lemon-vf2: comparison damecco.tex

equal deleted inserted replaced

-:f9608f3370b0
+:ba075f024c84
 many fields of application such as pattern analysis, computer vision
 questions and the analysis of chemical and biological systems has
 fostered the design of various algorithms for handling special graph
 structures.
-The idea of using state space representation and checking some
-conditions in each state to prune the search tree has made the VF2
-algorithm one of the state of the art graph matching algorithms for
-more than a decade. Recently, biological questions of ever increasing
-importance have required more efficient, specialized algorithms.
 This paper presents VF2++, a new algorithm based on the original VF2,
 which runs significantly faster on most test cases and performs
 especially well on special graph classes stemming from biological
 questions. VF2++ handles graphs of thousands of nodes in practically
 near linear time including preprocessing. Not only is it an improved
 version of VF2, but in fact, it is by far the fastest existing
-algorithm regarding biological graphs.
+algorithm especially on biological graphs.
 The reason for VF2++' superiority over VF2 is twofold. Firstly, taking
 into account the structure and the node labeling of the graph, VF2++
 determines a state order in which most of the unfruitful branches of
 the search space can be pruned immediately. Secondly, introducing more
 In the last decades, combinatorial structures, and especially graphs
 have been considered with ever increasing interest, and applied to the
 solution of several new and revised questions.  The expressiveness,
 the simplicity and the studiedness of graphs make them practical for
 modelling and appear constantly in several seemingly independent
-fields.  Bioinformatics and chemistry are amongst the most relevant
+fields, such as bioinformatics and chemistry.
-and most important fields.
 Complex biological systems arise from the interaction and cooperation
 of plenty of molecular components. Getting acquainted with such
-systems at the molecular level has primary importance, since
+systems at the molecular level is of primary importance, since
 protein-protein interaction, DNA-protein interaction, metabolic
 interaction, transcription factor binding, neuronal networks, and
-hormone signaling networks can be understood only this way.
+hormone signaling networks can be understood this way.
-For instance, a molecular structure can be considered as a graph,
+Many chemical and biological structures can easily be modeled
-whose nodes correspond to atoms and whose edges to chemical bonds. The
+as graphs, for instance, a molecular structure can be
-secondary structure of a protein can also be represented as a graph,
+considered as a graph, whose nodes correspond to atoms and whose
-where nodes are associated with aminoacids and the edges with hydrogen
+edges to chemical bonds. The similarity and dissimilarity of
-bonds. The nodes are often whole molecular components and the edges
+objects corresponding to nodes are incorporated to the model
-represent some relationships among them.  The similarity and
+by \emph{node labels}. Understanding such networks basically
-dissimilarity of objects corresponding to nodes are incorporated to
+requires finding specific subgraphs, thus calls for efficient
-the model by \emph{node labels}.  Many other chemical and biological
+graph matching algorithms.
-structures can easily be modeled in a similar way. Understanding such
-networks basically requires finding specific subgraphs, which can not
+Other real-world fields related to some
-avoid the application of graph matching algorithms.
+variants of graph matching include pattern recognition
-Finally, let some of the other real-world fields related to some
-variants of graph matching be briefly mentioned: pattern recognition
 and machine vision \cite{HorstBunkeApplications}, symbol recognition
 \cite{CordellaVentoSymbolRecognition}, face identification
 \cite{JianzhuangYongFaceIdentification}.  \\
 Subgraph and induced subgraph matching problems are known to be
 one of the few problems in NP neither known to be in P nor
 NP-Complete. Although polynomial time isomorphism algorithms are known
 for various graph classes, like trees and planar
 graphs\cite{PlanarGraphIso}, bounded valence
 graphs\cite{BondedDegGraphIso}, interval graphs\cite{IntervalGraphIso}
-or permutation graphs\cite{PermGraphIso}.
+or permutation graphs\cite{PermGraphIso}, and recently, an FPT algorithm has been presented for the coloured hypergraph isomorphism problem in \cite{ColoredHiperGraphIso}.
 In the following, some algorithms based on other approaches are
-summarized, which do not need any restrictions on the graphs. However,
+summarized, which do not need any restrictions on the graphs. Even though,
 an overall polynomial behaviour is not expectable from such an
-alternative, it may often have good performance, even on a graph class
+alternative, it may often have good practical performance, in fact,
-for which polynomial algorithm is known. Note that this summary
+it might be the best choice even on a graph class for which polynomial
-containing only exact matching algorithms is far not complete, neither
+algorithm is known.
-does it cover all the recent algorithms.
 The first practically usable approach was due to
-Ullmann\cite{Ullmann} which is a commonly used depth-first
+\emph{Ullmann}\cite{Ullmann} which is a commonly used depth-first
 search based algorithm with a complex heuristic for reducing the
 number of visited states. A major problem is its $\Theta(n^3)$ space
 complexity, which makes it impractical in the case of big sparse
 graphs.
 In a recent paper, Ullmann\cite{UllmannBit} presents an
 improved version of this algorithm based on a bit-vector solution for
 the binary Constraint Satisfaction Problem.
-The Nauty algorithm\cite{Nauty} transforms the two graphs to
+The \emph{Nauty} algorithm\cite{Nauty} transforms the two graphs to
 a canonical form before starting to check for the isomorphism. It has
 been considered as one of the fastest graph isomorphism algorithms,
 although graph categories were shown in which it takes exponentially
 many steps. This algorithm handles only the graph isomorphism problem.
 strategy and formulates the matching as a Constraint Satisfaction
 Problem to prune the search tree. The constraints are that the mapping
 has to be injective and edge-preserving, hence it is possible to
 handle new matching types as well.
-The \textbf{RI} algorithm\cite{RI} and its variations are based on a
+The \emph{RI} algorithm\cite{RI} and its variations are based on a
 state space representation. After reordering the nodes of the graphs,
 it uses some fast executable heuristic checks without using any
 complex pruning rules. It seems to run really efficiently on graphs
 coming from biology, and won the International Contest on Pattern
 Search in Biological Databases\cite{Content}.
 The currently most commonly used algorithm is the
-\textbf{VF2}\cite{VF2}, the improved version of VF\cite{VF}, which was
+\emph{VF2}\cite{VF2}, the improved version of \emph{VF}\cite{VF}, which was
 designed for solving pattern matching and computer vision problems,
 and has been one of the best overall algorithms for more than a
 decade. Although, it can't be up to new specialized algorithms, it is
 still widely used due to its simplicity and space efficiency. VF2 uses
 a state space representation and checks some conditions in each state
 to prune the search tree.
-Our first graph matching algorithm was the first version of VF2 which
+Meanwhile, another variant called \emph{VF2 Plus}\cite{VF2Plus} has
-recognizes the significance of the node ordering, more opportunities
-to increase the cutting efficiency and reduce its computational
-complexity. This project was initiated and sponsored by QuantumBio
-Inc.\cite{QUANTUMBIO} and the implementation --- along with a source
-code --- has been published as a part of LEMON\cite{LEMON} open source
-graph library.
-This paper introduces \textbf{VF2++}, a new further improved algorithm
-for the graph and (induced)subgraph isomorphism problem, which uses
-efficient cutting rules and determines a node order in which VF2 runs
-significantly faster on practical inputs.
-Meanwhile, another variant called \textbf{VF2 Plus}\cite{VF2Plus} has
 been published. It is considered to be as efficient as the RI
 algorithm and has a strictly better behavior on large graphs.  The
 main idea of VF2 Plus is to precompute a heuristic node order of the
 small graph, in which the VF2 works more efficiently.
+This paper introduces \emph{VF2++}, a new further improved algorithm
+for the graph and (induced)subgraph isomorphism problem, which uses
+efficient cutting rules and determines a node order in which VF2 runs
+significantly faster on practical inputs.
