Optimal parallel algorithms for proximate points, with applications

Abstract

Consider a set P of points in the plane sorted by χ-coordinate. A point p in P is said to be a proximate point if there exists a point q on the χ-axis such that p is the closest point to q over all points in P. The proximate 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 $\frac{n}{{\log \log n}}$Common-CRCW processors, and in O(log n) time using $\frac{n}{{\log n}}$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.

Work supported in part by NSF grant CCR-9522093, by ONR grant N00014-97-1-0526, and by a grant from the Casio Science Promotion Foundation.