Abstract
We define the notion of a “star unfolding” of the surface P of a convex polytope with n vertices and use it to construct an algorithm for computing a small superset of the set of all sequences of edges traversed by shortest paths on P. It requires O(n 6) time and produces O(n 6) sequences, thereby improving an earlier algorithm of Sharir that in O(n 8 log n) time produces O(n 7) sequences. A variant of our algorithm runs in O(n 5 log n) time and produces a more compact representation of size O(n 5) for the same set of O(n 6) sequences. In addition, we describe an O(n 10) time procedure for computing the geodesic diameter of P, which is the maximum possible separation of two points on P, with the distance measured along P, improving an earlier O(n 14 log n) algorithm of O'Rourke and Schevon.
Extended Abstract
Part of the work was carried out when the first two authors were at Courant Institute of Mathematical Sciences, New York University and the fourth author was at the Department of Computer Science, The Johns Hopkins University. The work of the second author was partially supported by an AT&T Bell Laboratories Ph.D. Scholarship. The work of the first two authors has also been partially supported by Dimacs (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center — NSF-STC88-09648. The third author's work is supported by NSF grant CCR-882194.
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© 1990 Springer-Verlag Berlin Heidelberg
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Agarwal, P.K., Aronov, B., O'Rourke, J., Schevon, C.A. (1990). Star unfolding of a polytope with applications. In: Gilbert, J.R., Karlsson, R. (eds) SWAT 90. SWAT 1990. Lecture Notes in Computer Science, vol 447. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52846-6_94
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DOI: https://doi.org/10.1007/3-540-52846-6_94
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