+This project was initiated and sponsored by QuantumBio
+Inc.\cite{QUANTUMBIO} and the implementation --- along with a source
+code --- has been published as a part of LEMON\cite{LEMON} open source
+graph library.
 \section{Problem Statement}
-This section provides a detailed description of the problems to be
+This section provides a formal description of the problems to be
 solved.
 \subsection{Definitions}
-Throughout the paper $G_{small}=(V_{small}, E_{small})$ and
+Throughout the paper $G_{1}=(V_{1}, E_{1})$ and
-$G_{large}=(V_{large}, E_{large})$ denote two undirected graphs.
+$G_{2}=(V_{2}, E_{2})$ denote two undirected graphs.
-\begin{definition}\label{sec:ismorphic}
-$G_{small}$ and $G_{large}$ are \textbf{isomorphic} if $\exists M:
-V_{small} \longrightarrow V_{large}$ bijection, for which the
-following is true:
-\begin{center}
-$\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow
-(M(u),M(v))\in{E_{large}}$
-\end{center}
-\end{definition}
-For the sake of simplicity in this paper subgraphs and induced
-subgraphs are defined in a more general way than usual:
 \begin{definition}
-$G_{small}$ is a \textbf{subgraph} of $G_{large}$ if $\exists I:
+$\mathcal{L}: (V_{1}\cup V_{2}) \longrightarrow K$ is a \textbf{node
-V_{small}\longrightarrow V_{large}$ injection, for which the
-following is true:
-\begin{center}
-$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Rightarrow (I(u),I(v))\in E_{large}$
-\end{center}
-\end{definition}
-\begin{definition}
-$G_{small}$ is an \textbf{induced subgraph} of $G_{large}$ if $\exists
-I: V_{small}\longrightarrow V_{large}$ injection, for which the
-following is true:
-\begin{center}
-$\forall u,v \in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow
-(I(u),I(v))\in E_{large}$
-\end{center}
-\end{definition}
-\begin{definition}
-$lab: (V_{small}\cup V_{large}) \longrightarrow K$ is a \textbf{node
 label function}, where K is an arbitrary set. The elements in K
 are the \textbf{node labels}. Two nodes, u and v are said to be
-\textbf{equivalent} if $lab(u)=lab(v)$.
+\textbf{equivalent} if $\mathcal{L}(u)=\mathcal{L}(v)$.
 \end{definition}
-When node labels are given, the matched nodes must have the same
+For the sake of simplicity, in this paper the graph, subgraph and induced subgraph isomorphisms are defined in a more general way.
-labels.  For example, the node labeled isomorphism is phrased by
+\begin{definition}\label{sec:ismorphic}
+$G_{1}$ and $G_{2}$ are \textbf{isomorphic} (by the node label $\mathcal{L}$) if $\exists \mathfrak{m}:
+V_{1} \longrightarrow V_{2}$ bijection, for which the
+following is true:
+\begin{center}
+$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
+$\forall u,v\in{V_{1}} : (u,v)\in{E_{1}} \Leftrightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in{E_{2}}$
+\end{center}
+\end{definition}
 \begin{definition}
-$G_{small}$ and $G_{large}$ are \textbf{isomorphic by the node label
+$G_{1}$ is a \textbf{subgraph} of $G_{2}$ (by the node label $\mathcal{L}$) if $\exists \mathfrak{m}:
-function lab} if $\exists M: V_{small} \longrightarrow V_{large}$
+V_{1}\longrightarrow V_{2}$ injection, for which the
-bijection, for which the following is true:
+following is true:
 \begin{center}
-$(\forall u,v\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow
+$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
-(M(u),M(v))\in{E_{large}})$ and $(\forall u\in{V_{small}} :
+$\mathbb{M}$
-lab(u)=lab(M(u)))$
+$\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Rightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$
 \end{center}
 \end{definition}
-Note that he other two definitions can be extended in the same way.
+\begin{definition}
+$G_{1}$ is an \textbf{induced subgraph} of $G_{2}$ (by the node label $\mathcal{L}$) if $\exists
+\mathfrak{m}: V_{1}\longrightarrow V_{2}$ injection, for which the
+following is true:
+\begin{center}
+$\forall u\in{V_{1}} : \mathcal{L}(u)=\mathcal{L}(\mathfrak{m}(u))$ and
+$\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Leftrightarrow
+(\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$
+\end{center}
+\end{definition}
 \subsection{Common problems}\label{sec:CommProb}
 The focus of this paper is on two extensively studied topics, the
 subgraph isomorphism and its variations. However, the following
 problems also appear in many applications.
 The \textbf{subgraph matching problem} is the following: is
-$G_{small}$ isomorphic to any subgraph of $G_{large}$ by a given node
+$G_{1}$ isomorphic to any subgraph of $G_{2}$ by a given node
 label?
 The \textbf{induced subgraph matching problem} asks the same about the
 existence of an induced subgraph.
 for the sake of simplicity, only the undirected case will be
 discussed.
 \subsection{Common notations}
-\indent Assume $G_{small}$ is searched in $G_{large}$.  The following
+\indent Assume $G_{1}$ is searched in $G_{2}$.  The following
 definitions and notations will be used throughout the whole paper.
 \begin{definition}
-A set $M\subseteq V_{small}\times V_{large}$ is called
+An injection $\mathfrak{m} : D \longrightarrow V_2$ is called (partial) \textbf{mapping}, where $D\subseteq V_1$.
-\textbf{mapping} if no node of $V_{small}\cup V_{large}$ appears
-in more than one pair in M.  That is, M uniquely associates some of
-the nodes in $V_{small}$ with some nodes of $V_{large}$ and vice
-versa.
 \end{definition}
+\begin{notation}
+$\mathfrak{D}(f)$ and $\mathfrak{R}(f)$ denote the domain and the range of a function $f$, respectively.
+\end{notation}
 \begin{definition}
-Mapping $M$ \textbf{covers} a node v if there exists a pair in M, which
+Mapping $\mathfrak{m}$ \textbf{covers} a node $u\in V_1\cup V_2$ if $u\in \mathfrak{D}(\mathfrak{m})\cup \mathfrak{R}(\mathfrak{m})$.
-contains v.
 \end{definition}
 \begin{definition}
-A mapping $M$ is $\mathbf{whole\ mapping}$ if $M$ covers all the
+A mapping $\mathfrak{m}$ is $\mathbf{whole\ mapping}$ if $\mathfrak{m}$ covers all the
-nodes in $V_{small}$.
+nodes of $V_{1}$, i.e. $\mathfrak{D}(\mathfrak{m})=V_1$.
 \end{definition}
+\begin{definition}
+Let \textbf{extend}$(\mathfrak{m},(u,v))$ denote the function $f : \mathfrak{D}(\mathfrak{m})\cup\{u\}\longrightarrow\mathfrak{R}(\mathfrak{m})\cup\{v\}$, for which $\forall w\in \mathfrak{D}(\mathfrak{m}) : \mathfrak{m}(w)=f(w)$ and $f(u)=v$ holds. Where $u\notin\mathfrak{D}(\mathfrak{m})$ and $v\notin\mathfrak{R}(\mathfrak{m})$, otherwise $extend(\mathfrak{m},(u,v))$ is undefined.
+\end{definition}
 \begin{notation}
-Let $\mathbf{M_{small}(s)} := \{u\in V_{small} : \exists v\in
+Throughout the paper, $\mathbf{PT}$ denotes a generic problem type
-V_{large}: (u,v)\in M(s)\}$ and $\mathbf{M_{large}(s)} := \{v\in
+which can be substituted by any of the $\mathbf{ISO}$, $\mathbf{SUB}$
-V_{large} : \exists u\in V_{small}: (u,v)\in M(s)\}$.
+and $\mathbf{IND}$ problems.
