---
title: Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time
url: https://www.emergentmind.com/papers/0512064
type: paper
arxiv_id: '0512064'
arxiv_url: https://arxiv.org/abs/0512064
published: '2005-12-15'
categories:
- cs.CG
---

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

## 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.