Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time

TL;DR

This paper introduces an almost linear time algorithm for finding shortest non-trivial cycles on orientable surfaces of bounded genus, significantly improving the efficiency over previous methods.

Contribution

It presents the first near-linear time algorithm for computing shortest non-contractible and non-separating cycles on such surfaces, solving a key open problem in computational topology.

Findings

Achieves O(n log n) runtime for the problem

Improves upon the previous O(n^{3/2}) algorithm

Uses universal-cover constructions and existing tools effectively

Abstract

We present an algorithm that computes a shortest non-contractible and a shortest non-separating cycle on an orientable combinatorial surface of bounded genus in O(n \log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon the current-best O(n^{3/2})-time algorithm by Cabello and Mohar (ESA 2005). Our algorithm uses universal-cover constructions to find short cycles and makes extensive use of existing tools from the field.