 \end{notation}
-\begin{notation}
-Let $\mathbf{Pair(M,v)}$ be the pair of $v$ in $M$ if such a node
-exist, otherwise $\mathbf{Pair(M,v)}$ is undefined, where $M$ is a mapping and $v\in V_{small}\cup V_{large}$.
-\end{notation}
-Note that if $\mathbf{Pair(M,v)}$ exists, then it is unique
-The definitions of the isomorphism types can be rephrased on the
-existence of a special whole mapping $M$, since it represents a
-bijection. For example
-\begin{center}
-$M\subseteq V_{small}\times V_{large}$ represents an induced subgraph
-isomorphism $\Leftrightarrow$ $M$ is whole mapping and $\forall u,v
-\in{V_{small}} : (u,v)\in{E_{small}} \Leftrightarrow
-(Pair(M,u),Pair(M,v))\in E_{large}$.
-\end{center}
-Throughout the paper, $\mathbf{PT}$ denotes a generic problem type
-which can be substituted by any of $\mathbf{ISO}$, $\mathbf{SUB}$
-and $\mathbf{IND}$.
 \begin{definition}
-Let M be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a
+Let $\mathfrak{m}$ be a mapping. A logical function $\mathbf{Cons_{PT}}$ is a
 \textbf{consistency function by } $\mathbf{PT}$ if the following
-holds. If there exists whole mapping $W$ of type $PT$ for which
+holds. If there exists a whole mapping $w$ satisfying the requirements of $PT$, for which $\mathfrak{m}$ is exactly $w$ restricted to $\mathfrak{D}(\mathfrak{m})$.
-$M\subseteq W$, then $Cons_{PT}(M)$ is true.
 \end{definition}
 \begin{definition}
-Let M be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a
+Let $\mathfrak{m}$ be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a
 \textbf{cutting function by } $\mathbf{PT}$ if the following
-holds. $\mathbf{Cut_{PT}(M)}$ is false if $M$ can be extended to a
+holds. $\mathbf{Cut_{PT}(\mathfrak{m})}$ is false if there exists a sequence of extend operations, which results in a whole mapping satisfying the requirements of $PT$.
-whole mapping $W$ of type $PT$.
 \end{definition}
 \begin{definition}
-$M$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$ if
+$\mathfrak{m}$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$ if
-$Cons_{PT}(M)$ is true.
+$Cons_{PT}(\mathfrak{m})$ is true.
 \end{definition}
 $Cons_{PT}$ and $Cut_{PT}$ will often be used in the following form.
 \begin{notation}
-Let $\mathbf{Cons_{PT}(p, M)}:=Cons_{PT}(M\cup\{p\})$, and
+Let $\mathbf{Cons_{PT}(p, \mathfrak{m})}:=Cons_{PT}(extend(\mathfrak{m},p))$, and
-$\mathbf{Cut_{PT}(p, M)}:=Cut_{PT}(M\cup\{p\})$, where
+$\mathbf{Cut_{PT}(p, \mathfrak{m})}:=Cut_{PT}(extend(\mathfrak{m},p))$, where
-$p\in{V_{small}\!\times\!V_{large}}$ and $M\cup\{p\}$ is mapping.
+$p\in{V_{1}\backslash\mathfrak{D}(\mathfrak{m}) \!\times\!V_{2}\backslash\mathfrak{R}(\mathfrak{m})}$.
 \end{notation}
 $Cons_{PT}$ will be used to check the consistency of the already
 covered nodes, while $Cut_{PT}$ is for looking ahead to recognize if
 no whole consistent mapping can contain the current mapping.
 \subsection{Overview of the algorithm}
 VF2 uses a state space representation of mappings, $Cons_{PT}$ for
 excluding inconsistency with the problem type and $Cut_{PT}$ for
-pruning the search tree.  Each state $s$ of the matching process can
+pruning the search tree.
-be associated with a mapping $M(s)$.
 Algorithm~\ref{alg:VF2Pseu} is a high level description of
-the VF2 matching algorithm.
+the VF2 matching algorithm. Each state of the matching process can
+be associated with a mapping $\mathfrak{m}$. The initial state
+is associated with a mapping $\mathfrak{m}$, for which
+$\mathfrak{D}(\mathfrak{m})=\emptyset$, i.e. it starts with an empty mapping.
 \begin{algorithm}
 \algtext*{EndIf}%ne nyomtasson end if-et
 \algtext*{EndFor}%ne
 \algtext*{EndProcedure}%ne nyomtasson ..
 \caption{\hspace{0.5cm}$A\ high\ level\ description\ of\ VF2$}\label{alg:VF2Pseu}
 \begin{algorithmic}[1]
-\Procedure{VF2}{State $s$, ProblemType $PT$} \If{$M(s$) covers
+\Procedure{VF2}{Mapping $\mathfrak{m}$, ProblemType $PT$}
-$V_{small}$} \State Output($M(s)$) \Else
+\If{$\mathfrak{m}$ covers
+$V_{1}$} \State Output($\mathfrak{m}$)
-\State Compute the set $P(s)$ of the pairs candidate for inclusion
+\Else
-in $M(s)$ \ForAll{$p\in{P(s)}$} \If{Cons$_{PT}$($p, M(s)$) $\wedge$
+\State Compute the set $P_\mathfrak{m}$ of the pairs candidate for inclusion
-$\neg$Cut$_{PT}$($p, M(s)$)} \State Compute the nascent state
+in $\mathfrak{m}$ \ForAll{$p\in{P_\mathfrak{m}}$} \If{Cons$_{PT}$($p,\mathfrak{m}$) $\wedge$
-$\tilde{s}$ by adding $p$ to $M(s)$ \State \textbf{call}
+$\neg$Cut$_{PT}$($p,\mathfrak{m}$)}
-VF2($\tilde{s}$, $PT$) \EndIf \EndFor \EndIf \EndProcedure
+\State \textbf{call}
+VF2($extend(\mathfrak{m},p)$, $PT$) \EndIf \EndFor \EndIf \EndProcedure
 \end{algorithmic}
 \end{algorithm}
-The initial state $s_0$ is associated with $M(s_0)=\emptyset$, i.e. it
+For the current mapping $\mathfrak{m}$, the algorithm computes $P_\mathfrak{m}$, the set of
-starts with an empty mapping.
+candidate node pairs for adding to the current mapping $\mathfrak{m}_s$.
-For each state $s$, the algorithm computes $P(s)$, the set of
+For each pair $p$ in $P_\mathfrak{m}$, $Cons_{PT}(p,\mathfrak{m})$ and
-candidate node pairs for adding to the current state $s$.
+$Cut_{PT}(p,\mathfrak{m})$ are evaluated. If the former is true and
+the latter is false, the whole process is recursively applied to
-For each pair $p$ in $P(s)$, $Cons_{PT}(p,M(s))$ and
+$extend(\mathfrak{m},p)$. Otherwise, $extend(\mathfrak{m},p)$ is not consistent by $PT$, or it
-$Cut_{PT}(p,M(s))$ are evaluated. If $Cons_{PT}(p,M(s))$ is true and
+can be proved that $\mathfrak{m}$ can not be extended to a whole mapping.
-$Cut_{PT}(p,M(s))$ is false, the successor state $\tilde{s}=s\cup
-\{p\}$ is computed, and the whole process is recursively applied to
-$\tilde{s}$. Otherwise, $\tilde{s}$ is not consistent by $PT$, or it
-can be proved that $s$ can not be extended to a whole mapping.
 In order to make sure of the correctness, see
 \begin{claim}
 Through consistent mappings, only consistent whole mappings can be
-reached, and all of the whole mappings are reachable through
+reached, and all the consistent whole mappings are reachable through
 consistent mappings.
 \end{claim}
-Note that a state may be reached in exponentially many different ways, since the
+Note that a mapping may be reached in exponentially many different ways, since the
-order of insertions into $M$ does not influence the nascent mapping.
