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

Yevgeny Schreiber*, Micha Sharir

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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)).

Original languageEnglish
Title of host publicationProceedings of the Twenty-Second Annual Symposium on Computational Geometry 2006, SCG'06
PublisherAssociation for Computing Machinery (ACM)
Number of pages10
ISBN (Print)1595933409, 9781595933409
StatePublished - 2006
Event22nd Annual Symposium on Computational Geometry 2006, SCG'06 - Sedona, AZ, United States
Duration: 5 Jun 20067 Jun 2006

Publication series

NameProceedings of the Annual Symposium on Computational Geometry


Conference22nd Annual Symposium on Computational Geometry 2006, SCG'06
Country/TerritoryUnited States
CitySedona, AZ


  • Continuous Dijkstra
  • Geodesics
  • Polytope Surface
  • Shortest Path
  • Shortest Path Map
  • Unfolding
  • Wavefront


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