Computing the All-Pairs Longest Chain in the Plane

Many problems on sequences and on special kinds of graphs involve the comput.ation of longesl. chains passing point.s in the plane. Given a set 5 of n point.s in the plane. we consider the problem of computing the matrix of longest chain lengths between all pairs of points in 5, and t.he matrix of "parent" point.ers that describes the n longest chain trees. We present a simple sequent.ial algorithm for computing these matrices. Our algorithm runs in 0(n 2 ) time, and hence is optimal. We also present a rather involved parallel algorithm t.hat computes these matrices in 0«logn)2) time using 0(n2 /logn) processors in the CREW PRAM model. These matrices enable us to report, in Ott) time, the length of a longest chain between any two points in 5 by using one processor, and t.he adual chain by using k processOni, where k is the number of points of 5 on that chain. The space complexit.), of the algorit.hms is 0{n2 ). K eyword,~: Comput.at.ional geomet.ry, CREW PRAM, increasing chains, longest paths, Monge mat.rices, parallel algorit.hms, sequential algorit.hms