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The computation of geodesic distances or paths between two points on triangulated meshes is a common operation in many computer graphics applications. In this article, we present an exact algorithm for the single-source all-vertices shortest path problem. Mitchell et al. [1987] proposed an O(n2 log n) method (MMP), based on Dijkstra's algorithm, where n is the complexity of the polyhedral surface. Then, Chen and Han [1990] (CH) improved the running time to O(n2). Interestingly Surazhsky et al. [2005] provided experimental evidence demonstrating that the MMP algorithm runs many times faster, in practice, than the CH algorithm. The CH algorithm encodes the structure of the set of shortest paths using a set of windows on the edges of the polyhedron. Our experiments showed that in many examples over 99% of the windows created by the CH algorithm are of no use to define a shortest path. So this article proposes to improve the CH algorithm by two separate techniques. One is to filter out useless windows using the current estimates of the distances to the vertices, the other is to maintain a priority queue like that achieved in Dijkstra's algorithm. Our experimental results suggest that the improved CH algorithm, in spite of an O(n2 log n) asymptotic time complexity, greatly outperforms the original CH algorithm in both time and space. Furthermore, it generally runs faster than the MMP algorithm and uses considerably less space.