+order of extensions does not influence the nascent mapping.
 However, one may observe
 \begin{claim}
 \label{claim:claimTotOrd}
-Let $\prec$ an arbitrary total ordering relation on $V_{small}$.  If
+Let $\prec$ be an arbitrary total ordering relation on $V_{1}$.  If
-the algorithm ignores each $p=(u,v) \in P(s)$, for which
+the algorithm ignores each $p=(u,v) \in P_\mathfrak{m}$, for which
 \begin{center}
-$\exists (\tilde{u},\tilde{v})\in P(s): \tilde{u} \prec u$,
+$\exists (\tilde{u},\tilde{v})\in P_\mathfrak{m}: \tilde{u} \prec u$,
 \end{center}
-then no state can be reached more than once, and each state associated
+then no mapping can be reached more than once, and each whole mapping remains reachable.
-with a whole mapping remains reachable.
 \end{claim}
 Note that the cornerstone of the improvements to VF2 is a proper
 choice of a total ordering.
-\subsection{The candidate set P(s)}
+\subsection{The candidate set}
 \label{candidateComputingVF2}
-Let $P(s)$ be the set of the candidate pairs for inclusion in $M(s)$.
+Let $P_\mathfrak{m}$ be the set of the candidate pairs for inclusion in $\mathfrak{m}$.
 \begin{notation}
-Let $\mathbf{T_{small}(s)}:=\{u \in V_{small} : u$ is not covered by
+Let $\mathbf{T_{1}(\mathfrak{m})}:=\{u \in V_{1}\backslash\mathfrak{D}(\mathfrak{m}) : \exists \tilde{u}\in{\mathfrak{D}(\mathfrak{m}): (u,\tilde{u})\in E_{1}}\}$, and
-$M(s)\wedge\exists \tilde{u}\in{V_{small}: (u,\tilde{u})\in E_{small}}
+$\mathbf{T_{2}(\mathfrak{m})} := \{v \in V_{2}\backslash\mathfrak{R}(\mathfrak{m}) : \exists\tilde{v}\in{\mathfrak{R}(\mathfrak{m}):(v,\tilde{v})\in E_{2}}\}$.
-\wedge \tilde{u}$ is covered by $M(s)\}$, and
-\\ $\mathbf{T_{large}(s)}\!:=\!\{v \in\!V_{large}\!:\!v$ is not
-covered by
-$M(s)\wedge\!\exists\tilde{v}\!\in\!{V_{large}\!:\!(v,\tilde{v})\in\!E_{large}}
-\wedge \tilde{v}$ is covered by $M(s)\}$
 \end{notation}
-The set $P(s)$ includes the pairs of uncovered neighbours of covered
+The set $P_\mathfrak{m}$ includes the pairs of uncovered neighbours of covered
 nodes, and if there is not such a node pair, all the pairs containing
 two uncovered nodes are added. Formally, let
 \[
-P(s)\!=\!
+P_\mathfrak{m}\!=\!
 \begin{cases}
-T_{small}(s)\times T_{large}(s)&\hspace{-0.15cm}\text{if }
+T_{1}(\mathfrak{m})\times T_{2}(\mathfrak{m})&\hspace{-0.15cm}\text{if }
-T_{small}(s)\!\neq\!\emptyset\!\wedge\!T_{large}(s)\!\neq
+T_{1}(\mathfrak{m})\!\neq\!\emptyset\ \text{and }T_{2}(\mathfrak{m})\!\neq
-\emptyset,\\ (V_{small}\!\setminus\!M_{small}(s))\!\times\!(V_{large}\!\setminus\!M_{large}(s))
+\emptyset,\\ (V_{1}\!\setminus\!\mathfrak{D}(\mathfrak{m}))\!\times\!(V_{2}\!\setminus\!\mathfrak{R}(\mathfrak{m}))
-&\hspace{-0.15cm}otherwise.
+&\hspace{-0.15cm}\text{otherwise}.
 \end{cases}
 \]
 \subsection{Consistency}
-Suppose $p=(u,v)$, where $u\in V_{small}$ and $v\in V_{large}$, $s$ is
+Suppose $p=(u,v)$, where $u\in V_{1}$ and $v\in V_{2}$, $\mathfrak{m}$ is a consistent mapping by
-a state of the matching procedure, $M(s)$ is consistent mapping by
+$PT$. $Cons_{PT}(p,\mathfrak{m})$ checks whether
-$PT$ and $lab(u)=lab(v)$.  $Cons_{PT}(p,M(s))$ checks whether
+including pair $p$ into $\mathfrak{m}$ leads to a consistent mapping by $PT$.
-including pair $p$ into $M(s)$ leads to a consistent mapping by $PT$.
 For example, the consistency function of induced subgraph isomorphism is as follows.
 \begin{notation}
-Let $\mathbf{\Gamma_{small} (u)}:=\{\tilde{u}\in V_{small} :
+Let $\mathbf{\Gamma_{1} (u)}:=\{\tilde{u}\in V_{1} :
-(u,\tilde{u})\in E_{small}\}$, and $\mathbf{\Gamma_{large}
+(u,\tilde{u})\in E_{1}\}$, and $\mathbf{\Gamma_{2}
-(v)}:=\{\tilde{v}\in V_{large} : (v,\tilde{v})\in E_{large}\}$, where $u\in V_{small}$ and $v\in V_{large}$.
+(v)}:=\{\tilde{v}\in V_{2} : (v,\tilde{v})\in E_{2}\}$, where $u\in V_{1}$ and $v\in V_{2}$.
 \end{notation}
-$M(s)\cup \{(u,v)\}$ is a consistent mapping by $IND$ $\Leftrightarrow
+$extend(\mathfrak{m},(u,v))$ is a consistent mapping by $IND$ $\Leftrightarrow
-(\forall \tilde{u}\in M_{small}: (u,\tilde{u})\in E_{small}
+(\forall \tilde{u}\in \mathfrak{D}(\mathfrak{m}): (u,\tilde{u})\in E_{1}
-\Leftrightarrow (v,Pair(M(s),\tilde{u}))\in E_{large})$. The
+\Leftrightarrow (v,\mathfrak{m}(\tilde{u}))\in E_{2})$. The
 following formulation gives an efficient way of calculating
 $Cons_{IND}$.
 \begin{claim}
-$Cons_{IND}((u,v),M(s)):=(\forall \tilde{v}\in \Gamma_{large}(v)
+$Cons_{IND}((u,v),\mathfrak{m}):=\mathcal{L}(u)\!\!=\!\!\mathcal{L}(v)\wedge(\forall \tilde{v}\in \Gamma_{2}(v)\cap\mathfrak{R}(\mathfrak{m}):(u,\mathfrak{m}^{-1}(\tilde{v}))\in E_{1})\wedge
-\ \cap\ M_{large}(s):\\(Pair(M(s),\tilde{v}),u)\in E_{small})\wedge
+(\forall \tilde{u}\in \Gamma_{1}(u)
-(\forall \tilde{u}\in \Gamma_{small}(u)
+\cap \mathfrak{D}(\mathfrak{m}):(v,\mathfrak{m}(\tilde{u}))\in E_{2})$ is a
-\ \cap\ M_{small}(s):(v,Pair(M(s),\tilde{u}))\in E_{large})$ is a
 consistency function in the case of $IND$.
 \end{claim}
 \subsection{Cutting rules}
-$Cut_{PT}(p,M(s))$ is defined by a collection of efficiently
+$Cut_{PT}(p,\mathfrak{m})$ is defined by a collection of efficiently
-verifiable conditions. The requirement is that $Cut_{PT}(p,M(s))$ can
+verifiable conditions. The requirement is that $Cut_{PT}(p,\mathfrak{m})$ can
-be true only if it is impossible to extended $M(s)\cup \{p\}$ to a
+be true only if it is impossible to extend $extend(\mathfrak{m},p)$ to a
 whole mapping.
