5种图同构算法的比较

A Performance Comparison of Five Algorithmsfor Graph Isomorphism

P. Foggia, C.Sansone, M. VentoDipartimento di Informatica e SistemisticaVia Claudio, 21 - I 80125 - Napoli, Italy{foggiapa, carlosan, vento}@unina.it

AbstractDespite the significant number of isomorphism algorithms presented in theliterature, till now no efforts have been done for characterizing their performance.Consequently, it is not clear how the behavior of those algorithms varies as thetype and the size of the graphs to be matched varies in case of real applications.In this paper we present a benchmarking activity for characterizing theperformance of a bunch of algorithms for exact graph isomorphism. To thispurpose we use a large database containing 10,000 couples of isomorphic graphswith different topologies (regular graphs, randomly connected graphs, boundedvalence graph), enriched with suitably modified versions of them for simulatingdistortions occurring in real cases. The size of the considered graphs ranges froma few nodes to about 1000 nodes.

1. IntroductionThe exact graph matching problem is of interest in a variety of different patternrecognition contexts. In fact, graphs are used to support structural descriptions aswell as for low level image representations.As it is well known, among the different types of graph matching algorithms,subgraph isomorphism is a NP-complete problem [10], while it is still an openquestion if also graph isomorphism is a NP-complete problem. So, timerequirements of brute force matching algorithms (either in case of isomorphism orsubgraph isomorphism) increase exponentially with the size of the input graphs,restricting the applicability of graph based techniques to problems implying graphswith a small number of nodes and edges.During the last decades significant research efforts have been devoted to improveperformances of the matching algorithms, both in terms of computational time andmemory requirements.Some algorithms reduce the computational complexity of the matching process byimposing topological restrictions on the graphs [1, 11, 14, 22]An alternative approach for reducing the matching complexity is that of using anadequate representation of the searching process and pruning unprofitable paths inthe search space. In this way, no restrictions must be imposed on the structure of theinput graphs and the obtained algorithms can be applied in more general cases.One of the pioneer papers ascribable to this area [6], proposes an isomorphismalgorithm which performs suitable transformations on the input graphs, in order tofind a different representation for which the matching is computationally moreconvenient. However, it has been shown [15] that the conjecture on which thismethod is based is not always true, so limiting the applicability of the algorithm.A procedure that significantly reduces the size of the search space is the backtrackingalgorithm proposed by Ullmann in [21]. This algorithm is devised for both graphisomorphism and subgraph isomorphism and is still today one of the most commonlyused for exact graph matching. This is confirmed by the fact that in Messmer [16] itis compared with other algorithms and it results the more convenient in terms ofmatching time in case of one-to-one matching.Another backtracking algorithm is the one presented in [20] by Schmidt and Druffel.It uses the information contained in the distance matrix representation of a graph toestablish an initial partition of the graph nodes. This distance matrix information isthen used in a backtracking procedure to reduce the search tree of possible mappings.A more recent algorithm, known as VF, is based on a depth-first search strategy, witha set of rules to efficiently prune the search tree. Such rules in case of isomorphismare shown in [5].As regards the graph isomorphism problem, it is also necessary to mention theMcKay's Nauty algorithm [17]. It is based on a set of transformations that reduce thegraphs to be matched to a canonical form on which the testing of the isomorphism issignificantly faster. Even if Nauty is considered one of the fastest graph isomorphismalgorithms today available, it has been shown that there are categories of graphs forwhich it requires exponential time [19].Another possible approach to the isomorphism problem is the one presented in [2];instead of reducing the complexity of matching two graphs, the authors attempt toreduce the overall computational cost when matching a sample graph against a largeset of prototypes. The method performs the matching in quadratic time with the sizeof the input graph and independently on the number of prototypes. It is obviouslyconvenient in applications requiring the matching of a graph against a database, butthe memory required to store the pre-processed database grows exponentially withthe size of the graphs, making the method suitable only for small graphs. So one ofthe authors concludes in [16] that in case of one-to-one matching other algorithms(in particular, in [9] the Ullmann’s one is cited) are more suitable.All the above cited algorithms are exact ones, i.e. they find the exact solution to thematching problem, if any. Besides them, other techniques (as those based onnon-deterministic paradigms [4, 7, 13]), able to find approximate solutions to thematching problem have been proposed, especially in the recent past. We do notexplicitly consider them in this paper, since they are really so powerful to reduce thecomplexity (in most cases) from exponential to polynomial, but, as said, they do notguarantee finding an exact and optimal solution.Despite the fact that a new algorithm for graph matching is really useful only if it canbe demonstrated that its performances are better than those obtainable by the existing