Isomorphism Algorithm Research Articles

An enhanced classical approach to graph isomorphism using continuous-time quantum walk

Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O(N7) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O(N5) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.

A Partition Method for Graph Isomorphism

Abstract Complexity of the graph isomorphism algorithms mainly depends on matching time which is directly related to efficiency of their partition methods. This paper proposed a partition method by sorted sequences of length-L path numbers, and divided cells of partition into 3 categories: not similar; completely similar; similar but not completely. The method was tested on several types of graphs with different order, each type with the same order 100 graphs. The results indicate that not similar cells can be refined by adding path length to other types or trivial cells if the vertex is not similar with all other vertices. For almost all asymmetric graphs, the path length is a small value, e.g., for 6-regular graphs with 100∼1000 vertices the average path length is 4 to get all cells to trivial ones.

A polynomial algorithm for submap isomorphism of general maps

Combinatorial maps explicitly encode orientations of edges around vertices, and have been used in many fields. In this paper, we address the problem of searching for patterns in model maps by putting forward the concept of symbol graph. A symbol graph will be constructed and stored for each model map in the preprocessing. Furthermore, an algorithm for submap isomorphism is presented based on symbol sequence searching in the symbol graphs. The computational complexity of this algorithm is quadratic in the worst case if we neglect the preprocessing step.

Hierarchical graphs for rule-based modeling of biochemical systems

BackgroundIn rule-based modeling, graphs are used to represent molecules: a colored vertex represents a component of a molecule, a vertex attribute represents the internal state of a component, and an edge represents a bond between components. Components of a molecule share the same color. Furthermore, graph-rewriting rules are used to represent molecular interactions. A rule that specifies addition (removal) of an edge represents a class of association (dissociation) reactions, and a rule that specifies a change of a vertex attribute represents a class of reactions that affect the internal state of a molecular component. A set of rules comprises an executable model that can be used to determine, through various means, the system-level dynamics of molecular interactions in a biochemical system.ResultsFor purposes of model annotation, we propose the use of hierarchical graphs to represent structural relationships among components and subcomponents of molecules. We illustrate how hierarchical graphs can be used to naturally document the structural organization of the functional components and subcomponents of two proteins: the protein tyrosine kinase Lck and the T cell receptor (TCR) complex. We also show that computational methods developed for regular graphs can be applied to hierarchical graphs. In particular, we describe a generalization of Nauty, a graph isomorphism and canonical labeling algorithm. The generalized version of the Nauty procedure, which we call HNauty, can be used to assign canonical labels to hierarchical graphs or more generally to graphs with multiple edge types. The difference between the Nauty and HNauty procedures is minor, but for completeness, we provide an explanation of the entire HNauty algorithm.ConclusionsHierarchical graphs provide more intuitive formal representations of proteins and other structured molecules with multiple functional components than do the regular graphs of current languages for specifying rule-based models, such as the BioNetGen language (BNGL). Thus, the proposed use of hierarchical graphs should promote clarity and better understanding of rule-based models.

Complexity classes of equivalence problems revisited

To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms. To determine if two graphs are cospectral (have the same eigenvalues), we compute their characteristic polynomials and see if they are equal; the characteristic polynomial is a complete invariant for cospectrality. Finally, an equivalence relation may be decidable in P without either a complete invariant or canonical form. Blass and Gurevich (1984) asked whether these conditions on equivalence relations—having an FP canonical form, having an FP complete invariant, and being in P—are distinct. They showed that this question requires non-relativizing techniques to resolve. We extend their results, and give new connections to probabilistic and quantum computation.

Algorithm for cataloging topologies in the Common Information Model (CIM)

This paper presents a novel algorithm for finding the catalog of topologies in a power system’s model in the Common Information Model (CIM) format. The algorithm prepares the models of complex, large-scale power systems (e.g., Smart Grids with renewable energy sources) and allows analytic Distribution Management System (DMS) functions to achieve high performance and optimize the power system operation in real time. It utilizes the Ullmann graph isomorphism algorithm to find unique topologies. In addition, it is optimized for parallel execution on 64-bit, multi-processor computers. Its ability to handle large amounts of data was verified on detailed, real-life electric power system data.

