Computing intersections and arrangements for red-blue curve segments in parallel

Abstract

Let $A$ and $B$ be two sets of ``well-behaved'' (i.e., continuous and x-monotone) curve segments in the plane, where no two segments in $A$ (similarly, $B$) intersect. In this paper we show how to report all points of intersection between segments in $A$ and segments in $B$, and how to construct the arrangement defined by the segments in $A\\cup B$ in parallel using the concurrent-read-exclusive-write (CREW-) PRAM model. The algorithms perform a work of $O(n\\log n+k)$ using $p\\leq n+k/\\log n$ ($p\\leq n/\\log n+k/\\log ^2 n$, resp.,) processors if we assume that the handling of segments is ``cheap'', e.g., if two segments intersect at most a constant number of times, where $n$ is the total number of segments and $k$ is the number of points of intersection. If we only assume that a single processor can compute an arbitrary point of intersection between two segments in constant time, the performed work increases to $O(n\\log n+m(k+p))$, where $m$ is the maximal number of points of intersection between two segments. We also show how to count the number of points of intersection between segments in $A$ and segments in $B$ in time $O(\\log n)$ using $n$ processors on a CREW-PRAM if two curve segments intersect at most twice.