 As an example, the cutting function of induced subgraph isomorphism is presented.
 \begin{notation}
-Let $\mathbf{\tilde{T}_{small}}(s):=(V_{small}\backslash
+Let $\mathbf{\tilde{T}_{1}}(\mathfrak{m}):=(V_{1}\backslash
-M_{small}(s))\backslash T_{small}(s)$, and
+\mathfrak{D}(\mathfrak{m}))\backslash T_{1}(\mathfrak{m})$, and
-\\ $\mathbf{\tilde{T}_{large}}(s):=(V_{large}\backslash
+\\ $\mathbf{\tilde{T}_{2}}(\mathfrak{m}):=(V_{2}\backslash
-M_{large}(s))\backslash T_{large}(s)$.
+\mathfrak{R}(\mathfrak{m}))\backslash T_{2}(\mathfrak{m})$.
 \end{notation}
 \begin{claim}
-$Cut_{IND}((u,v),M(s)):= |\Gamma_{large} (v)\ \cap\ T_{large}(s)| <
+$Cut_{IND}((u,v),\mathfrak{m}):= |\Gamma_{2} (v)\ \cap\ T_{2}(\mathfrak{m})| <
-|\Gamma_{small} (u)\ \cap\ T_{small}(s)| \vee |\Gamma_{large}(v)\cap
+|\Gamma_{1} (u)\ \cap\ T_{1}(\mathfrak{m})| \vee |\Gamma_{2}(v)\cap
-\tilde{T}_{large}(s)| < |\Gamma_{small}(u)\cap
+\tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}(u)\cap
-\tilde{T}_{small}(s)|$ is a cutting function by $IND$.
+\tilde{T}_{1}(\mathfrak{m})|$ is a cutting function by $IND$.
 \end{claim}
 \section{The VF2++ Algorithm}
 Although any total ordering relation makes the search space of VF2 a
 tree, its choice turns out to dramatically influence the number of
 visited states. The goal is to determine an efficient one as quickly
 Note that a weaker version of the cutting rules and the more efficient
 candidate set calculating were described in \cite{VF2Plus}, too.
 It should be noted that all the methods described in this section are
-extendable to handle directed graphs and edge labels as well.
+extendable to handle directed graphs and edge labels as well.\newline
 The basic ideas and the detailed description of VF2++ are provided in
 the following.
-\subsection{Preparations}
-\begin{claim}
-\label{claim:claimCoverFromLeft}
-The total ordering relation uniquely determines a node order, in which
-the nodes of $V_{small}$ will be covered by VF2. From the point of
-view of the matching procedure, this means, that always the same node
-of $G_{small}$ will be covered on the d-th level.
-\end{claim}
-\begin{definition}
-An order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{small}|)})$ of
-$V_{small}$ is \textbf{matching order} if exists $\prec$ total
-ordering relation, s.t. the VF2 with $\prec$ on the d-th level finds
-pair for $u_{\sigma(d)}$ for all $d\in\{1,..,|V_{small}|\}$.
-\end{definition}
-\begin{claim}\label{claim:MOclaim}
-A total ordering is matching order iff the nodes of every component
-form an interval in the node sequence, and every node connects to a
-previous node in its component except the first node of each component.
-\end{claim}
-To summing up, a total ordering always uniquely determines a matching
-order, and every matching order can be determined by a total ordering,
-however, more than one different total orderings may determine the
-same matching order.
-\subsection{Idea behind the algorithm}
 The goal is to find a matching order in which the algorithm is able to
 recognize inconsistency or prune the infeasible branches on the
 highest levels and goes deep only if it is needed.
 \begin{notation}
-Let $\mathbf{Conn_{H}(u)}:=|\Gamma_{small}(u)\cap H\}|$, that is the
+Let $\mathbf{Conn_{H}(u)}:=|\Gamma_{1}(u)\cap H\}|$, that is the
-number of neighbours of u which are in H, where $u\in V_{small} $ and
+number of neighbours of u which are in H, where $u\in V_{1} $ and
-$H\subseteq V_{small}$.
+$H\subseteq V_{1}$.
 \end{notation}
-The principal question is the following. Suppose a state $s$ is
+The principal question is the following. Suppose a mapping $\mathfrak{m}$ is
-given. For which node of $T_{small}(s)$ is the hardest to find a
+given. For which node of $T_{1}(\mathfrak{m})$ is the hardest to find a
-consistent pair in $G_{large}$? The more covered neighbours a node in
+consistent pair in $G_{2}$? The more covered neighbours a node in
-$T_{small}(s)$ has --- i.e. the largest $Conn_{M_{small}(s)}$ it has
+$T_{1}(\mathfrak{m})$ has --- i.e. the largest $Conn_{\mathfrak{D}(\mathfrak{m})}$ it has
 ---, the more rarely satisfiable consistency constraints for its pair
 are given.
 In biology, most of the graphs are sparse, thus several nodes in
-$T_{small}(s)$ may have the same $Conn_{M_{small}(s)}$, which makes
+$T_{1}(\mathfrak{m})$ may have the same $Conn_{\mathfrak{D}(\mathfrak{m})}$, which makes
 reasonable to define a secondary and a tertiary order between them.
 The observation above proves itself to be as determining, that the
 secondary ordering prefers nodes with the most uncovered neighbours
-among which have the same $Conn_{M_{small}(s)}$ to increase
+among which have the same $Conn_{\mathfrak{D}(\mathfrak{m})}$ to increase
-$Conn_{M_{small}(s)}$ of uncovered nodes so much, as possible.  The
+$Conn_{\mathfrak{D}(\mathfrak{m})}$ of uncovered nodes so much, as possible.  The
 tertiary ordering prefers nodes having the rarest uncovered labels.
 Note that the secondary ordering is the same as the ordering by $deg$,
 which is a static data in front of the above used.
 These rules can easily result in a matching order which contains the
 nodes of a long path successively, whose nodes may have low $Conn$ and
-is easily matchable into $G_{large}$. To avoid that, a BFS order is
+is easily matchable into $G_{2}$. To avoid that, a BFS order is
 used, which provides the shortest possible paths.
 \newline
 In the following, some examples on which the VF2 may be slow are
 described, although they are easily solvable by using a proper
 matching order.
 \begin{example}
-Suppose $G_{small}$ can be mapped into $G_{large}$ in many ways
+Suppose $G_{1}$ can be mapped into $G_{2}$ in many ways
-without node labels. Let $u\in V_{small}$ and $v\in V_{large}$.
+without node labels. Let $u\in V_{1}$ and $v\in V_{2}$.
 \newline
-$lab(u):=black$
+$\mathcal{L}(u):=black$
 \newline
-$lab(v):=black$
+$\mathcal{L}(v):=black$
 \newline
-$lab(\tilde{u}):=red \ \forall \tilde{u}\in (V_{small}\backslash
+$\mathcal{L}(\tilde{u}):=red \ \forall \tilde{u}\in (V_{1}\backslash
 \{u\})$
 \newline
-$lab(\tilde{v}):=red \ \forall \tilde{v}\in (V_{large}\backslash
+$\mathcal{L}(\tilde{v}):=red \ \forall \tilde{v}\in (V_{2}\backslash
 \{v\})$
 \newline
-Now, any mapping by the node label $lab$ must contain $(u,v)$, since
+Now, any mapping by $\mathcal{L}$ must contain $(u,v)$, since
-$u$ is black and no node in $V_{large}$ has a black label except
+$u$ is black and no node in $V_{2}$ has a black label except
 $v$. If unfortunately $u$ were the last node which will get covered,
 VF2 would check only in the last steps, whether $u$ can be matched to
 $v$.
 \newline
 However, had $u$ been the first matched node, u would have been
 matched immediately to v, so all the mappings would have been
 precluded in which node labels can not correspond.
