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\journal{Discrete Applied Mathematics}

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\title{Improved Algorithms for Matching Biological Graphs}

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\author{Alp{\'a}r J{\"u}ttner and P{\'e}ter Madarasi}

\address{Dept of Operations Research, ELTE}

\begin{abstract}

Subgraph isomorphism is a well-known NP-Complete problem, while its

special case, the graph isomorphism problem is one of the few problems

in NP neither known to be in P nor NP-Complete. Their appearance in

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.

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 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

efficient - nevertheless still easier to compute - cutting rules

reduces the chance of going astray even further.

In addition to the usual subgraph isomorphism, specialized versions

for induced subgraph isomorphism and for graph isomorphism are

presented. VF2++ has gained a runtime improvement of one order of

magnitude respecting induced subgraph isomorphism and a better

asymptotical behaviour in the case of graph isomorphism problem.

After having provided the description of VF2++, in order to evaluate

its effectiveness, an extensive comparison to the contemporary other

algorithms is shown, using a wide range of inputs, including both real

life biological and chemical datasets and standard randomly generated

graph series.

The work was motivated and sponsored by QuantumBio Inc., and all the

developed algorithms are available as the part of the open source

LEMON graph and network optimization library

(http://lemon.cs.elte.hu).

\end{abstract}

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\section{Introduction}

\label{sec:intro}

In the last decades, combinatorial structures, and especially graphs

have been considered with ever increasing interest, and applied to the

solution of several new and revised questions.  The expressiveness,

the simplicity and the studiedness of graphs make them practical for

modelling and appear constantly in several seemingly independent

fields, such as bioinformatics and chemistry.

Complex biological systems arise from the interaction and cooperation

of plenty of molecular components. Getting acquainted with such

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 this way.

Many chemical and biological structures can easily be modeled

as graphs, for instance, a molecular structure can be

considered as a graph, whose nodes correspond to atoms and whose

edges to chemical bonds. The similarity and dissimilarity of

objects corresponding to nodes are incorporated to the model

by \emph{node labels}. Understanding such networks basically

requires finding specific subgraphs, thus calls for efficient

graph matching algorithms.

Other real-world fields related to some

variants of graph matching include pattern recognition

and machine vision \cite{HorstBunkeApplications}, symbol recognition

\cite{CordellaVentoSymbolRecognition}, face identification

\cite{JianzhuangYongFaceIdentification}.  \\

Subgraph and induced subgraph matching problems are known to be

NP-Complete\cite{SubgraphNPC}, while the graph isomorphism problem is

one of the few problems in NP neither known to be in P nor

NP-Complete. Although polynomial time isomorphism algorithms are known

for various graph classes, like trees and planar

graphs\cite{PlanarGraphIso}, bounded valence

graphs\cite{BondedDegGraphIso}, interval graphs\cite{IntervalGraphIso}

or permutation graphs\cite{PermGraphIso}, 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. Even though,

an overall polynomial behaviour is not expectable from such an

alternative, it may often have good practical performance, in fact,

it might be the best choice even on a graph class for which polynomial

algorithm is known.

The first practically usable approach was due to

\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 \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.

The \emph{LAD} algorithm\cite{Lad} uses a depth-first search

strategy and formulates the matching as a Constraint Satisfaction

Problem to prune the search tree. The constraints are that the mapping

has to be injective and edge-preserving, hence it is possible to

handle new matching types as well.

The \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

\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.

Meanwhile, another variant called \emph{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 formal description of the problems to be

solved.

\subsection{Definitions}

Throughout the paper $G_{1}=(V_{1}, E_{1})$ and

$G_{2}=(V_{2}, E_{2})$ denote two undirected graphs.

\begin{definition}

$\mathcal{L}: (V_{1}\cup V_{2}) \longrightarrow K$ 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 $\mathcal{L}(u)=\mathcal{L}(v)$.

\end{definition}

For the sake of simplicity, in this paper the graph, subgraph and induced subgraph isomorphisms are defined in a more general way.

\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}

$G_{1}$ is a \textbf{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\\

$\forall u,v \in{V_{1}} : (u,v)\in{E_{1}} \Rightarrow (\mathfrak{m}(u),\mathfrak{m}(v))\in E_{2}$

\end{center}

\end{definition}

\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

$G_{1}$ 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.

The \textbf{graph isomorphism problem} can be defined as induced

subgraph matching problem where the sizes of the two graphs are equal.

In addition to existence, it may be needed to show such a subgraph, or

it may be necessary to list all of them.

