쌍별 색 개선과 효율적인 백트래킹을 이용한 빠른 그래프 동형 알고리즘

학위논문(박사) -- 서울대학교대학원 : 공과대학 컴퓨터공학부, 2021.8. 구건모.Graph isomorphism is a core problem in graph analysis of various domains including social networks, bioinformatics, chemistry, and so on. As real-world graphs are getting bigger and bigger, applications demand practically fast algorithms that can run on large-scale graphs. Existing approaches, however, show limited performances on large-scale real-world graphs either in time or space. Also, graph isomorphism query processing is often required in many applications, which is a natural generalization of graph isomorphism for multiple graphs. In this thesis we present fast algorithms for graph isomorphism and graph isomorphism query processing. First, we present a new approach to graph isomorphism, which is the framework of pairwise color refinement and efficient backtracking. Within the framework, we introduce three efficient techniques, which together lead to a much faster and scalable algorithm for graph isomorphism. Experiments on real-world datasets show that our algorithm outperforms state-of-the-art solutions by up to several orders of magnitude in terms of running time. Second, We develop an efficient algorithm for graph isomorphism query processing. We use a two-level index using degree sequences and color-label distributions. Experimental results on real datasets show that our 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.그래프 동형 문제는 소셜 네트워크 서비스, 생물정보학, 화학정보학 등등 다양한 응용 분야에서 그래프 분석을 위해 다루고 있는 핵심 문제이다. 실생활에서 다루는 그래프 데이터의 크기가 커져 감에 따라, 대용량의 그래프를 처리할 수 있는 그래프 동형 알고리즘의 필요성이 높아지고 있다. 그러나 현재 존재하는 그래프 동형 알고리즘들은 대용량의 그래프에 대해서 시간 혹은 공간 측면에서 한계를 보여준다. 응용 분야 중에서는 여러 개의 그래프들 중에서 하나의 쿼리 그래프와 동형인 그래프를 모두 찾는 문제, 즉 그래프 동형 쿼리 프로세싱을 종종 요구하기도 한다. 본 논문에서는 대용량의 실제 그래프 데이터에 대해서 그래프 동형 문제와 그래프 동형 쿼리 프로세싱 문제를 빠르게 푸는 알고리즘들을 제안한다. 첫 번째로, 본 논문에서는 그래프 동형 문제를 위한 빠르고 확장성 있는 알고리즘을 제안한다. 이를 위해 쌍별 색 개선(pairwise color refinement)과 효율적인 백트래킹으로 구성된 프레임워크를 소개한다. 이 프레임워크 내에서 세 가지 효율적인 테크닉을 사용한다. 실제 그래프 데이터에 대한 실험을 통해 본 알고리즘이 현존하는 가장 빠른 알고리즘들보다 평균 수천 배 빠름을 보였다. 두 번째로, 본 논문에서는 그래프 동형 쿼리 프로세싱을 위한 효율적인 알고리즘을 개발한다. 본 알고리즘은 차수열과 색-레이블 분포를 이용한 인덱스를 이용한다. 실제 그래프 데이터에 대한 실험을 통해 본 알고리즘이 현존하는 알고리즘들보다 인덱싱 시간에서는 항상 평균 수천 배 빠르고, 쿼리 처리 시간에서는 중$\cdot$대용량의 그래프들에 대해서 평균 수십 배 빠르게 동작하는 것을 보였다.1. Introduction 1 1.1. Background 1 1.2. Organization 3 2. Preliminaries 4 2.1. Notation 4 2.2. Problem Definitions 6 2.3. Related Work 7 3. Graph Isomorphism 9 3.1. Algorithm Overview 12 3.2. Pairwise Color Refinement and Binary Cell Mapping 13 3.3. Compressed Candidate Space 16 3.4. Backtracking and Partial Failing Sets 21 3.5. Performance Evaluation 31 3.5.1. Comparing with Existing Solutions 35 3.5.2. Effectiveness of Individual Techniques 39 3.5.3. Analysis with Varying Degrees of Similarity 42 3.5.4. Sensitivity Analysis 46 4. Graph Isomorphism Query Processing 48 4.1. Canonical Coloring 51 4.2. Index Construction 56 4.3. Query Processing 59 4.4. Performance Evaluation 63 4.4.1. Varying Number of Hops 67 4.4.2. Varying Number of Data Graphs 74 5. Conclusion 78 5.1. Summary 78 5.2. Future Directions 79 요약 95박

Scalable Graph Isomorphism: Combining Pairwise Color Refinement and Backtracking via Compressed Candidate Space

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 stateof-the-art algorithms by up to orders of magnitude in terms of running time. I1

Scalable graph isomorphism: Combining pairwise color refinement and backtracking via compressed candidate space

© 2021 IEEE.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.N

Scalable graph isomorphism: Combining pairwise color refinement and backtracking via compressed candidate space

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