Structured recursive separator decompositions for planar graphs in linear time

  • Authors:

  • Philip N. Klein;Shay Mozes;Christian Sommer

  • Affiliations:

  • Brown University, Providence, RI, USA;MIT, Cambridge, MA, USA;MIT, Cambridge, MA, USA

  • Venue:
  • Proceedings of the forty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2013

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Abstract

Given a triangulated planar graph G on n vertices and an integer rr--division of G with few holes is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called holes), and such that, for each region, the total number of vertices on these faces is O(√ r). We provide an algorithm for computing r--divisions with few holes in linear time. In fact, our algorithm computes a structure, called decomposition tree, which represents a recursive decomposition of G that includes r--divisions for essentially all values of r. In particular, given an exponentially increasing sequence {vec r} = (r1,r2,...), our algorithm can produce a recursive {vec r}--division with few holes in linear time. r--divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st--cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).