It should be noted that some authors misleadingly refer to the term

\emph{subgraph isomorphism problem} as an \emph{induced subgraph

  isomorphism problem}.

The following sections give the descriptions of VF2, VF2++, VF2 Plus

and a particular comparison.

\section{The VF2 Algorithm}

This algorithm is the basis of both the VF2++ and the VF2 Plus.  VF2

is able to handle all the variations mentioned in Section

  \ref{sec:CommProb}.  Although it can also handle directed graphs,

for the sake of simplicity, only the undirected case will be

discussed.

\subsection{Common notations}

\indent Assume $G_{1}$ is searched in $G_{2}$.  The following

definitions and notations will be used throughout the whole paper.

\begin{definition}

An injection $\mathfrak{m} : D \longrightarrow V_2$ is called (partial) \textbf{mapping}, where $D\subseteq V_1$.

\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 $\mathfrak{m}$ \textbf{covers} a node $u\in V_1\cup V_2$ if $u\in \mathfrak{D}(\mathfrak{m})\cup \mathfrak{R}(\mathfrak{m})$.

\end{definition}

\begin{definition}

A mapping $\mathfrak{m}$ is $\mathbf{whole\ mapping}$ if $\mathfrak{m}$ covers all the

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 $u\in V_1\setminus\mathfrak{D}(\mathfrak{m})$ and $v\in V_2\setminus\mathfrak{R}(\mathfrak{m})$, otherwise $extend(\mathfrak{m},(u,v))$ is undefined.

\end{definition}

\begin{notation}

Throughout the paper, $\mathbf{PT}$ denotes a generic problem type

which can be substituted by any of the $\mathbf{ISO}$, $\mathbf{SUB}$

and $\mathbf{IND}$ problems.

\end{notation}

\begin{definition}

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 a whole mapping $w$ satisfying the requirements of $PT$, for which $\mathfrak{m}$ is exactly $w$ restricted to $\mathfrak{D}(\mathfrak{m})$.

\end{definition}

\begin{definition} 

Let $\mathfrak{m}$ be a mapping. A logical function $\mathbf{Cut_{PT}}$ is a

\textbf{cutting function by } $\mathbf{PT}$ if the following

holds. $\mathbf{Cut_{PT}(\mathfrak{m})}$ is false if there exists a sequence of extend operations, which results in a whole mapping satisfying the requirements of $PT$.

\end{definition}

\begin{definition}

$\mathfrak{m}$ is said to be \textbf{consistent mapping by} $\mathbf{PT}$ if

  $Cons_{PT}(\mathfrak{m})$ is true.

\end{definition}

$Cons_{PT}$ and $Cut_{PT}$ will often be used in the following form.

\begin{notation}

Let $\mathbf{Cons_{PT}(p, \mathfrak{m})}:=Cons_{PT}(extend(\mathfrak{m},p))$, and

$\mathbf{Cut_{PT}(p, \mathfrak{m})}:=Cut_{PT}(extend(\mathfrak{m},p))$, where

$p\in{V_{1}\backslash\mathfrak{D}(\mathfrak{m}) \!\times\!V_{2}\backslash\mathfrak{R}(\mathfrak{m})}$.

\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.

Algorithm~\ref{alg:VF2Pseu} is a high level description of

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}{Mapping $\mathfrak{m}$, ProblemType $PT$}

  \If{$\mathfrak{m}$ covers

    $V_{1}$} \State Output($\mathfrak{m}$)

  \Else

  \State Compute the set $P_\mathfrak{m}$ of the pairs candidate for inclusion

  in $\mathfrak{m}$ \ForAll{$p\in{P_\mathfrak{m}}$} \If{Cons$_{PT}$($p,\mathfrak{m}$) $\wedge$

    $\neg$Cut$_{PT}$($p,\mathfrak{m}$)}

    \State \textbf{call}

  VF2($extend(\mathfrak{m},p)$, $PT$) \EndIf \EndFor \EndIf \EndProcedure

\end{algorithmic}

\end{algorithm}

For the current mapping $\mathfrak{m}$, the algorithm computes $P_\mathfrak{m}$, the set of

candidate node pairs for adding to the current mapping $\mathfrak{m}_s$.

