, Volume 64, Issue 3, pp 329–361 | Cite as

Beyond Good Partition Shapes: An Analysis of Diffusive Graph Partitioning

  • Henning MeyerhenkeEmail author
  • Thomas Sauerwald


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.


Diffusive graph partitioning Relaxed cut optimization Disturbed diffusion Random walks 



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


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  2. 2.Department 1: Algorithms & ComplexityMax-Planck Institute for Computer ScienceSaarbrückenGermany

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