Parallel algorithms for all minimum link paths and link center problems

Abstract

The link metric, defined on a constrained region R of the plane, sets the distance between a pair of points in R equal the minimum number of segments or links that are needed to construct a path in R between the points. The minimum link path problem is to compute a path consisting of minimum number of links between two points in R, when R is the inside of a simple polygon P of size ns. Recently Chandru et al. [1] proposed a parallel algorithm for computing minimum link path between two points inside P and it runs in O(log n log log n) time using O(n) processors. Here we show that minimum link paths from a point to all vertices of P can be computed in O(log² n log log n) time using O(n) processors. Using this result we propose a parallel algorithm for computing the link center of P. The link center of P is the set of points x inside P such that the link distance from x to any other point in P is minimized. The algorithm runs in O(log² n log log n) time using O(n²) processors. We also show that a triangle in the approximate link center can be computed in O(log³ n log log n) time using O(n) processors. The complexity results of this paper are with respect to the CREW-PRAM model of computation.