For each pair $p$ in $P_\mathfrak{m}$, $Cons_{PT}(p,\mathfrak{m})$ and

$Cut_{PT}(p,\mathfrak{m})$ are evaluated. If the former is true and

the latter is false, the whole process is recursively applied to

$extend(\mathfrak{m},p)$. Otherwise, $extend(\mathfrak{m},p)$ is not consistent by $PT$, or it

can be proved that $\mathfrak{m}$ 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 the consistent whole mappings are reachable through

consistent mappings.

\end{claim}

Note that a mapping may be reached in exponentially many different ways, since the

order of extensions does not influence the nascent mapping.

However, one may observe

\begin{claim}

\label{claim:claimTotOrd}

Let $\prec$ be an arbitrary total ordering relation on $V_{1}$.  If

the algorithm ignores each $p=(u,v) \in P_\mathfrak{m}$, for which

\begin{center}

$\exists (\tilde{u},\tilde{v})\in P_\mathfrak{m}: \tilde{u} \prec u$,

\end{center}

then no mapping can be reached more than once, and each 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}

\label{candidateComputingVF2}

Let $P_\mathfrak{m}$ be the set of the candidate pairs for inclusion in $\mathfrak{m}$.

\begin{notation}

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

 $\mathbf{T_{2}(\mathfrak{m})} := \{v \in V_{2}\backslash\mathfrak{R}(\mathfrak{m}) : \exists\tilde{v}\in{\mathfrak{R}(\mathfrak{m}):(v,\tilde{v})\in E_{2}}\}$.

\end{notation}

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_\mathfrak{m}\!=\!

  \begin{cases} 

   T_{1}(\mathfrak{m})\times T_{2}(\mathfrak{m})&\hspace{-0.15cm}\text{if }

   T_{1}(\mathfrak{m})\!\neq\!\emptyset\ \text{and }T_{2}(\mathfrak{m})\!\neq

   \emptyset,\\ (V_{1}\!\setminus\!\mathfrak{D}(\mathfrak{m}))\!\times\!(V_{2}\!\setminus\!\mathfrak{R}(\mathfrak{m}))

   &\hspace{-0.15cm}\text{otherwise}.

  \end{cases}

\]

\subsection{Consistency}

Suppose $p=(u,v)$, where $u\in V_{1}$ and $v\in V_{2}$, $\mathfrak{m}$ is a consistent mapping by

$PT$. $Cons_{PT}(p,\mathfrak{m})$ checks whether

including pair $p$ into $\mathfrak{m}$ leads to a consistent mapping by $PT$.

For example, the consistency function of induced subgraph isomorphism is as follows.

\begin{notation}

Let $\mathbf{\Gamma_{1} (u)}:=\{\tilde{u}\in V_{1} :

(u,\tilde{u})\in E_{1}\}$, and $\mathbf{\Gamma_{2}

  (v)}:=\{\tilde{v}\in V_{2} : (v,\tilde{v})\in E_{2}\}$, where $u\in V_{1}$ and $v\in V_{2}$.

\end{notation}

$extend(\mathfrak{m},(u,v))$ is a consistent mapping by $IND$ $\Leftrightarrow

(\forall \tilde{u}\in \mathfrak{D}(\mathfrak{m}): (u,\tilde{u})\in E_{1}

\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),\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

  (\forall \tilde{u}\in \Gamma_{1}(u)

  \cap \mathfrak{D}(\mathfrak{m}):(v,\mathfrak{m}(\tilde{u}))\in E_{2})$ is a

  consistency function in the case of $IND$.

\end{claim}

\subsection{Cutting rules}

$Cut_{PT}(p,\mathfrak{m})$ is defined by a collection of efficiently

verifiable conditions. The requirement is that $Cut_{PT}(p,\mathfrak{m})$ can

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}_{1}}(\mathfrak{m}):=(V_{1}\backslash

\mathfrak{D}(\mathfrak{m}))\backslash T_{1}(\mathfrak{m})$, and

\\ $\mathbf{\tilde{T}_{2}}(\mathfrak{m}):=(V_{2}\backslash

\mathfrak{R}(\mathfrak{m}))\backslash T_{2}(\mathfrak{m})$.

\end{notation}

\begin{claim}

$Cut_{IND}((u,v),\mathfrak{m}):= |\Gamma_{2} (v)\ \cap\ T_{2}(\mathfrak{m})| <

  |\Gamma_{1} (u)\ \cap\ T_{1}(\mathfrak{m})| \vee |\Gamma_{2}(v)\cap

  \tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}(u)\cap

  \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

as possible.