 \end{example}
 \begin{example}
-Suppose there is no node label given, $G_{small}$ is a small graph and
+Suppose there is no node label given, $G_{1}$ is a small graph and
-can not be mapped into $G_{large}$ and $u\in V_{small}$.
+can not be mapped into $G_{2}$ and $u\in V_{1}$.
 \newline
-Let $G'_{small}:=(V_{small}\cup
+Let $G'_{1}:=(V_{1}\cup
-\{u'_{1},u'_{2},..,u'_{k}\},E_{small}\cup
+\{u'_{1},u'_{2},..,u'_{k}\},E_{1}\cup
 \{(u,u'_{1}),(u'_{1},u'_{2}),..,(u'_{k-1},u'_{k})\})$, that is,
-$G'_{small}$ is $G_{small}\cup \{ a\ k$ long path, which is disjoint
+$G'_{1}$ is $G_{1}\cup \{ a\ k$ long path, which is disjoint
-from $G_{small}$ and one of its starting points is connected to $u\in
+from $G_{1}$ and one of its starting points is connected to $u\in
-V_{small}\}$.
+V_{1}\}$.
 \newline
-Is there a subgraph of $G_{large}$, which is isomorph with
+Is there a subgraph of $G_{2}$, which is isomorph with
-$G'_{small}$?
+$G'_{1}$?
 \newline
 If unfortunately the nodes of the path were the first $k$ nodes in the
 matching order, the algorithm would iterate through all the possible k
-long paths in $G_{large}$, and it would recognize that no path can be
+long paths in $G_{2}$, and it would recognize that no path can be
-extended to $G'_{small}$.
+extended to $G'_{1}$.
 \newline
-However, had it started by the matching of $G_{small}$, it would not
+However, had it started by the matching of $G_{1}$, it would not
 have matched any nodes of the path.
 \end{example}
 These examples may look artificial, but the same problems also appear
 in real-world instances, even though in a less obvious way.
+\subsection{Preparations}
+\begin{claim}
+\label{claim:claimCoverFromLeft}
+The total ordering relation uniquely determines a node order, in which
+the nodes of $V_{1}$ will be covered by VF2. From the point of
+view of the matching procedure, this means, that always the same node
+of $G_{1}$ will be covered on the d-th level.
+\end{claim}
+\begin{definition}
+An order $(u_{\sigma(1)},u_{\sigma(2)},..,u_{\sigma(|V_{1}|)})$ of
+$V_{1}$ is \textbf{matching order} if exists $\prec$ total
+ordering relation, s.t. the VF2 with $\prec$ on the d-th level finds
+pair for $u_{\sigma(d)}$ for all $d\in\{1,..,|V_{1}|\}$.
+\end{definition}
+\begin{claim}\label{claim:MOclaim}
+A total ordering is matching order iff the nodes of every component
+form an interval in the node sequence, and every node connects to a
+previous node in its component except the first node of each component.
+\end{claim}
+To summing up, a total ordering always uniquely determines a matching
+order, and every matching order can be determined by a total ordering,
+however, more than one different total orderings may determine the
+same matching order.
 \subsection{Total ordering}
-Instead of the total ordering relation, the matching order will be
+The matching order will be searched directly.
-searched directly.
 \begin{notation}
-Let \textbf{F$_\mathcal{M}$(l)}$:=|\{v\in V_{large} :
+Let \textbf{F$_\mathcal{M}$(l)}$:=|\{v\in V_{2} :
-l=lab(v)\}|-|\{u\in V_{small}\backslash \mathcal{M} : l=lab(u)\}|$ ,
+l=\mathcal{L}(v)\}|-|\{u\in V_{1}\backslash \mathcal{M} : l=\mathcal{L}(u)\}|$ ,
-where $l$ is a label and $\mathcal{M}\subseteq V_{small}$.
+where $l$ is a label and $\mathcal{M}\subseteq V_{1}$.
 \end{notation}
 \begin{definition}Let $\mathbf{arg\ max}_{f}(S) :=\{u\in S : f(u)=max_{v\in S}\{f(v)\}\}$ and $\mathbf{arg\ min}_{f}(S) := arg\ max_{-f}(S)$, where $S$ is a finite set and $f:S\longrightarrow \mathbb{R}$.
 \end{definition}
 \algtext*{EndWhile}
 \algtext*{EndFor}
 \caption{\hspace{0.5cm}$The\ method\ of\ VF2++\ for\ determining\ the\ node\ order$}\label{alg:VF2PPPseu}
 \begin{algorithmic}[1]
 \Procedure{VF2++order}{} \State $\mathcal{M}$ := $\emptyset$
-\Comment{matching order} \While{$V_{small}\backslash \mathcal{M}
+\Comment{matching order} \While{$V_{1}\backslash \mathcal{M}
 \neq\emptyset$} \State $r\in$ arg max$_{deg}$ (arg
-min$_{F_\mathcal{M}\circ lab}(V_{small}\backslash
+min$_{F_\mathcal{M}\circ \mathcal{L}}(V_{1}\backslash
 \mathcal{M})$)\label{alg:findMin} \State Compute $T$, a BFS tree with
 root node $r$.  \For{$d=0,1,...,depth(T)$} \State $V_d$:=nodes of the
 $d$-th level \State Process $V_d$ \Comment{See Algorithm
 \ref{alg:VF2PPProcess1}} \EndFor
 \EndWhile \EndProcedure
 \algtext*{EndProcedure}%ne nyomtasson ..
 \algtext*{EndWhile}
 \caption{\hspace{.5cm}$The\ method\ for\ processing\ a\ level\ of\ the\ BFS\ tree$}\label{alg:VF2PPProcess1}
 \begin{algorithmic}[1]
 \Procedure{VF2++ProcessLevel}{$V_{d}$} \While{$V_d\neq\emptyset$}
-\State $m\in$ arg min$_{F_\mathcal{M}\circ\ lab}($ arg max$_{deg}($arg
+\State $m\in$ arg min$_{F_\mathcal{M}\circ\ \mathcal{L}}($ arg max$_{deg}($arg
 max$_{Conn_{\mathcal{M}}}(V_{d})))$ \State $V_d:=V_d\backslash m$
 \State Append node $m$ to the end of $\mathcal{M}$ \State Refresh
 $F_\mathcal{M}$ \EndWhile \EndProcedure
 \end{algorithmic}
 \end{algorithm}
 Algorithm~\ref{alg:VF2PPPseu} is a high level description of the
 matching order procedure of VF2++. It computes a BFS tree for each
-component in ascending order of their rarest $lab$ and largest $deg$,
+component in ascending order of their rarest node labels and largest $deg$,
 whose root vertex is the component's minimal
 node. Algorithm~\ref{alg:VF2PPProcess1} is a method to process a level of the BFS tree, which appends the nodes of the current level in descending
 lexicographic order by $(Conn_{\mathcal{M}},deg,-F_\mathcal{M})$ separately
 to $\mathcal{M}$, and refreshes $F_\mathcal{M}$ immediately.
 provides a matching order.
 \subsection{Cutting rules}
 \label{VF2PPCuttingRules}
-This section presents the cutting rule of VF2++ in the case of IND, which is improved
+This section presents the cutting rules of VF2++, which are improved by using extra information coming from the node labels.
