Optimal Algorithms to Detect Null-Homologous Cycles on 2-Manifolds

Abstract

Given a cycle of length k on a triangulated 2-manifold, we determine if it is null-homologous (bounds a surface) in O(n+k) optimal time and space where n is the size of the triangulation. Further, with a preprocessing step of O(n) time we answer the same query for any cycle of length k in O(g+k) time, g the genus of the 2-manifold. This is optimal for k < g.