Computing Faces in Segment and Simplex Arrangements (Preliminary Version)

Abstract For a set S of n line segments in the plane, we give thefirst work-optimal deterministic parallel algorithm for constructing their arrangement. It runs in O(log2 n) time using O(n log n + k) work in the EREW PRAM model, where kis the number of intersecting line segment pairs, and provides a fairly simple divide-and-conquer alternative to theoptimal sequential "plane-sweep " algorithm of Chazelle and Edelsbrunner. Moreover, our method can be used to out-put all k intersecting pairs while using only O(n) workingspace, which solves an open problem posed by Chazelle and Edelsbrunner. We also describe a sequential algorithm forcomputing a single face in an arrangement of n line seg-ments that runs in O(nff2(n) log n) time, which improves ona previous O(n log2 n) time algorithm.For collections of simplices in IR d, we give methods for constructing a set of m = O(nd\Gamma 1 logc n+k) cells of constantdescriptive complexity that covers their arrangement, where c? 1 is a constant and k is the number of faces in thearrangement. The construction is performed sequentially in O(m) time, or in O(log n) time using O(m) work in theEREW PRAM model. The covering can be augmented to answer point location queries in O(log n) time. In additionto supplying the first parallel methods for these problems, we improve on the previous best sequential methods by reducingthe query times (from O(log2 n) in IR3 and O(log3 n) in IRd, d? 3), and also the size and construction cost of the covering(from O(nd\Gamma 1+ffl + k))