We present an algorithm that computes a shortest noncontractible 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 is based on universal-cover constructions to find short cycles and makes extensive use of existing tools from the field. disk and Colin de Verdière and Lazarus [7] showed how to decompose a given surface into parts of genus 0. Later works by Erickson and Whittlesey [11] and Colin de Verdière and Erickson [5] deal with the computation of short generators of the fundamental group and shortest paths and cycles in given homotopy classes. A central open question that arose in this context is: (P) How to compute a shortest non-trivial cycle on a topological surface efficiently?
Recently, REF presents an algorithm that computes a shortest nontrivial cycle in O(n log n) on an orientable combinatorial surface of bounded genus.
Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time
SCG '06
Mathematics Computer Science