Map of geometric minimal cuts with applications

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 = (VH, EH) 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(n3). Based on this observation, the first approach enumerates all rectilinear geometric minimal cuts and computes their L∞Hausdorff Voronoi diagram, which is equivalent to the L∞ 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∞ Hausdorff Voronoi diagram of rectangles that provides the map of the MGMC problem and plane sweep algorithms for its construction.