Optimal parallel algorithms for proximate points, with applications - (Extended abstract)

Consider a set P of points in the plane sorted by c-coordinate. A point p in P is said to be a proximate point if there exists a point q on the cc-axis such that p is the closest point to q over all points. in P. The primate points problem is to determine all proximate points in P. We propose optimal sequential and parallel algorithms for the proximate points problem. Our sequential algorithm runs in O(n) time. Our parallel algorithms run in O(log log n) time using n/log log n Common-CRCW processors, and in O(log n) time using n/log n EREW processors. We show that both parallel algorithms are work-time optimal; the EREW algorithm is also time-optimal. As it turns out, the proximate points problem finds interesting and highly nontrivial applications to pattern analysis, digital geometry, and image processing.