Fast Parallel All-Subgraph Enumeration Using Multicore Machines

Hindawi Publishing Corporation Scientific Programming Volume 2015, Article ID 901321, 11 pages http://dx.doi.org/10.1155/2015/901321 Research Article Fast Parallel All-Subgraph Enumeration Using Multicore Machines Saeed Shahrivari and Saeed Jalili Computer Engineering Department, Tarbiat Modares University (TMU), Tehran 14115-111, Iran Correspondence should be addressed to Saeed Jalili; Received 28 January 2014; Revised 21 November 2014; Accepted 21 November 2014 Academic Editor: Przemyslaw Kazienko Copyright © 2015 S. Shahrivari and S. Jalili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Enumerating all subgraphs of an input graph is an important task for analyzing complex networks. Valuable information can be extracted about the characteristics of the input graph using all-subgraph enumeration. Notwithstanding, the number of subgraphs grows exponentially with growth of the input graph or by increasing the size of the subgraphs to be enumerated. Hence, all-subgraph enumeration is very time consuming when the size of the subgraphs or the input graph is big. We propose a parallel solution named Subenum which in contrast to available solutions can perform much faster. Subenum enumerates subgraphs using edges instead of vertices, and this approach leads to a parallel and load-balanced enumeration algorithm that can have efficient execution on current multicore and multiprocessor machines. Also, Subenum uses a fast heuristic which can effectively accelerate non-isomorphism subgraph enumeration. Subenum can efficiently use external memory, and unlike other subgraph enumeration methods, it is not associated with the main memory limits of the used machine. Hence, Subenum can handle large input graphs and subgraph sizes that other solutions cannot handle. Several experiments are done using real-world input graphs. Compared to the available solutions, Subenum can enumerate subgraphs several orders of magnitude faster and the experimental results show that the performance of Subenum scales almost linearly by using additional processor cores. 1. Introduction Enumerating subgraphs of a given size has been shown to be a very useful task in the area of complex network analysis. Subgraphs can be used to identify building blocks and functional and nonfunctional characteristics in social, biological, chemical, and technological graphs [1]. An interesting application is subgraph mining which can be used to extract functional properties. A good example is finding network motifs, which are defined as connected subgraphs that occur significantly more frequently than expected [2]. One of the best known approaches for finding network motifs is to enumerate all subgraphs and then extract significant motifs after omitting frequent subgraphs that occur in random networks [3]. There are also many other applications in areas like data mining, statistics, systems biology, chemoinformatics, social networks, telecommunications, and web mining. Although subgraph enumeration is a useful task, it is a computational challenging problem [4]. Enumeration can be classified into two distinct problems: enumerating all labeled subgraphs and enumerating nonisomorphic subgraphs, that is, subgraphs that have identical structure but different vertex labels. In the first problem, all of the subgraphs of a given size should be enumerated. On the other hand, in the second problem which is much more important, all of the nonisomorphic subgraphs of a given size must be enumerated. Both problems are very time consuming because the number of both labeled and nonisomorphic subgraphs increases exponentially by giving a bigger subgraph size or a larger input graph for subgraph enumeration. As the size of the input graph increases, the number of subgraphs of size 𝑘 increases exponentially (in the worst case 𝐶(𝑛, 𝑘) for a complete graph) [5]. The number of nonisomorphic subgraphs, which can be calculated using the Polya enumeration theorem [6], also increases exponentially as 𝑘 increases. Therefore, by increasing the subgraphs size or the input graph’s size, subgraph enumeration will take more time. When nonisomorphic subgraphs are enumerated, the problem becomes more complicated because an additional mechanism must be used to identify isomorphic subgraphs. 2 Scientific Programming There is no known polynomial algorithm for subgraph isomorphism problem yet, and this overcomplicates the subgraph enumeration problem [7]. Due to the complex nature of subgraph enumeration problem, it is a very challenging and time-consuming problem. Available sequential algorithms tend to take a lot of time to do the job [3]. Hence, a good solution is to use parallel and distributed systems to accelerate subgraph enumeration [8]. Several other recent works targeting parallel subgraph enumeration have been proposed recently [8]. However, most of the related works are based on message passing interface (MPI) and hence are designed to work on cluster computing systems [8, 9]. In contrast, our goal is to provide a fast and easy to use tool for subgraph enumeration on commodity multicore and multiprocessor machines and to the best of our knowledge it has not yet been done. For this reason, we present a parallel solution, named Subenum, which is designed for faster and more scalable subgraph enumeration on multicore and multiprocessor machines. Subenum provides fast and efficient methods for counting and dumping both all and just nonisomorphic subgraphs. Subenum’s strength compared to other similar works can be classified into three categories. First, we have presented a new edge-based parallel subgraph enumeration algorithm named PSE, which is an improved version of the well-known sequential ESU algorithm. PSE provides a parallel and loadbalanced approach for subgraph enumeration. The second strength is using a custom polynomial-time heuristic for detecting isomorphic subgraphs. The last strength is using a combination of external sorting and the nauty canonical labeling algorithm which enables Subenum to enumerate nonisomorphic subgraphs even when the number of subgraphs is so big that they cannot be stored in the main memory. For evaluating the performance of Subenum we have performed several experiments on real-world graphs from different areas like social network, biological networks, software engineering, and electrical circuits. During the experiments, we compared Subenum’s performance to stateof-the-art algorithms and implementations. Experimental results show that Subenum provides a parallel, load-balanced, and effective solution for all-subgraph enumeration problem. Compared to the fastest available tools for nonisomorphic subgraph enumeration, Subenum enumerates subgraphs several times faster and is able to reduce execution time from days to hours. In addition, Sube (...truncated)