Mesh partitioning is an indispensable tool for efficient parallel numerical simulations. Its goal is to minimize communication between the processes of a simulation while achieving load balance. Established graph-based partitioning tools yield a high solution quality; however, their scalability is limited. Geometric approaches usually scale better, but their solution quality may be unsatisfactory for "non-trivial" mesh topologies. In this paper, we present a scalable version of k-means that is adapted to yield balanced clusters. Balanced k-means constitutes the core of our new partitioning algorithm Geographer. Bootstrapping of initial centers is performed with space-filling curves, leading to fast convergence of the subsequent balanced k-means algorithm. Our experiments with up to 16 384 MPI processes on numerous benchmark meshes show the following: (i) Geographer produces partitions with a lower communication volume than state-of-the-art geometric partitioners from the Zoltan package; (ii) Geographer scales well on large inputs; (iii) a Delaunay mesh with a few billion vertices and edges can be partitioned in a few seconds.
The bubble framework introduced by Diekmann al @cite_20 achieves well-shaped partitions by repeatedly selecting center vertices and growing blocks around them using constrained breadth-first search. This concept is similar to the @math -means problem (cf. below), except that cluster mem -ber -ship and centers are computed with graph-theoretic instead of geometric distances. Due to the discrete nature of graph distances, the center selection can be computationally demanding.
@cite_20
1995006434 2038164234 2041869086 1890589357
In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS C, a key component of a practical successful graph partitioner ( in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS C, which computes the similarity measure used in Bubble-FOS C and is therefore the most crucial component. By relating FOS C to random walks, we obtain precise characterizations of the behavior of FOS C on tori and hypercubes. Besides leading to new knowledge on FOS C (and therefore also on Bubble-FOS C), these characterizations have been recently used for the analysis of load balancing algorithms ( in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS C, at least one of the two parts is connected. Using the aforementioned relation between FOS C and random walks, we prove that in vertex-transitive graphs both parts must be connected components. The NP-hard graph partitioning problem is an important subtask in load balancing and many other applications. It requires the division of a graph's vertex set into P equally sized subsets such that some objective function is optimized. State-of-the-art libraries addressing this problem show several deficiencies: they are hard to parallelize, focus on small edge-cuts instead of few boundary vertices, and often produce disconnected partitions. This work introduces our novel graph partitioning and repartitioning heuristic Bubble-FOS C. In contrast to other libraries, Bubble-FOS C does not try to minimize the edge-cut explicitly, but focuses instead implicitly on good partition shapes. The shapes are optimized by diffusion processes that are embedded into an iterative framework. This approach incorporates a high degree of parallelism. Besides describing the evolution process that led to the new diffusion scheme FOS C used by Bubble-FOS C, we reveal some of FOS C's properties and propose a number of enhancements for a fast and reliable implementation. Our experiments, in which we compare sequential and parallel Bubble-FOS C implementations to the state-of-the-art libraries Metis and Jostle, reveal that our new heuristic is slower, but generates high-quality solutions that are often superior. 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 NP-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 [H. Meyerhenke, B. Monien, S. Schamberger, Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid, in: Proceedings of the 20th IEEE International Parallel and Distributed Processing Symposium, IPDPS'06, IEEE Computer Society, 2006, p. 57 (CD)], which optimizes partition shapes by a diffusive 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 different diffusive 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 different 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 significant number of partitionings of six widely used benchmark graphs. The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on @math nodes with two equal-sized clusters, with an between-class edge probability of @math and a within-class edge probability of @math . Although most of the literature on this model has focused on the case of increasing degrees (ie. @math as @math ), the sparse case @math is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborov 'a based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if @math and @math , then conjectured that it is possible to cluster in a way correlated with the true partition if @math , and impossible if @math for some sufficiently large @math . We prove half of their prediction, showing that it is indeed impossible to cluster if @math . Following , our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.