{tikzpicture}
 \end{center}
-\vspace*{-0.8cm}
-\caption{}\label{fig:randISO5}
-\end{figure}
-
-\begin{figure}[H]
+\end{subfigure}
+%\hspace{1cm}
+\begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 10$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 10$},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}]
+  format/1000 sep = \space}]
 %\addplot+[only marks] table {proteinsOrig.txt};
 \addplot table {randGraph/iso/vf2pIso10_1.txt};
 \addplot[mark=triangle*,mark size=1.8pt,color=red] table
@@ -1464,17 +1477,16 @@
 \end{axis}
 \end{tikzpicture}
 \end{center}
-\vspace*{-0.8cm}
-\caption{}\label{fig:randISO10}
-\end{figure}
-
-\begin{figure}[H]
+\end{subfigure}
+%%\hspace{1cm}
+\hspace*{-1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 15$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 15$},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}]
+  format/1000 sep = \space}]
 %\addplot+[only marks] table {proteinsOrig.txt};
 \addplot table {randGraph/iso/vf2pIso15_1.txt};
 \addplot[mark=triangle*,mark size=1.8pt,color=red] table
@@ -1482,17 +1494,14 @@
 \end{axis}
 \end{tikzpicture}
 \end{center}
-\vspace*{-0.8cm}
-\caption{}\label{fig:randISO15}
-\end{figure}
-
-\begin{figure}[H]
+     \end{subfigure}
+     \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 35$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 35$},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}]
+  format/1000 sep = \space}]
 %\addplot+[only marks] table {proteinsOrig.txt};
 \addplot table {randGraph/iso/vf2pIso35_1.txt};
 \addplot[mark=triangle*,mark size=1.8pt,color=red] table
@@ -1500,32 +1509,25 @@
 \end{axis}
 \end{tikzpicture}
 \end{center}
-\vspace*{-0.8cm}
-\caption{}\label{fig:randISO35}
-\end{figure}
-
-\begin{figure}[H]
-\begin{center}
+\end{subfigure}
+\begin{subfigure}[b]{0.55\textwidth}
+\hspace*{-1.5cm}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 45$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 45$},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}]
+  format/1000 sep = \space}]
 %\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}
+\end{subfigure}
+\hspace*{-1.5cm}
+\begin{subfigure}[b]{0.55\textwidth}
 \begin{tikzpicture}
-\begin{axis}[title={Random ISO, $\delta = 100$},xlabel={target size},ylabel={time (ms)},legend entries={VF2 Plus,VF2++},grid
+\begin{axis}[title={Random ISO, $\delta = 100$},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}]
@@ -1535,12 +1537,21 @@
         {randGraph/iso/vf2ppIso100_1.txt};
 \end{axis}
 \end{tikzpicture}
-\end{center}
+\end{subfigure}
 \vspace*{-0.8cm}
-\caption{}\label{fig:randISO100}
+\caption{IND on graphs having an average degree of
+  5.}\label{fig:randISO}
 \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
@@ -1571,7 +1582,8 @@
 
 
 \begin{figure}[H]
-\vspace*{-0.8cm}
+\vspace*{-1.5cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
@@ -1602,7 +1614,7 @@
 \end{tikzpicture}
 \end{center}
 \end{subfigure}
-\hspace{1cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
@@ -1634,7 +1646,7 @@
 \end{center}
 \end{subfigure}
 \begin{subfigure}[b]{0.55\textwidth}
-          
+\hspace*{-1.5cm}
 \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
@@ -1647,6 +1659,7 @@
 \end{axis}
 \end{tikzpicture}
      \end{subfigure}
+     \hspace*{-1.5cm}
      \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
@@ -1667,6 +1680,7 @@
 
 \begin{figure}[H]
 \begin{center}
+\hspace*{-2cm}
 \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
@@ -1699,9 +1713,11 @@
 
 
 \begin{figure}[H]
-\vspace*{-0.8cm}
+\vspace*{-1.5cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
+\hspace*{-0.5cm}
 \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
@@ -1717,6 +1733,7 @@
      \end{subfigure}
      \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
+     \hspace*{-0.5cm}
 \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
@@ -1730,7 +1747,7 @@
 \end{tikzpicture}
 \end{center}
 \end{subfigure}
-\hspace{1cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
@@ -1761,8 +1778,8 @@
 \end{tikzpicture}
 \end{center}
 \end{subfigure}
+\hspace*{-1.5cm}
 \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
@@ -1795,6 +1812,7 @@
 
 \begin{figure}[H]
 \begin{center}
+\hspace*{-2cm}
 \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
@@ -1834,7 +1852,8 @@
 
 
 \begin{figure}[H]
-\vspace*{-0.8cm}
+\vspace*{-1.5cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
@@ -1865,7 +1884,7 @@
 \end{tikzpicture}
 \end{center}
 \end{subfigure}
-\hspace{1cm}
+\hspace*{-1.5cm}
 \begin{subfigure}[b]{0.55\textwidth}
 \begin{center}
 \begin{tikzpicture}
@@ -1896,8 +1915,8 @@
 \end{tikzpicture}
 \end{center}
 \end{subfigure}
+\hspace*{-1.5cm}
 \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
@@ -1930,6 +1949,7 @@
 
 \begin{figure}[H]
 \begin{center}
+\hspace*{-2cm}
 \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