Planar separators and parallel polygon triangulation

We show how to construct an O(root n)-separator decomposition of a planar graph G in O(n) time. Such a decomposition defines a binary tree, where each node corresponds to a subgraph of G and stores an O(root n)-separator of that subgraph. We also show how to construct an O(n(epsilon))-way decomposition tree in parallel in O(log n) time so that each node corresponds to a subgraph of G and stores an O(n(12+epsilon))-separator of that subgraph. We demonstrate the utility of such a separator decomposition by showing how it can be used in the design of a parallel algorithm for triangulating a simple polygon deterministically in O(log n) time using O(n/log n) processors on a CRCW PRAM. (C) 1995 Academic Press, Inc.