Constant-time O(1) all pairs geodesic distance query on triangle meshes

Geodesic plays an important role in geometric computation and analysis. Rather than the widely studied <i>single source all destination</i> discrete geodesic problem, very little work has been reported on the <i>all pairs</i> geodesic distance query So far, the best known result is due to Cook IV and Wenk [2009], who pre-computed the pairwise geodesic between any two mesh vertices in <i>O</i>(<i>n</i><sup>5</sup>2<sup>α(<i>n</i>)</sup> log<i>n</i>) time complexity and <i>O</i>(<i>n</i><sup>4</sup>) space complexity, where <i>n</i> is the number of mesh vertices and α(<i>n</i>) the inverse Ackermann function. Then the geodesic distance between any pair of points on the mesh edges can be computed in <i>O</i>(<i>m</i> + log<i>n</i>) time, where <i>m</i> is the number of edges crossed by the geodesic path. Although Cook IV and Wenk's algorithm is able to compute the <i>exact</i> geodesic the high computational cost limits its applications to real-world models which usually contain thousands of vertices.

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