An optimal parallel algorithm for planar cycle separators

We present an optimal parallel algorithm for computing a cycle separator of an n-vertex embedded planar undirected graph in O(log n) time on n/log n processors. As a consequence, we also obtain an improved parallel algorithm for constructing a depth-first search tree rooted at any given vertex in a connected planar undirected graph in O(log²n) time on n/log n processors. The best previous algorithms for computing depth-first search trees and cycle separators achieved the same time complexities, but with n processors. Our algorithms run on a parallel random access machine that permits concurrent reads and concurrent writes in its shared memory and allows an arbitrary processor to succeed in case of a write conflict.