TY - GEN

T1 - An optimal-time algorithm for shortest paths on a convex polytope in three dimensions

AU - Schreiber, Yevgeny

AU - Sharir, Micha

PY - 2006

Y1 - 2006

N2 - We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(n log n) time and requires O(n log n) space, where n is the number of edges of P. The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11], and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront" from s along ∂P. This is effected by generalizing the concept of conforming subdivision of the free space used in [11], and adapting it for the case of a convex polytope in ℝ3, allowing the algorithm to accomplish the propagation in discrete steps, between the "transparent" edges of the subdivision. The algorithm constructs a dynamic version of Mount's data structure [16] that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path π can be reported in additional O(k) time, where k is the number of polytope edges crossed by π. The algorithm generalizes to the case of m source points to yield an implicit representation of the geodesic Voronoi diagram of m sites on the surface of P, in time O((n + m) log(n + m)), so that the site closest to a query point can be reported in time O(log(n + m)).

AB - We present an optimal-time algorithm for computing (an implicit representation of) the shortest-path map from a fixed source s on the surface of a convex polytope P in three dimensions. Our algorithm runs in O(n log n) time and requires O(n log n) space, where n is the number of edges of P. The algorithm is based on the O(n log n) algorithm of Hershberger and Suri for shortest paths in the plane [11], and similarly follows the continuous Dijkstra paradigm, which propagates a "wavefront" from s along ∂P. This is effected by generalizing the concept of conforming subdivision of the free space used in [11], and adapting it for the case of a convex polytope in ℝ3, allowing the algorithm to accomplish the propagation in discrete steps, between the "transparent" edges of the subdivision. The algorithm constructs a dynamic version of Mount's data structure [16] that implicitly encodes the shortest paths from s to all other points of the surface. This structure allows us to answer single-source shortest-path queries, where the length of the path, as well as its combinatorial type, can be reported in O(log n) time; the actual path π can be reported in additional O(k) time, where k is the number of polytope edges crossed by π. The algorithm generalizes to the case of m source points to yield an implicit representation of the geodesic Voronoi diagram of m sites on the surface of P, in time O((n + m) log(n + m)), so that the site closest to a query point can be reported in time O(log(n + m)).

KW - Continuous Dijkstra

KW - Geodesics

KW - Polytope Surface

KW - Shortest Path

KW - Shortest Path Map

KW - Unfolding

KW - Wavefront

UR - http://www.scopus.com/inward/record.url?scp=33748060967&partnerID=8YFLogxK

U2 - 10.1145/1137856.1137862

DO - 10.1145/1137856.1137862

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AN - SCOPUS:33748060967

SN - 1595933409

SN - 9781595933409

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 30

EP - 39

BT - Proceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06

PB - Association for Computing Machinery (ACM)

T2 - 22nd Annual Symposium on Computational Geometry 2006, SCG'06

Y2 - 5 June 2006 through 7 June 2006

ER -