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We describe a randomized algorithm for computing the trapezoidal decomposition of a simple polygon. Its expected running time is linear in the size of the polygon. By a well-known and simple linear time reduction, this implies a linear time algorithm for triangulating a simple polygon. Our algorithm is considerably simpler than Chazelle’s [3] celebrated optimal deterministic algorithm. The new algorithm can be viewed as a combination of Chazelle’s algorithm and of simple nonoptimal randomized algorithms due to Clarkson et al. [6], [7], [9] and to Seidel [20]. As in Chazelle’s algorithm, it is indispensable to include a bottom-up preprocessing phase, in addition to the actual top-down construction. An essential new idea is the use of random sampling on subchains of the initial polygonal chain, rather than on individual edges as is normally done.

In this paper we present an O(1/a logn)-time parallel algorithm for computing the convex hull ofn points in R3. This algorithm usesO(@#@ n1+a) processors on a CREW PRAM, for any constant 0 < a =1. So far, all adequately documented parallel algorithms proposed for this problem use time at least O(log2n). In addition, the algorithm presented here is the first parallel algorithm for the three-dimensional convex hull problem that is not based on the serial divide-and-conquer algorithm of Preparata and Hong, whose crucial operation is the merging of the convex hulls of two linearly separated point sets. The contributions of this paper are therefore (i) an O(logn)-time parallel algorithm for the three-dimensional convex hull problem, and (ii) a parallel algorithm for this problem that does not follow the traditional paradigm.