Maximum common subgraph isomorphism algorithms and their applications in molecular science: a review

AbstractThe intuitive description of small and large molecules using graphs has led to an increasing interest in the application of graph concepts for describing, analyzing, and comparing small molecules as well as proteins. Graph theory is a well‐studied field and many applications are present in various scientific disciplines. Recent literature describes a number of successful applications to biological problems. One of the most applied concepts aims at finding a maximal common subgraph (MCS) isomorphism between two graphs. We review exact MCS algorithms, especially designed for graphs obtained from small and large molecules, and give an overview of their successful applications. © 2011 John Wiley & Sons, Ltd. WIREs Comput Mol Sci 2011 1 68–79 DOI: 10.1002/wcms.5This article is categorized under: Structure and Mechanism > Molecular Structures Computer and Information Science > Chemoinformatics Computer and Information Science > Computer Algorithms and Programming

Building species trees from larger parts of phylogenomic databases

Gene trees are leaf-labeled trees inferred from molecular sequences. Because of gene duplication events arising in genomes, some species host several copies of the same gene, hence individual gene trees usually have several leaves labeled with identical species names. Dealing with such multi-labeled gene trees (MUL trees) is a substantial problem in phylogenomics, e.g. current supertree methods do not handle MUL trees, which restricts studies aimed at building the Tree of Life to a very small core of mono-copy genes. We propose to tackle this problem by mainly transforming a collection of MUL trees into a collection of trees, each containing single copies of labels. To achieve that aim, we provide several fast algorithmic building stones and describe how they fit in a general framework to build a species tree. First, we propose to separately preprocess each MUL tree in order to remove its redundant parts with respect to speciation events. For this purpose, we present a tree isomorphism algorithm for MUL trees to reduce redundant parts of these trees. Second, we show how the speciation signal contained in a MUL tree can be represented by a linear set of triplets. When this set is topologically coherent (compatible), we show that it can be used to produce a single-copy gene tree to replace the MUL tree while preserving the information it contains on speciation events. As an alternative approach, we propose to extract from each MUL tree a maximum size subtree that is free of duplication events. The algorithms are finally applied in a supertree analysis of hogenom, a database of homologous genes from fully sequenced genomes.

On orthogonal ray graphs

An orthogonal ray graph is an intersection graph of horizontal and vertical rays (half-lines) in the x y -plane. An orthogonal ray graph is a 2-directional orthogonal ray graph if all the horizontal rays extend in the positive x -direction and all the vertical rays extend in the positive y -direction. We first show that the class of orthogonal ray graphs is a proper subset of the class of unit grid intersection graphs. We next provide several characterizations of 2-directional orthogonal ray graphs. Our first characterization is based on forbidden submatrices. A characterization in terms of a vertex ordering follows immediately. Next, we show that 2-directional orthogonal ray graphs are exactly those bipartite graphs whose complements are circular arc graphs. This characterization implies polynomial-time recognition and isomorphism algorithms for 2-directional orthogonal ray graphs. It also leads to a characterization of 2-directional orthogonal ray graphs by a list of forbidden induced subgraphs. We also show a characterization of 2-directional orthogonal ray trees, which implies a linear-time algorithm to recognize such trees. Our results settle an open question of deciding whether a ( 0 , 1 ) -matrix can be permuted to avoid the submatrices [ 1 0 0 1 ] and [ 1 0 1 1 ] .

Bit-vector algorithms for binary constraint satisfaction and subgraph isomorphism

A solution to a binary constraint satisfaction problem is a set of discrete values, one in each of a given set of domains, subject to constraints that allow only prescribed pairs of values in specified pairs of domains. Solutions are sought by backtrack search interleaved with a process that removes from domains those values that are currently inconsistent with provisional choices already made in the course of search. For each value in a given domain, a bit-vector shows which values in another domain are or are not permitted in a solution. Bit-vector representation of constraints allows bit-parallel, therefore fast, operations for editing domains during search. This article revises and updates bit-vector algorithms published in the 1970's, and introduces focus search, which is a new bit-vector algorithm relying more on search and less on domain-editing than previous algorithms. Focus search is competitive within a limited family of constraint satisfaction problems. Determination of subgraph isomorphism is a specialized binary constraint satisfaction problem for which bit-vector algorithms have been widely used since the 1980s, particularly for matching molecular structures. This article very substantially updates the author's 1976 subgraph isomorphism algorithm, and reports experimental results with random and real-life data.

