Space-efficient algorithms for reachability in surface-embedded graphs

This work presents a log-space reduction which compresses an n-vertex directed acyclic graph with m(n) sources embedded on a surface of genus g(n), to a graph with O(m(n) + g(n)) vertices while preserving reachability between a given pair of vertices. Applying existing algorithms to this reduced graph yields new deterministic algorithms with improved space bounds as well as improved simultaneous time-space bounds for the reachability problem over a large class of directed acyclic graphs. Specifically, it significantly extends the class of surface-embedded graphs with log-space reachability algorithms: from planar graphs with O(log n) sources, to graphs with 2(O(root log n)) sources embedded on a surface of genus 2(O(root log n)). Additionally, it yields an O(n(1-epsilon)) space algorithm with polynomial running time for reachability over graphs with O(n(1-epsilon)) sources embedded on surfaces of genus O(n(1-epsilon)).