-by using extra information coming from the node labels. For other problem types, the rules can be formulated similarly.
 \begin{notation}
-Let $\mathbf{\Gamma_{small}^{l}(u)}:=\{\tilde{u} : lab(\tilde{u})=l
+Let $\mathbf{\Gamma_{1}^{l}(u)}:=\{\tilde{u} : \mathcal{L}(\tilde{u})=l
-\wedge \tilde{u}\in \Gamma_{small} (u)\}$ and
+\wedge \tilde{u}\in \Gamma_{1} (u)\}$ and
-$\mathbf{\Gamma_{large}^{l}(v)}:=\{\tilde{v} : lab(\tilde{v})=l \wedge
+$\mathbf{\Gamma_{2}^{l}(v)}:=\{\tilde{v} : \mathcal{L}(\tilde{v})=l \wedge
-\tilde{v}\in \Gamma_{large} (v)\}$, where $u\in V_{small}$, $v\in
+\tilde{v}\in \Gamma_{2} (v)\}$, where $u\in V_{1}$, $v\in
-V_{large}$ and $l$ is a label.
+V_{2}$ and $l$ is a label.
 \end{notation}
+\subsubsection{Induced subgraph isomorphism}
 \begin{claim}
-\[LabCut_{IND}((u,v),M(s))\!:=\!\!\!\!\!\bigvee_{l\ is\ label}\!\!\!\!\!\!\!|\Gamma_{large}^{l} (v) \cap T_{large}(s)|\!<\!|\Gamma_{small}^{l}(u)\cap T_{small}(s)|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{large}^{l}(v)\cap \tilde{T}_{large}(s)| < |\Gamma_{small}^{l}(u)\cap \tilde{T}_{small}(s)|\] is a cutting function by IND.
+\[LabCut_{IND}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!<\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] is a cutting function by IND.
 \end{claim}
+\subsubsection{Graph isomorphism}
+\begin{claim}
-\subsection{Implementation details}
+\[LabCut_{ISO}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!\neq\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\  \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| \neq |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] is a cutting function by ISO.
+\end{claim}
+\subsubsection{Subgraph isomorphism}
+\begin{claim}
+\[LabCut_{SU\!B}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!<\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\] is a cutting function by SUB.
+\end{claim}
+\section{Implementation details}
 This section provides a detailed summary of an efficient
 implementation of VF2++.
 \subsubsection{Storing a mapping}
 After fixing an arbitrary node order ($u_0, u_1, ..,
-u_{|G_{small}|-1}$) of $G_{small}$, an array $M$ is usable to store
+u_{|G_{1}|-1}$) of $G_{1}$, an array $M$ is usable to store
 the current mapping in the following way.
 \[
 M[i] =
 \begin{cases}
-v & if\ (u_i,v)\ is\ in\ the\ mapping\\ INVALID &
+v & if\ (u_i,v)\ is\ in\ the\ mapping\\ INV\!ALI\!D &
 if\ no\ node\ has\ been\ mapped\ to\ u_i,
 \end{cases}
 \]
-where $i\in\{0,1, ..,|G_{small}|-1\}$, $v\in V_{large}$ and $INVALID$
+where $i\in\{0,1, ..,|G_{1}|-1\}$, $v\in V_{2}$ and $INV\!ALI\!D$
 means "no node".
 \subsubsection{Avoiding the recurrence}
 The recursion of Algorithm~\ref{alg:VF2Pseu} can be realized
 as a \textit{while loop}, which has a loop counter $depth$ denoting the
 all-time depth of the recursion. Fixing a matching order, let $M$
 denote the array storing the all-time mapping. Based on Claim~\ref{claim:claimCoverFromLeft},
-$M$ is $INVALID$ from index $depth$+1 and not $INVALID$ before
+$M$ is $INV\!ALI\!D$ from index $depth$+1 and not $INV\!ALI\!D$ before
 $depth$. $M[depth]$ changes
 while the state is being processed, but the property is held before
 both stepping back to a predecessor state and exploring a successor
 state.
 section for details.
 \subsubsection{Calculating the candidates for a node}
 Being aware of Claim~\ref{claim:claimCoverFromLeft}, the
 task is not to maintain the candidate set, but to generate the
-candidate nodes in $G_{large}$ for a given node $u\in V_{small}$.  In
+candidate nodes in $G_{2}$ for a given node $u\in V_{1}$.  In
 case of any of the three problem types and a mapping $M$ if a node $v\in
-V_{large}$ is a potential pair of $u\in V_{small}$, then $\forall
+V_{2}$ is a potential pair of $u\in V_{1}$, then $\forall
-u'\in V_{small} : (u,u')\in
+u'\in V_{1} : (u,u')\in
-E_{small}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in
+E_{1}\ and\ u'\ is\ covered\ by\ M\ \Rightarrow (v,Pair(M,u'))\in
-E_{large}$. That is, each covered neighbour of $u$ has to be mapped to
+E_{2}$. That is, each covered neighbour of $u$ has to be mapped to
 a covered neighbour of $v$.
 Having said that, an algorithm running in $\Theta(deg)$ time is
 describable if there exists a covered node in the component containing
 $u$, and a linear one otherwise.
 For using lookup tables, the node labels are associated with the
 numbers $\{0,1,..,|K|-1\}$, where $K$ is the set of the labels. It
 enables $F_\mathcal{M}$ to be stored in an array. At first, the node order
 $\mathcal{M}=\emptyset$, so $F_\mathcal{M}[i]$ is the number of nodes
-in $V_{small}$ having label i, which is easy to compute in
+in $V_{1}$ having label $i$, which is easy to compute in
-$\Theta(|V_{small}|)$ steps.
+$\Theta(|V_{1}|)$ steps.
-Representing $\mathcal{M}\subseteq V_{small}$ as an array of
+Representing $\mathcal{M}\subseteq V_{1}$ as an array of
-size $|V_{small}|$, both the computation of the BFS tree, and processing its levels by Algorithm~\ref{alg:VF2PPProcess1} can be done inplace by swapping nodes.
+size $|V_{1}|$, both the computation of the BFS tree, and processing its levels by Algorithm~\ref{alg:VF2PPProcess1} can be done inplace by swapping nodes.
 \subsubsection{Cutting rules}
 In Section~\ref{VF2PPCuttingRules}, the cutting rules were
-described using the sets $T_{small}$, $T_{large}$, $\tilde T_{small}$
+described using the sets $T_{1}$, $T_{2}$, $\tilde T_{1}$
-and $\tilde T_{large}$, which are dependent on the all-time mapping
+and $\tilde T_{2}$, which are dependent on the all-time mapping
 (i.e. on the all-time state). The aim is to check the labeled cutting
 rules of VF2++ in $\Theta(deg)$ time.
 Firstly, suppose that these four sets are given in such a way, that
 checking whether a node is in a certain set takes constant time,
 e.g. they are given by their 0-1 characteristic vectors. Let $L$ be an
 initially zero integer lookup table of size $|K|$. After incrementing
-$L[lab(u')]$ for all $u'\in \Gamma_{small}(u) \cap T_{small}(s)$ and
+$L[\mathcal{L}(u')]$ for all $u'\in \Gamma_{1}(u) \cap T_{1}(\mathfrak{m})$ and
-decrementing $L[lab(v')]$ for all $v'\in\Gamma_{large} (v) \cap
+decrementing $L[\mathcal{L}(v')]$ for all $v'\in\Gamma_{2} (v) \cap
-T_{large}(s)$, the first part of the cutting rules is checkable in
+T_{2}(s)$, the first part of the cutting rules is checkable in
 $\Theta(deg)$ time by considering the proper signs of $L$. Setting $L$
 to zero takes $\Theta(deg)$ time again, which makes it possible to use
 the same table through the whole algorithm. The second part of the
 cutting rules can be verified using the same method with $\tilde
-T_{small}$ and $\tilde T_{large}$ instead of $T_{small}$ and
+T_{1}$ and $\tilde T_{2}$ instead of $T_{1}$ and
-$T_{large}$. Thus, the overall complexity is $\Theta(deg)$.
+$T_{2}$. Thus, the overall complexity is $\Theta(deg)$.
