Computing the L1-diameter and center of a simple rectilinear polygon in parallel

Abstract

The diameter of a set S of points is the maximal distance between a pair of points in S. The center of S is the set of points that minimise the distance to their furthest neighbom’s. The problem of finding the diameter and center of a simple polygon with n vertices for different distance measures has been studied extensively in recent years. There are algorithms that run in linear time if the geodesic Euclidean metric is used and O(n log n) time if the link metric is used. In this paper we consider the L1-metric inside a simple rectilinear polygon P, i.e. the distance between two points in P is defined as the length of a shortest rectilinear path connecting them. We give an 0(lo8 n) time algorithm to compute the L1-diameter and center on a EREW-PRAM with n/log n processors if a triangulation of the polygon is provided.