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Computing the Map of Geometric Minimal Cuts

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Abstract

In this paper we consider the following problem of computing a map of geometric minimal cuts (called MGMC problem): Given a graph G=(V,E) and a planar rectilinear embedding of a subgraph H=(V H ,E H ) of G, compute the map of geometric minimal cuts induced by axis-aligned rectangles in the embedding plane. The MGMC problem is motivated by the critical area extraction problem in VLSI designs and finds applications in several other fields. In this paper, we propose a novel approach based on a mix of geometric and graph algorithm techniques for the MGMC problem. Our approach first shows that unlike the classic min-cut problem on graphs, the number of all rectilinear geometric minimal cuts is bounded by a low polynomial, O(n 3). Our algorithm for identifying geometric minimal cuts runs in O(n 3logn(loglogn)3) expected time which can be reduced to O(nlogn(loglogn)3) when the maximum size of the cut is bounded by a constant, where n=|V H |. Once geometric minimal cuts are identified we show that the problem can be reduced to computing the L ∞ Hausdorff Voronoi diagram of axis aligned rectangles. We present the first output-sensitive algorithm to compute this diagram which runs in O((N+K)log2 NloglogN) time and O(Nlog2 N) space, where N is the number of rectangles and K is the complexity of the Hausdorff Voronoi diagram. Our approach settles several open problems regarding the MGMC problem.

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Notes

  1. A map means a partition of the embedding plane (as in trapezoidal map) into cells so that all points in the same cell share the same “closest” geometric minimal cut.

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Acknowledgements

The authors would like to thank three anonymous referees for their thoughtful comments and suggestions which significantly improve the presentation of the paper.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY, 14260, USA

    Jinhui Xu & Lei Xu

  2. Faculty of Informatics, Università della Svizzera Italiana, Lugano, 6904, Switzerland

    Evanthia Papadopoulou

Authors
  1. Jinhui Xu
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  2. Lei Xu
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  3. Evanthia Papadopoulou
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Correspondence to Jinhui Xu.

Additional information

The research of J. Xu and L. Xu was supported in part by NSF through a CAREER Award CCF-0546509 and grants IIS-0713489 and IIS-1115220. The research of E. Papadopoulou was supported in part by the Swiss National Science Foundation grant 200021-127137.

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Xu, J., Xu, L. & Papadopoulou, E. Computing the Map of Geometric Minimal Cuts. Algorithmica 68, 805–834 (2014). https://doi.org/10.1007/s00453-012-9704-9

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