A new diffusion-based multilevel algorithm for computing graph partitions of very high quality

Abstract

Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. High-quality partitions are important for many applications, whose objective functions are often N P-hard to optimize. Most state-of-the-art graph partitioning libraries use a variant of the Kernighan-Lin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called Bubble-FOS/C (Meyerhenke et al., IPDPS'06), which optimizes partition shapes by a diusive mechanism. However, it is too slow for practical use, despite its high solution quality.In this paper, besides proving that Bubble-FOS/C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a dierent diusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with Bubble-FOS/C in a multilevel framework based on two dierent hierarchy construction methods, we obtain our new graph partitioning heuristic DibaP. Compared to Bubble-FOS/C, DibaP shows a considerable acceleration, while retaining the positive properties of the slower algorithm. Experiments with popular benchmark graphs show that DibaP computes consistently better results than the state-of-the-art libraries METIS and JOSTLE. Moreover, with our new algorithm, we have improved the best known edge-cut values for a signicant number of partitionings of six widely used benchmark graphs.

A Distributed Algorithm for Large-Scale Graph Partitioning

“…Although the most important metric for edge-cut graph partitioning is the size of the edge-cut (or energy), a number of studies [Hendrickson 1998] show that this metric alone is not enough to measure the partitioning quality. Several metrics are, therefore, defined and used in the literature [Meyerhenke et al 2008[Meyerhenke et al , 2009, among which we selected the following in our evaluations:…”

“…The very large scale of the graphs we target poses a major challenge. Although numerous algorithms are known for graph partitioning [Enright et al 2002;Kumar 1999a, 1998;Kernighan and Lin 1970;Meyerhenke et al 2008Meyerhenke et al , 2009Schulz 2012, 2011], including parallel ones, most of the techniques involved assume a form of cheap random access to the entire graph. In contrast to this, large-scale graphs do not fit into the main memory of a single computer; in fact, they often do not fit on a single local file system either.…”

JA-BE-JA: A Distributed Algorithm for Balanced Graph Partitioning

“…Several metrics are, therefore, defined and used in the literature [7], [8], among which we selected the following ones in our evaluations:…”

“…The very large scale of the graphs we target poses a major challenge. Although a very large number of algorithms are known for graph partitioning [3], [4], [5], [6], [7], [8], [9], [10], including parallel ones, most of the techniques involved assume a form of cheap random access to the entire graph. In contrast to this, large scale graphs do not fit into the main memory of a single computer, in fact, they often do not fit on a single local file system either.…”

Advanced Coarsening Schemes for Graph Partitioning

“…The basic idea can be traced back to multigrid solvers for solving elliptic partial differential equations [31] but more recent practical methods are based on mostly graph-theoretic aspects of, in particular, edge contraction and local search. Well-known software packages based on this approach include Jostle [33], Metis [26], DiBaP [17], and Scotch [18].…”