SING: Subgraph search In Non-homogeneous Graphs

BackgroundFinding the subgraphs of a graph database that are isomorphic to a given query graph has practical applications in several fields, from cheminformatics to image understanding. Since subgraph isomorphism is a computationally hard problem, indexing techniques have been intensively exploited to speed up the process. Such systems filter out those graphs which cannot contain the query, and apply a subgraph isomorphism algorithm to each residual candidate graph. The applicability of such systems is limited to databases of small graphs, because their filtering power degrades on large graphs.ResultsIn this paper, SING (Subgraph search In Non-homogeneous Graphs), a novel indexing system able to cope with large graphs, is presented. The method uses the notion of feature, which can be a small subgraph, subtree or path. Each graph in the database is annotated with the set of all its features. The key point is to make use of feature locality information. This idea is used to both improve the filtering performance and speed up the subgraph isomorphism task.ConclusionsExtensive tests on chemical compounds, biological networks and synthetic graphs show that the proposed system outperforms the most popular systems in query time over databases of medium and large graphs. Other specific tests show that the proposed system is effective for single large graphs.

On the Impossibility of a Quantum Sieve Algorithm for Graph Isomorphism

It is known that any quantum algorithm for graph isomorphism that works within the framework of the hidden subgroup problem (HSP) must perform highly entangled measurements across $\Omega(n\log n)$ coset states. One of the only known models for how such a measurement could be carried out efficiently is Kuperberg's algorithm for the HSP in the dihedral group, in which quantum states are adaptively combined and measured according to the decomposition of tensor products into irreducible representations. This “quantum sieve” starts with coset states and works its way down toward representations whose probabilities differ depending on, for example, whether the hidden subgroup is trivial or nontrivial. In this paper we show that no such approach can produce a polynomial-time quantum algorithm for graph isomorphism. Specifically, we consider the natural reduction of graph isomorphism to the HSP over the wreath product $S_n\wr\mathbb{Z}_2$. Using a recently proved bound on the irreducible characters of $S_n$, we show that no algorithm in this family can solve graph isomorphism in less than $\mathrm{e}^{\Omega(\sqrt{n})}$ time, no matter what adaptive rule it uses to select and combine quantum states. In particular, algorithms of this type can offer essentially no improvement over the best known classical algorithms, which run in time $\mathrm{e}^{O(\sqrt{n\log n})}$.

Quantum algorithms for Simon's problem over nonabelian groups

Daniel Simon's 1994 discovery of an efficient quantum algorithm for finding “hidden shifts” of Z 2 n provided the first algebraic problem for which quantum computers are exponentially faster than their classical counterparts. In this article, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G , this is the problem of recovering an involution m = ( m 1 ,…, m n ) ∈ G n from an oracle f with the property that f ( x ⋅ y ) = f ( x ) ⇔ y ∈ {1, m }. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form G n , where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two. Although groups of the form G n have a simple product structure, they share important representation--theoretic properties with the symmetric groups S n , where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called “standard method” requires highly entangled measurements on the tensor product of many coset states. In this article, we provide quantum algorithms with time complexity 2 O (√ n ) that recover hidden involutions m = ( m 1 ,… m n ) ∈ G n where, as in Simon's problem, each m i is either the identity or the conjugate of a known element m which satisfies κ( m ) = −κ(1) for some κ ∈ Ĝ . Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the “missing harmonic” approach of Moore and Russell. These are the first nontrivial HSP algorithms for group families that require highly entangled multiregister Fourier sampling.

Automatic design of application-specific reconfigurable processor extensions with UPaK synthesis kernel

This article presents a new tool for automatic design of application-specific reconfigurable processor extensions based on UPaK (Abstract U nified Pa tterns Based Synthesis K ernel for Hardware and Software Systems). We introduce a complete design flow that identifies new instructions, selects specific instructions and schedules a considered application on the newly created reconfigurable architecture. The identified extensions are implemented as specialized sequential or parallel instructions. These instructions are executed on a reconfigurable unit implementing all merged patterns. Our method uses specially developed algorithms for subgraph isomorphism that are implemented as graph matching constraints. These constraints together with separate algorithms are able to efficiently identify computational patterns and carry out application mapping and scheduling. Our methods can handle both time-constrained and resource-constrained scheduling. Experimental results show that the presented method provides high coverage of application graphs with small number of patterns and ensures high application execution speedup both for sequential and parallel application execution with reconfigurable processor extensions implementing selected patterns.