The main reason for VF2++' superiority over VF2 is twofold. Firstly,

taking into account the structure and the node labeling of the graph,

VF2++ determines a state order in which most of the unfruitful

branches of the search space can be pruned immediately. Secondly,

introducing more efficient --- nevertheless still easier to compute

--- cutting rules reduces the chance of going astray even further.

In addition to the usual subgraph isomorphism, specialized versions

for induced subgraph isomorphism and for graph isomorphism have been

designed. VF2++ has gained a runtime improvement of one order of

magnitude respecting induced subgraph isomorphism and a better

asymptotical behaviour in the case of graph isomorphism problem.

Note that a weaker version of the cutting rules and the more efficient

candidate set calculating were described in \cite{VF2Plus}, too.

It should be noted that all the methods described in this section are

extendable to handle directed graphs and edge labels as well.\newline

The basic ideas and the detailed description of VF2++ are provided in

the following.

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 $\mathbf{Conn_{H}(u)}:=|\Gamma_{1}(u)\cap H\}|$, that is the

number of neighbours of u which are in H, where $u\in V_{1} $ and

$H\subseteq V_{1}$.

\end{notation}

The principal question is the following. Suppose a mapping $\mathfrak{m}$ is

given. For which node of $T_{1}(\mathfrak{m})$ is the hardest to find a

consistent pair in $G_{2}$? The more covered neighbours a node in

$T_{1}(\mathfrak{m})$ 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

$T_{1}(\mathfrak{m})$ 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 $Conn_{\mathfrak{D}(\mathfrak{m})}$ to increase

$Conn_{\mathfrak{D}(\mathfrak{m})}$ 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 $G_{2}$. 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 $G_{1}$ can be mapped into $G_{2}$ in many ways

without node labels. Let $u\in V_{1}$ and $v\in V_{2}$.

\newline

$\mathcal{L}(u):=black$

\newline

$\mathcal{L}(v):=black$

\newline

$\mathcal{L}(\tilde{u}):=red \ \forall \tilde{u}\in (V_{1}\backslash

\{u\})$

\newline

$\mathcal{L}(\tilde{v}):=red \ \forall \tilde{v}\in (V_{2}\backslash

\{v\})$

\newline

Now, any mapping by $\mathcal{L}$ must contain $(u,v)$, since

$u$ 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, $G_{1}$ is a small graph and

can not be mapped into $G_{2}$ and $u\in V_{1}$.

\newline

Let $G'_{1}:=(V_{1}\cup

\{u'_{1},u'_{2},..,u'_{k}\},E_{1}\cup

\{(u,u'_{1}),(u'_{1},u'_{2}),..,(u'_{k-1},u'_{k})\})$, that is,

$G'_{1}$ is $G_{1}\cup \{ a\ k$ long path, which is disjoint

from $G_{1}$ and one of its starting points is connected to $u\in

V_{1}\}$.

\newline

Is there a subgraph of $G_{2}$, which is isomorph with

$G'_{1}$?

\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 $G_{2}$, and it would recognize that no path can be

extended to $G'_{1}$.

\newline

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}

The matching order will be searched directly.

\begin{notation}

Let \textbf{F$_\mathcal{M}$(l)}$:=|\{v\in V_{2} :

l=\mathcal{L}(v)\}|-|\{u\in V_{1}\backslash \mathcal{M} : l=\mathcal{L}(u)\}|$ ,

where $l$ is a label and $\mathcal{M}\subseteq V_{1}$.

\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}

\begin{algorithm}

\algtext*{EndIf}

\algtext*{EndProcedure}

\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_{1}\backslash \mathcal{M}

  \neq\emptyset$} \State $r\in$ arg max$_{deg}$ (arg

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

\end{algorithmic}

\end{algorithm}

\begin{algorithm}

\algtext*{EndIf}

\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$_{F_\mathcal{M}\circ\ \mathcal{L}}($ 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 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.

Claim~\ref{claim:MOclaim} shows that Algorithm~\ref{alg:VF2PPPseu}

provides a matching order.

\subsection{Cutting rules}

\label{VF2PPCuttingRules}

This section presents the cutting rules of VF2++, which are improved by using extra information coming from the node labels.

\begin{notation}

Let $\mathbf{\Gamma_{1}^{l}(u)}:=\{\tilde{u} : \mathcal{L}(\tilde{u})=l

\wedge \tilde{u}\in \Gamma_{1} (u)\}$ and

$\mathbf{\Gamma_{2}^{l}(v)}:=\{\tilde{v} : \mathcal{L}(\tilde{v})=l \wedge

\tilde{v}\in \Gamma_{2} (v)\}$, where $u\in V_{1}$, $v\in

V_{2}$ and $l$ is a label.

