Given a set of data graphs and a query graph, graph isomorphism query processing is the problem of finding all the data graphs that are isomorphic to the query graph. Graph isomorphism query processing is a core problem in graph analysis of various application domains. In existing approaches, index construction or query processing takes much time as the graph sizes increase. In this paper, we propose an efficient algorithm for graph isomorphism query processing. We introduce the color-label distribution which represents the canonical coloring of a vertex-labeled graph. Based on degree sequences and color-label distributions, we introduce a two-level index, which helps us efficiently solve graph isomorphism query processing. Experimental results on real datasets show that the proposed algorithm is orders of magnitude faster than the state-of-the-art algorithms in terms of index construction time, and it runs faster than existing algorithms in terms of query processing time as the graph sizes increase.

Graph isomorphism is a core problem in graph analysis of various application domains. Given two graphs, the graph isomorphism problem is to determine whether there exists an isomorphism between them. As real-world graphs are getting bigger and bigger, applications demand practically fast algorithms that can run on large-scale graphs. However, existing approaches such as graph canonization and subgraph isomorphism show limited performances on large-scale graphs either in time or space. In this paper, we propose a new approach to graph isomorphism, which is the framework of pairwise color refinement and efficient backtracking. The main features of our approach are: (1) pairwise color refinement and binary cell mapping (2) compressed CS (candidate space), and (3) partial failing set, which together lead to a much faster and scalable algorithm for graph isomorphism. Extensive experiments with real-world datasets show that our approach outperforms state-of-the-art algorithms by up to orders of magnitude in terms of running time.

최근 다양한 분야에서 NP-hard에 속하는 그래프 문제를 효율적으로 해결하는 것이 점차 중요해지고 있다. 그래프 분석에서 가장 핵심적인 문제 중 하나는 부분 그래프 동형(subgraph isomorphism)문제이다. 부분 그래프 동형 문제란 데이터 그래프와 쿼리 그래프가 주어졌을 때, 데이터 그래프에서 쿼리그래프의 모든 임베딩(embedding)을 찾는 문제이다. 이 문제를 해결하는 최신 알고리즘인 DAF에서는 후보 수 순서(candidate-size order)와 경로 수 순서(path-size order)라는 두 가지 동적 매칭 순서를 제시하였다. 하지만 구체적으로 어떤 경우에 두 가지 동적 매칭 순서의 성능 차이가 일어나는지는 언급되지 않았다. 본 논문에서는 다양한 그래프에서 두 가지 동적 매칭 순서의 성능을 비교하여 쿼리 그래프의 트리 형태 경로(tree-like path)의 평균 길이가 길어지거나 데이터 그래프의 레이블(label) 개수가 줄어들수록 경로 수 순서의 효율성이 높아지는 것을 밝혔다.

Cartesian tree matching is the problem of finding every substring of a given text which has the same Cartesian tree as that of a given pattern. In this paper we propose fast algorithms for single and multiple pattern Cartesian tree matching by introducing new representations and encodings. For single pattern Cartesian tree matching, we present the framework of a binary filtration method and an efficient verification technique. Any exact string matching algorithm can be used as a filtration for Cartesian tree matching in our framework. For multiple pattern Cartesian tree matching, we present two fingerprinting methods, i.e., the parent-distance encoding and the binary encoding. By combining an efficient fingerprinting method and a conventional multiple string matching algorithm, we can efficiently solve multiple pattern Cartesian tree matching. By experiments we show that our matching algorithms provide good performances for both single and multiple pattern Cartesian tree matching.

Cartesian tree matching is the problem of finding all sub-strings in a given text which have the same Cartesian trees as that of a given pattern. In this paper, we deal with Cartesian tree matching for the case of multiple patterns. We present two fingerprinting methods, i.e., the parent-distance encoding and the binary encoding. By combining an efficient fingerprinting method and a conventional multiple string matching algorithm, we can efficiently solve multiple pattern Cartesian tree matching. We propose three practical algorithms for multiple pattern Cartesian tree matching based on the Wu-Manber algorithm, the Rabin-Karp algorithm, and the Alpha Skip Search algorithm, respectively. In the experiments we compare our solutions against the previous algorithm [18]. Our solutions run faster than the previous algorithm as the pattern lengths increase. Especially, our algorithm based on Wu-Manber runs up to 33 times faster.

Subgraph matching (or subgraph isomorphism) is one of the fundamental problems in graph analysis. Extensive research has been done to develop practical solutions for subgraph matching. The state-of-the-art algorithms such as \textsfCFL-Match and \textsfTurbo\textsubscriptiso convert a query graph into a spanning tree for obtaining candidates for each query vertex and obtaining a good matching order with the spanning tree. However, by using the spanning tree instead of the original query graph, it could lead to lower pruning power and a sub-optimal matching order. Another limitation is that they perform redundant computation in search without utilizing the knowledge learned from past computation. In this paper, we introduce three novel concepts to address these inherent limitations: 1) dynamic programming between a directed acyclic graph (DAG) and a graph, 2) adaptive matching order with DAG ordering, and 3) pruning by failing sets, which together lead to a much faster algorithm \textsfDAF for subgraph matching. Extensive experiments with real datasets show that \textsfDAF outperforms the fastest existing solution by up to orders of magnitude in terms of recursive calls as well as in terms of the elapsed time.

Given a text T and a pattern P, the order-preserving matching problem is to find all substrings in T which have the same relative orders as P. Order-preserving matching has been an active research area since it was introduced by Kubica et al. [13] and Kim et al. [11]. In this paper we present two algorithms for the multiple order-preserving matching problem, one of which runs in sublinear time on average and the other in linear time on average. Both algorithms run much faster than the previous algorithms.