-An other integer lookup table storing the number of covered neighbours
+Another integer lookup table storing the number of covered neighbours
-of each node in $G_{large}$ gives all the information about the sets
+of each node in $G_{2}$ gives all the information about the sets
-$T_{large}$ and $\tilde T_{large}$, which is maintainable in
+$T_{2}$ and $\tilde T_{2}$, which is maintainable in
 $\Theta(deg)$ time when a pair is added or substracted by incrementing
 or decrementing the proper indices. A further improvement is that the
-values of $L[lab(u')]$ in case of checking $u$ is dependent only on
+values of $L[\mathcal{L}(u')]$ in case of checking $u$ are dependent only on
-$u$, i.e. on the size of the mapping, so for each $u\in V_{small}$ an
+$u$, i.e. on the size of the mapping, so for each $u\in V_{1}$ an
 array of pairs (label, number of such labels) can be stored to skip
 the maintaining operations. Note that these arrays are at most of size
-$deg$. Skipping this trick, the number of covered neighbours has to be
+$deg$.
-stored for each node of $G_{small}$ as well to get the sets
-$T_{small}$ and $\tilde T_{small}$.
+Using similar techniques, the consistency function can be evaluated in
-Using similar tricks, the consistency function can be evaluated in
 $\Theta(deg)$ steps, as well.
-\section{The VF2 Plus Algorithm}
-The VF2 Plus algorithm is a recently improved version of VF2. It was
-compared with the state of the art algorithms in \cite{VF2Plus} and
-has proved itself to be competitive with RI, the best algorithm on
-biological graphs.  \\ A short summary of VF2 Plus follows, which uses
-the notation and the conventions of the original paper.
-\subsection{Ordering procedure}
-VF2 Plus uses a sorting procedure that prefers nodes in $V_{small}$
-with the lowest probability to find a pair in $V_{small}$ and the
-highest number of connections with the nodes already sorted by the
-algorithm.
-\begin{definition}
-$(u,v)$ is a \textbf{feasible pair} if $lab(u)=lab(v)$ and
-$deg(u)\leq deg(v)$, where $u\in{V_{small}}$ and $ v\in{V_{large}}$.
-\end{definition}
-$P_{lab}(L):=$ a priori probability to find a node with label $L$ in
-$V_{large}$
-\newline
-$P_{deg}(d):=$ a priori probability to find a node with degree $d$ in
-$V_{large}$
-\newline
-$P(u):=P_{lab}(L)*\bigcup_{d'>d}P_{deg}(d')$\\ $M$ is the set of
-already sorted nodes, $T$ is the set of nodes candidate to be
-selected, and $degreeM$ of a node is the number of its neighbours in
-$M$.
-\begin{algorithm}
-\algtext*{EndIf}%ne nyomtasson end if-et
-\algtext*{EndFor}%nenyomtasson ..
-\algtext*{EndProcedure}%ne nyomtasson ..
-\algtext*{EndWhile}
-\caption{}\label{alg:VF2PlusPseu}
-\begin{algorithmic}[1]
-\Procedure{VF2 Plus order}{} \State Select the node with the lowest
-$P$.  \If {more nodes share the same $P$} \State select the one with
-maximum degree \EndIf \If {more nodes share the same $P$ and have the
-max degree} \State select the first \EndIf \State Put the selected
-node in the set $M$. \label{alg:putIn} \State Put all its unsorted
-neighbours in the set $T$.  \If {$M\neq V_{small}$} \State From set
-$T$ select the node with maximum $degreeM$.  \If {more nodes have
-maximum $degreeM$} \State Select the one with the lowest $P$ \EndIf
-\If {more nodes have maximum $degreeM$ and $P$} \State Select the
-first.  \EndIf \State \textbf{goto \ref{alg:putIn}.}  \EndIf
-\EndProcedure
-\end{algorithmic}
-\end{algorithm}
-Using these notations, Algorithm~\ref{alg:VF2PlusPseu}
-provides the description of the sorting procedure.
-Note that $P(u)$ is not the exact probability of finding a consistent
-pair for $u$ by choosing a node of $V_{large}$ randomly, since
-$P_{lab}$ and $P_{deg}$ are not independent, though calculating the
-real probability would take quadratic time, which may be reduced by
-using fittingly lookup tables.
 \section{Experimental results}
-This section compares the performance of VF2++ and VF2 Plus. Both
+This section compares the performance of VF2++ and VF2 Plus. According to
-algorithms have run faster with orders of magnitude than VF2, thus its
+our experience, both algorithms run faster than VF2 with orders of
-inclusion was not reasonable.
+magnitude, thus its inclusion was not reasonable.
+The algorithms were implemented in C++ using the open source
+LEMON graph and network optimization library\cite{LEMON}. The test were carried out on a linux based system with an Intel i7 X980 3.33 GHz CPU and 6 GB of RAM.
 \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
 and an average degree of less than 3. The protein dataset contains
 graphs having 500-10 000 nodes and an average degree of 4, while the
 contact map dataset contains graphs with 150-800 nodes and an average
 degree of 20.  \\
-In the following, the induced subgraph isomorphism and the graph
+In the following, both the induced subgraph isomorphism and the graph
 isomorphism will be examined.
 This dataset provides graph pairs, between which all the induced subgraph isomorphisms have to be found. For runtime results, please see Figure~\ref{fig:bioIND}.
 In an other experiment, the nodes of each graph in the database had been
 graph, that of the small graph dramatically influences the hardness of
 a given problem too, so the overall picture is provided by examining
 small graphs of various size.
 For each chart, a number $0<\rho< 1$ has been fixed, and the following
-has been executed 150 times. Generating a large graph $G_{large}$ of an average degree of $\delta$,
+has been executed 150 times. Generating a large graph $G_{2}$ of an average degree of $\delta$,
-choose 10 of its induced subgraphs having $\rho\ |V_{large}|$ nodes,
+choose 10 of its induced subgraphs having $\rho\ |V_{2}|$ nodes,
 and for all the 10 subgraphs find a mapping by using both the graph
 matching algorithms.  The $\delta = 5, 10, 35$ and $\rho = 0.05, 0.1,
 0.3, 0.6, 0.8, 0.95$ cases have been examined, see
 Figure~\ref{fig:randIND5}, \ref{fig:randIND10} and
 \ref{fig:randIND35}.
 \section{Conclusion}
-In this paper, after providing a short summary of the recent
+This paper presented VF2++, a new graph matching algorithm based on VF2, called VF2++, and analyzed it from a practical viewpoint.
-algorithms, a new graph matching algorithm based on VF2, called VF2++,
-has been presented and analyzed from a practical viewpoint.
 Recognizing the importance of the node order and determining an
 efficient one, VF2++ is able to match graphs of thousands of nodes in
 near practically linear time including preprocessing. In addition to
 the proper order, VF2++ uses more efficient consistency and cutting
 rules which are easy to compute and make the algorithm able to prune
 most of the unfruitful branches without going astray.
 In order to show the efficiency of the new method, it has been
-compared to VF2 Plus, which is the best concurrent algorithm based on
+compared to VF2 Plus\cite{VF2Plus}, which is the best contemporary algorithm.
-\cite{VF2Plus}.
+.
 The experiments show that VF2++ consistently outperforms VF2 Plus on
 biological graphs. It seems to be asymptotically faster on protein and
 on contact map graphs in the case of induced subgraph isomorphism,
 while in the case of graph isomorphism, it has definitely better
 asymptotic behaviour on protein graphs.
 Regarding random sparse graphs, not only has VF2++ proved itself to be
-faster than VF2 Plus, but it has a practically linear behaviour both
+faster than VF2 Plus, but it also has a practically linear behaviour both
-in the case of induced subgraph- and graph isomorphism, as well.
+in the case of induced subgraph- and graph isomorphism.
 %% The Appendices part is started with the command \appendix;
 %% appendix sections are then done as normal sections

changeset 19	b9a8744c5efc
parent 17	909bcb203f25
child 20	80d56dee41d9