Star Unfolding of a Polytope with Applications (1993) [30 citations — 2 self]
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author = {Pankaj K. Agarwal and Pankaj K. Agarwal and Joseph O'Rourke and Boris Aronov and Boris Aronov and Catherine A. Schevon and Catherine A. Schevon},
title = {Star Unfolding of a Polytope with Applications},
journal = {SIAM J. Comput},
year = {1993},
volume = {26},
pages = {1689--1713}
}
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Abstract:
We define the notion of a star unfolding of the surface P of a convex polytope with n vertices, and use it to solve several problems related to shortest paths on P. The first algorithm computes the edge sequences traversed by shortest paths on P in time O(n 6 fi(n) log n), where fi(n) is an extremely slowly-growing function. A much simpler O(n 6 ) time algorithm that finds a small superset of all such edge sequences is also sketched. Second, we describe an O(n 8 log n) time procedure for computing the geodesic diameter of P: the maximum possible separation of two points on P, with the distance measured along P. Finally, we describe an algorithm that preprocesses P into a data structure that can efficiently answer the queries of the form: "Given two points, what is the length of the shortest path connecting them?" Given a parameter n 2 s n 4 , it can preprocess P, in time O(n 4 s 1+ffi ) for any ffi ? 0, into a data structure of size O(n 4 s 1+ffi ), so tha...
