Finding one tight cycle

A cycle on a combinatorial surface is tight if it as short as possible in its (free) homotopy class. We describe an algorithm to compute a single tight, noncontractible, essentially simple cycle on a given orientable combinatorial surface in O(n log n) time. The only method previously known for this problem was to compute the globally shortest noncontractible or nonseparating cycle in O(min{g3, n} n log n) time, where g is the genus of the surface. As a consequence, we can compute the shortest cycle freely homotopic to a chosen boundary cycle in O (n log n) time, a tight octagonal decomposition in O(gn log n) time, and a shortest contractible cycle enclosing a nonempty set of faces in O(n log2 n) time.