Computing the Map of Geometric Minimal Cuts

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.