A parallel algorithm for finding a separator in planar graphs

We present a randomized parallel algorithm for finding a simple cycle separator in a planar graph. The size of the separator is O(√n) and it separates the graph so that the largest part contains at most 2/8 ¿ n vertices. Our algorithm takes T = O(log2(n)) time and P = O(n + f1+ε) processors, where n is the number of vertices, f is the number of faces and ε is any positive constant. The algorithm is based on the solution of Lipton and Tarjan [8] for the sequential case which takes O(n) time. Combining our algorithm with the Pan and Reif [12] algorithm, enables us to find a BFS of planar graph in time O(log3(n)) using n1.5/log(n) processors. Using a variation of our algorithm we can construct a simple cycle separator of size O(d ¿ √f) were d is maximum face size.