##
###
THE HAUSDORFF VORONOI DIAGRAM OF POLYGONAL OBJECTS: A DIVIDE AND CONQUER APPROACH

EVANTHIA PAPADOPOULOU, D. T. LEE

2004
*
International journal of computational geometry and applications
*

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conquer algorithm for the construction of this diagram that improves upon previous results. As a byproduct
## more »

... introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram. The Hausdorff Voronoi diagram finds direct application in the problem of computing the critical area of a VLSI Layout, a measure reflecting the sensitivity of a VLSI design to random manufacturing defects, described in a companion paper. 13 a Two polygons P, Q are called non-crossing if their convex hulls admit at most two supporting segments appearing on the convex hull of P ∪ Q (see Def. 4 and Def. 6). November 5, 2004 10:36 WSPC/Guidelines draft The Hausdorff Voronoi Diagram of Polygonal Objects: A Divide and Conquer Approach 3 O((n + K) log n), where K was the number of interacting b pairs of shapes. The plane sweep approach was generalized in Ref. [13] for arbitrary clusters of points and the Euclidean metric with time complexity as given below. In this paper we list the structural properties of the Hausdorff Voronoi diagram and provide tighter combinatorial bounds and algorithms. Specifically, we show that for arbitrary polygons or arbitrary clusters of points, the size of the Hausdorff Voronoi diagram is O(n+m), where m is O(n 2 ) and reflects the number of crossings among shapes in S (see Theorem 1 for a precise definition of m). In the case of noncrossing polygons, not necessarily disjoint, Hausdorff Voronoi regions are shown to remain connected and thus, the size of the Hausdorff Voronoi diagram is shown to be O(n). That is, the connectivity and linearity of the diagram is established for a more general class of polygonal objects than the ones shown in Refs. [1, 5] . This bound automatically improves the time complexity of the algorithm of Ref. [5] to O(n 2 ). We present a divide and conquer algorithm to construct the Hausdorff Voronoi diagram of time complexity O(M + n log 2 n + (m + K) log n), where n is the number of points on the convex hulls of shapes in S, K = Σ P ∈S K(P ), M = Σ P ∈S M (P ), where K(P ) is the number of shapes enclosed in the minimum enclosing circle of P , and M (P ) is the number of convex hull points q ∈ Q that are interacting with P that is, q is enclosed in the minimum enclosing circle of P and either Q is entirely enclosed in the minimum enclosing circle of P or Q is crossing with P . The algorithm assumes an O(|S| log n) preprocessing time to compute the convex hulls of input shapes. Note that this is an improvement over the bound given in a preliminary version of this paper. 15 In addition we introduce the Hausdorff hull, a structure that relates to the Hausdorff Voronoi diagram in the same way as a convex hull relates to the ordinary Voronoi diagram (see Def. 8), and show that it can be computed in O(n log n) time. In the companion paper, 13 we refine the O(n + m) bound on the size of the Hausdorff Voronoi digram and show that it is tight in the worst case. We also present a simple plane sweep algorithm of comparable time complexity O(M a + (n + m + K a ) log m), where M a and K a are defined similarly to M and K with the difference that they are defined over the anchor circle of P , a specially defined enclosing circle, generally different from the minimum enclosing circle of P . Our motivation for studying the Hausdorff Voronoi diagram comes from an application in VLSI manufacturing, namely VLSI yield prediction, as explained in our companion paper. 13 The problem statement is repeated here for completeness.

doi:10.1142/s0218195904001536
fatcat:jitnfhrk5ngupnuyl7bt7oxpsi