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 timespace 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(√log n)} sources embedded on a surface of genus 2^{O(√log n)}. Additionally, it yields an O(n^1-ϵ) space algorithm with polynomial running time for reachability over graphs with O(n^1-ϵ) sources embedded on surfaces of genus O(n^1-ϵ).