Abstract We describe algorithms to compute the shortest path homo-topic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface.Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface withcomplexity n, genus g> = 2, and no boundary, we constructin O(n2 log n) time a tight octagonal decomposition of thesurface--a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface intoa complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest pathhomotopic to a given path of complexity k in O(gnk) time, orthe shortest cycle homotopic to a given cycle of complexity k in O(gnk log(nk)) time. A similar algorithm computesshortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colinde Verdi`ere and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in thecomplexity of the surface and the input curves, regardless of the surface geometry. 1 Introduction We consider the following topological version of theshortest path problem in geometric spaces: Given a path or cycle fl on an arbitrary topological surface, findthe shortest path or cycle that can be obtained fro
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