Peeling and Nibbling the Cactus: Subexponential-Time Algorithms for Counting Triangulations and Related Problems

Given a set of n points S in the plane, a triangulation T of S is a maximal set of non-crossing segments with endpoints in S. We present an algorithm that computes the number of triangulations on a given set of n points in time n(11+o(1)) √ , significantly improving the previous best running time of O(2nn2) by Alvarez and Seidel [SoCG 2013]. Our main tool is identifying separators of size O( √ n) of a triangulation in a canonical way. The definition of the separators are based on the decomposition of the triangulation into nested layers (“cactus graphs”). Based on the above algorithm, we develop a simple and formal framework to count other non-crossing straight-line graphs in nO( √ n) time. We demonstrate the usefulness of the framework by applying it to counting non-crossing Hamilton cycles, spanning trees, perfect matchings, 3-colorable triangulations, connected graphs, cycle decompositions, quadrangulations, 3-regular graphs, and more. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems