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A new diffusionbased multilevel algorithm for computing graph partitions of very high quality
 In Proc. 22nd IPDPS
, 2008
"... Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph p ..."
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Cited by 32 (9 self)
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Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph partitioning libraries use a variant of the KernighanLin (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 BubbleFOS/C (Meyerhenke et al., IPDPS'06), 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 BubbleFOS/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 BubbleFOS/C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph
The Cover Time of Deterministic Random Walks
, 2010
"... The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this “deterministic random walk ” covers all vertices (or all edges). We present general techniques to der ..."
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Cited by 11 (2 self)
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The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this “deterministic random walk ” covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.
Beyond Good Shapes: Diffusionbased Graph Partitioning is Relaxed Cut Optimization?
"... Abstract. In this paper we study the prevalent problem of graph partitioning by analyzing the diffusionbased partitioning heuristic BUBBLEFOS/C, a key component of the practically successful partitioner DIBAP [14]. Our analysis reveals that BUBBLEFOS/C, which yields wellshaped partitions in exp ..."
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Cited by 2 (2 self)
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Abstract. In this paper we study the prevalent problem of graph partitioning by analyzing the diffusionbased partitioning heuristic BUBBLEFOS/C, a key component of the practically successful partitioner DIBAP [14]. Our analysis reveals that BUBBLEFOS/C, which yields wellshaped partitions in experiments, computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). It therefore provides the first substantial theoretical insights (beyond intuition) why BUBBLEFOS/C (and therefore indirectly DIBAP) yields good experimental results. Moreover, we show that in bisections computed by BUBBLEFOS/C, at least one of the two parts is connected. Using arguments based on random walk techniques, we prove that in vertextransitive graphs actually both parts must be connected components each. All these results may help to eventually bridge the gap between practical and theoretical graph partitioning.
CNS0708307. Parts of this work were performed while the authors were affiliated with the
"... the date of receipt and acceptance should be inserted later Abstract In this paper we study the prevalent problem of graph partitioning by analyzing the diffusionbased partitioning heuristic BubbleFOS/C, a key component of a practical successful graph partitioner (Meyerhenke et al., J. Parallel an ..."
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the date of receipt and acceptance should be inserted later Abstract In this paper we study the prevalent problem of graph partitioning by analyzing the diffusionbased partitioning heuristic BubbleFOS/C, a key component of a practical successful graph partitioner (Meyerhenke et al., J. Parallel and Distrib. Computing, 69(9):750–761, 2009). We begin by studying the disturbed diffusion scheme FOS/C, which computes the similarity measure used inBubbleFOS/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 BubbleFOS/C), these characterizations have been recently used for the analysis of load balancing algorithms (Berenbrink et al., SODA’11). We then regard BubbleFOS/C, which has been shown in previous experiments to produce solutions with good partition shapes and other