Fast Algorithm for Graph Isomorphism Testing

López-Presa, José Luis; Anta, Antonio Fernández

doi:10.1007/978-3-642-02011-7_21

Cited by 17 publications

(34 citation statements)

References 12 publications

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“…It was shown in [12] that conauto-1.0 is able to solve the GI problem in polynomial time with high probability if at least one of the two input graphs is a random graph G(n, p) for p ∈ [ω(ln 4 n/n ln ln n), 1 − ω(ln 4 n/n ln ln n)]. Using a similar analysis, it is not hard to show a similar result for the complexity of conauto-2.0 solving the GA problem.…”

Section: Correctness Of Ead and Bjmentioning

confidence: 92%

“…Conflict propagation is used by bliss [5] to prune brother nodes when one of them generates a conflict which was not found in the corresponding node of the first path. Conflicts may be detected at the nodes of the search tree, or during the refinement process as done by conauto [12] (for GI) and saucy [8].…”

Section: Related Workmentioning

confidence: 99%

“…It is easy to adapt prior analyses [12] to show that conauto-2.0 has asymptotic time complexity O(n 3 ) with high probability when processing a random graph…”

Section: Contributionsmentioning

confidence: 99%

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Novel Techniques for Automorphism Group Computation

López-Presa

Chiroque

Anta

2013

Experimental Algorithms

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Abstract. Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. In this work we propose four novel techniques to speed up algorithms that solve the GA problem by exploring a search tree. They increase the performance of the algorithm by allowing to reduce the depth of the search tree, and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of the input graph. We also describe how the techniques have been incorporated into the GA algorithm conauto, as conauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. We have experimentally evaluated the impact of each of the above techniques with several graph families. We have observed that each of the techniques by itself significantly reduces the number of processed nodes of the search tree in some subset of graphs, which justifies the use of each of them. Then, when they are applied together, their effect is combined, leading to reductions in the number of processed nodes in most graphs. This is also reflected in a reduction of the running time, which is substantial in some graph families.

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Section: Correctness Of Ead and Bjmentioning

confidence: 92%

Section: Related Workmentioning

confidence: 99%

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Novel Techniques for Automorphism Group Computation

López-Presa

Chiroque

Anta

2013

Experimental Algorithms

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“…Indeed, there are at least two recent tools-bliss [22] and saucy [14,15]-showing impressive results in discovering automorphisms for large and sparse graphs. Furthermore, canonical labeling techniques can be combined with three search methods as in another very recent algorithm conauto [26] that uses precomputed graph automorphisms to prune the search tree. Another popular strategy often employed in GI is to consider graph vertex invariants; indeed, many examples of invariants are available in the literature [18] and they are also essential for canonical labeling.…”

Section: Exact Algorithms For Graph Isomorphismmentioning

confidence: 99%

“…We show evidence that the new approach competes well with state-of-the-art algorithms on numerous graph classes. While certain crafted graphs might be approached more effectively only using very recent canonical labeling techniques [22,26], these methods employ more complex mathematics based on refining work which began 3 decades ago [29]. Furthermore, GI-Ext is also very useful to detect large isomorphic sub-structures induced in non-isomorphic graphs-e.g.…”

Section: Introductionmentioning

confidence: 99%

Isomorphism Testing via Polynomial-Time Graph Extensions

Porumbel

2010

J Math Model Algor

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This paper deals with algorithms for detecting graph isomorphism (GI) properties. The GI literature consists of numerous research directions, from highly theoretical studies (e.g. defining the GI complexity class) to very practical applications (pattern recognition, image processing). We first present the context of our work and provide a brief overview of various algorithms developed in such disparate contexts. Compared to well-known NP-complete problems, GI is only rarely tackled with general-purpose combinatorial optimization techniques; however, classical search algorithms are commonly applied to graph matching (GM). We show that, by specifically focusing on exploiting isomorphism properties, classical GM heuristics can become very useful for GI. We introduce a polynomial graph extension procedure that provides a graph coloring (labeling) capable of rapidly guiding a simple-but-effective heuristic toward the solution. The resulting algorithm (GIExt) is quite simple, very fast and practical: it solves GI within a time in the region of O(|V | 3 ) for numerous graph classes, including difficult (dense and regular) graphs with up to 20.000 vertices and 200.000.000 edges. GI-Ext can compete with recent state-of-the-art GI algorithms based on well-established GI techniques (e.g. canonical labeling) refined over the last three decades. In addition, GI-Ext also solves certain GM problems, e.g. it detects important isomorphic structures induced in non-isomorphic graphs.

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A polynomial‐time algorithm for simple undirected graph isomorphism

Chen

Huang

et al. 2019

Concurrency and Computation

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The graph isomorphism problem is to determine two finite graphs that are isomorphic which is not known with a polynomial-time solution. This paper solves the simple undirected graph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial-time solution to check if two simple undirected graphs are isomorphic or not. Three new representation methods of a graph as vertex/edge adjacency matrix and triple tuple are proposed. A duality of edge and vertex and a reflexivity between vertex adjacency matrix and edge adjacency matrix were first introduced to present the core idea. Beyond this, the mathematical approval is based on an equivalence between permutation and bijection. Because only addition and multiplication operations satisfy the commutative law, we propose a permutation theorem to check fast whether one of two sets of arrays is a permutation of another or not. The permutation theorem was mathematically approved by Integer Factorization Theory, Pythagorean Triples Theorem, and Fundamental Theorem of Arithmetic. For each of two n-ary arrays, the linear and squared sums of elements were respectively calculated to produce the results.

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Fast Algorithm for Graph Isomorphism Testing

Cited by 17 publications

References 12 publications

Novel Techniques for Automorphism Group Computation

Novel Techniques for Automorphism Group Computation

Isomorphism Testing via Polynomial-Time Graph Extensions

A polynomial‐time algorithm for simple undirected graph isomorphism

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