An improved parallel algorithm for a geometric matching problem with applications

Let B be a set of n(b) blue points and R a set of n(r) red points in the plane, where n(b) + n(r) = n. A blue point b and a red point r can be matched ifr dominates b, that is, if x(b) less than or equal to x(r) and y(b) less than or equal to y(r). We consider the problem of finding a maximum cardinality matching between the points in B and the points in R. We give an adaptive parallel divide-and-conquer algorithm to solve this problems in O(log(3) n/ log k) time using the CREW PRAM with O(kn(2)/ log n). It follows that finding a maximum clique in a circular-arc graph and finding the minimum number of colors to color a trapezoid graph can be solved within these resource bounds.