Identification of family-specific residue packing motifs and their use for structure-based protein function prediction: I. Method development

Protein function prediction is one of the central problems in computational biology. We present a novel automated protein structure-based function prediction method using libraries of local residue packing patterns that are common to most proteins in a known functional family. Critical to this approach is the representation of a protein structure as a graph where residue vertices (residue name used as a vertex label) are connected by geometrical proximity edges. The approach employs two steps. First, it uses a fast subgraph mining algorithm to find all occurrences of family-specific labeled subgraphs for all well characterized protein structural and functional families. Second, it queries a new structure for occurrences of a set of motifs characteristic of a known family, using a graph index to speed up Ullman's subgraph isomorphism algorithm. The confidence of function inference from structure depends on the number of family-specific motifs found in the query structure compared with their distribution in a large non-redundant database of proteins. This method can assign a new structure to a specific functional family in cases where sequence alignments, sequence patterns, structural superposition and active site templates fail to provide accurate annotation.

SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM

The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prove that our algorithms will work on all possible instances. In this paper, we also give a new construction for cospectral graphs.

Subsea: an efficient heuristic algorithm for subgraph isomorphism

We present a novel approach to the problem of finding all subgraphs and induced subgraphs of a (target) graph which are isomorphic to another (pattern) graph. To attain efficiency we use a special representation of the pattern graph. We also combine our search algorithm with some known bisection algorithms. Experimental comparison with other algorithms was performed on several types of graphs. The comparison results suggest that the approach provided here is most effective when all instances of a subgraph need to be found.

A New Approach to Graph Recognition and Applications to Distance-Hereditary Graphs

Algorithms used in data mining and bioinformatics have to deal with huge amount of data efficiently. In many applications, the data are supposed to have explicit or implicit structures. To develop efficient algorithms for such data, we have to propose possible structure models and test if the models are feasible. Hence, it is important to make a compact model for structured data, and enumerate all instances efficiently. There are few graph classes besides trees that can be used for a model. In this paper, we investigate distance-hereditary graphs. This class of graphs consists of isometric graphs and hence contains trees and cographs. First, a canonical and compact tree representation of the class is proposed. The tree representation can be constructed in linear time by using prefix trees. Usually, prefix trees are used to maintain a set of strings. In our algorithm, the prefix trees are used to maintain the neighborhood of vertices, which is a new approach unlike the lexicographically breadth-first search used in other studies. Based on the canonical tree representation, efficient algorithms for the distance-hereditary graphs are proposed, including linear time algorithms for graph recognition and graph isomorphism and an efficient enumeration algorithm. An efficient coding for the tree representation is also presented; it requires ⌈3.59n⌉ bits for a distance-hereditary graph of n vertices and 3n bits for a cograph. The results of coding improve previously known upper bounds (both are 2 O(n log n)) of the number of distance-hereditary graphs and cographs to 2⌈3.59n⌉ and 23n , respectively.

External memory layout vs. schematic

The circuit represented by a VLSI layout must be verified by checking it against the schematic circuit as an important part of the functional verification step. This involves two central problems of matching the circuit graphs with each other (graph isomorphism) and extracting a higher level of circuit from a given level by finding subcircuits in the circuit graph (subgraph isomorphism). Modern day VLSI layouts contain millions of devices. Hence the memory requirements of the data structures required by tools for verifying them become huge and can easily exceed the amount of internal memory available on a computer. In such a scenario, a program not aware of the memory hierarchy performs badly because of its unorganized input/output operations (I/Os) as the speed of a disk access is about a million times slower than accessing a main memory location. In this article, we present I/O-efficient algorithms for the graph isomorphism and subgraph isomorphism problems in the context of verification of VLSI layouts. Experimental results show the need and utility of I/O-efficient algorithms for handling problems with large memory requirements.

Taming verification hardness

Graphs are widely used to model complicated data semantics in many applications. In this paper, we aim to develop efficient techniques to retrieve graphs, containing a given query graph, from a large set of graphs. Considering the problem of testing subgraph isomorphism is generally NP-hard, most of the existing techniques are based on the framework of filtering -and- verification to reduce the precise computation costs; consequently various novel feature-based indexes have been developed. While the existing techniques work well for small query graphs, the verification phase becomes a bottleneck when the query graph size increases. Motivated by this, in the paper we firstly propose a novel and efficient algorithm for testing subgraph isomorphism, QuickSI. Secondly, we develop a new feature-based index technique to accommodate QuickSI in the filtering phase. Our extensive experiments on real and synthetic data demonstrate the efficiency and scalability of the proposed techniques, which significantly improve the existing techniques.