\end{notation}

\subsubsection{Induced subgraph isomorphism}

\begin{claim}

\[LabCut_{IND}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!<\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\ \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| < |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] is a cutting function by IND.

\end{claim}

\subsubsection{Graph isomorphism}

\begin{claim}

\[LabCut_{ISO}((u,v),\mathfrak{m}):=\bigvee_{l\ is\ label}|\Gamma_{2}^{l} (v) \cap T_{2}(\mathfrak{m})|\!\neq\!|\Gamma_{1}^{l}(u)\cap T_{1}(\mathfrak{m})|\  \vee\]\[\bigvee_{l\ is\ label} \newline |\Gamma_{2}^{l}(v)\cap \tilde{T}_{2}(\mathfrak{m})| \neq |\Gamma_{1}^{l}(u)\cap \tilde{T}_{1}(\mathfrak{m})|\] 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_{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\\ INV\!ALI\!D &

   if\ no\ node\ has\ been\ mapped\ to\ u_i,

  \end{cases}

\]

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 $INV\!ALI\!D$ 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.

The necessary part of the candidate set is easily maintainable or

computable by following

Section~\ref{candidateComputingVF2}. A much faster method

has been designed for biological- and sparse graphs, see the next

section for details.

\subsubsection{Calculating the candidates for a node}

Being aware of Claim~\ref{claim:claimCoverFromLeft}, the

task is not to maintain the candidate set, but to generate the

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 $\mathfrak{m}$, if a node $v\in

V_{2}$ is a potential pair of $u\in V_{1}$, then $\forall

u'\in \mathfrak{D}(\mathfrak{m}) : (u,u')\in

E_{1}\Rightarrow (v,\mathfrak{m}(u'))\in

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.

\subsubsection{Determining the node order}

This section describes how the node order preprocessing method of

VF2++ can efficiently be implemented.

For using lookup tables, the node labels are associated with the

numbers $\{0,1,..,|K|-1\}$, where $K$ is the set of the labels. It

enables $F_\mathcal{M}$ to be stored in an array. At first, the node order

$\mathcal{M}=\emptyset$, so $F_\mathcal{M}[i]$ is the number of nodes

in $V_{1}$ having label $i$, which is easy to compute in

$\Theta(|V_{1}|)$ steps.

Representing $\mathcal{M}\subseteq V_{1}$ as an array of

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 $T_{1}$, $T_{2}$, $\tilde T_{1}$

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

$L[\mathcal{L}(u')]$ for all $u'\in \Gamma_{1}(u) \cap T_{1}(\mathfrak{m})$ and

decrementing $L[\mathcal{L}(v')]$ for all $v'\in\Gamma_{2} (v) \cap

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_{1}$ and $\tilde T_{2}$ instead of $T_{1}$ and

$T_{2}$. Thus, the overall complexity is $\Theta(deg)$.

Another integer lookup table storing the number of covered neighbours

of each node in $G_{2}$ gives all the information about the sets

$T_{2}$ 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 $L[\mathcal{L}(u')]$ in case of checking $u$ are dependent only on

$u$, i.e. on the size of the mapping, so for each $u\in V_{1}$ 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

$deg$.

Using similar techniques, the consistency function can be evaluated in

$\Theta(deg)$ steps, as well.

\section{Experimental results}

This section compares the performance of VF2++ and VF2 Plus. According to

our experience, both algorithms run faster than VF2 with orders of

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

Bank\cite{ProteinDataBank}.

The molecule dataset contains small graphs with less than 100 nodes

and an average degree of less than 3. The protein dataset contains

graphs having 500-10 000 nodes and an average degree of 4, while the

contact map dataset contains graphs with 150-800 nodes and an average

degree of 20.  \\

In the following, 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

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 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 almost in 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

    format/1000 sep = \thinspace}] %\addplot+[only marks] table

    {proteinsOrig.txt}; \addplot[mark=*,mark size=1.2pt,color=blue]

    table {Orig/Proteins.256.txt}; \addplot[mark=triangle*,mark

      size=1.8pt,color=red] table {VF2PPLabel/Proteins.256.txt};

  \end{axis}

  \end{tikzpicture}

\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}

\end{subfigure}

\end{center}

\vspace*{-0.5cm}

\caption{\normalsize{Induced subgraph isomorphism on biological graphs}}\label{fig:bioIND}

\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 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}