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Beyond Good Partition Shapes: An Analysis of Diffusive Graph Partitioning

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Abstract

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 (Meyerhenke et al. 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 (Berenbrink et al. 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.

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Notes

  1. Here, the maximum degree of G is defined as \(\operatorname {deg}(G):=\max_{u\in V}\operatorname{deg}(u)\).

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Acknowledgements

The authors thank Christoph Buchheim, Burkhard Monien, Peter Sanders, and Christian Schulz for helpful discussions.

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Authors and Affiliations

  1. Institute of Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Henning Meyerhenke

  2. Department 1: Algorithms & Complexity, Max-Planck Institute for Computer Science, Saarbrücken, Germany

    Thomas Sauerwald

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  1. Henning Meyerhenke
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  2. Thomas Sauerwald
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Correspondence to Henning Meyerhenke.

Additional information

Parts of this paper have been published in a preliminary form in the Proceedings of the 17th and 21st International Symposium on Algorithms and Computation (ISAAC 2006 and ISAAC 2010) [32, 35].

This work was partially supported by German Research Foundation (DFG) Research Training Group GK-693 of the Paderborn Institute for Scientific Computation and by DFG Priority Programme 1307 Algorithm Engineering. H. Meyerhenke was also partially supported by the CASS-MT Center led by Pacific Northwest National Laboratory and NSF Grant CNS-0708307. Parts of this work were performed while the authors were affiliated with the Department of Computer Science, University of Paderborn, Germany, and while H.M. was affiliated with Georgia Institute of Technology, Atlanta, Georgia, USA.

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Meyerhenke, H., Sauerwald, T. Beyond Good Partition Shapes: An Analysis of Diffusive Graph Partitioning. Algorithmica 64, 329–361 (2012). https://doi.org/10.1007/s00453-012-9666-y

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