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Map of Geometric Minimal Cuts with Applications*

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Abstract

This chapter considers the following problem of computing a map of geometric minimal cuts (called the MGMC problem): Given a graph G = (V, E) and a planar 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. This chapter surveys two different approaches for the MGMC problem based on a mix of geometric and graph algorithm techniques that can be regarded complementary. It is first shown 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). Based on this observation, the first approach enumerates all rectilinear geometric minimal cuts and computes their \(L_{\infty }\) Hausdorff Voronoi diagram, which is equivalent to the \(L_{\infty }\) Hausdorff Voronoi diagram of axis-aligned rectangles. The second approach is based on higher-order Voronoi diagrams and identifies necessary geometric minimal cuts and their Hausdorff Voronoi diagram in an iterative manner. The embedding in the latter approach includes arbitrary polygons. This chapter also presents the structural properties of the \(L_{\infty }\) Hausdorff Voronoi diagram of rectangles that provides the map of the MGMC problem and plane sweep algorithms for its construction.

*The work of the first author was supported in part by the Swiss National Science Foundation SNSF project 200021-127137. The work of the last two authors was supported in part by NSF through a CAREER Award CCF-0546509 and two grants IIS-0713489 and IIS-1115220.

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Notes

  1. 1.

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

  2. 2.

    The generator of a cut is a portion of the farthest Voronoi diagram of the elements constituting the cut.

  3. 3.

    A 45∘ ray is a ray of slope ± 1.

  4. 4.

    A biconnected component of a graph G is a maximal set of edges, such that any two edges in the set lie on a common simple cycle.

  5. 5.

    An articulation point (resp. bridge) of a graph G is a vertex (resp. edge) whose removal disconnects G.

  6. 6.

    The (directed) Hausdorff distance from a set A to a set B is \(h(A,B) =\max _{a\in A}\min _{b\in B}\{d(a,b)\}\). The (undirected) Hausdorff distance between A and B is \(d_{h}(A,B) =\max \{ h(A,B),h(B,A)\}\).

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Author information

Authors and Affiliations

  1. Faculty of Informatics, Università della Svizzera italiana, Via Giuseppe Buffi 13, 6904, Lugano, Switzerland

    Evanthia Papadopoulou

  2. Department of Computer Science and Engineering, University at Buffalo, The State University of New York, 201 Bell Hall, 14260, Buffalo, NY, USA

    Jinhui Xu

  3. Department of Computer Science and Engineering, University at Buffalo, The State University of New York, 201 Bell Hall, 14260, Buffalo, NY, USA

    Lei Xu

Authors
  1. Evanthia Papadopoulou
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  2. Jinhui Xu
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  3. Lei Xu
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Corresponding authors

Correspondence to Evanthia Papadopoulou , Jinhui Xu or Lei Xu .

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Eng, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Computer Science, University of Texas, Dallas, Richardson, Texas, USA

    Ding-Zhu Du

  3. Dept. Comp. Sci. & Engineering, University of California, San Diego, La Jolla, California, USA

    Ronald L. Graham

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Papadopoulou, E., Xu, J., Xu, L. (2013). Map of Geometric Minimal Cuts with Applications* . In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_27

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