Coarse-Grained Parallel Geometric Search

We present a parallel algorithm for solving the next element search problem on a set of line segments, using a BSP like model referred to as the Coarse Grained Multicomputer (CGM). The algorithm requires O(1) communication rounds (h-relations with h = O(n/p)), O((n/p) log n) local computation, and O((n/p) log p) memory per processor, assuming n/p ≥ p. Our result implies solutions to the point location, trapezoidal decomposition and polygon triangulation problems. A simplified version for axis parallel segments requires only O(n/p) memory per processor, and we discuss an implementation of this version. As in a previous paper by Develliers and Fabri [13], our algorithm is based on a distributed implementation of segment trees which are of size O(n log n). This paper improves on [13] in several ways: (1) It studies the more general next element search problem which also solves, e.g., planar point location. (2) The algorithms require only O((n/p) log n) local computation instead of O(log p ∗ (n/p) log n). (3) The algorithms require only O((n/p) log p) local memory instead of O((n/p) log n).