The weighted region problem: finding shortest paths through a weighted planar subdivision

  • Authors:

  • Joseph S. B. Mitchell;Christos H. Papadimitriou

  • Affiliations:

  • Cornell Univ., Ithaca, NY;Univ. of California at San Diego, La Jolla

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1991

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Abstract

The problem of determining shortest paths through a weighted planar polygonal subdivision with n vertices is considered. Distances are measured according to a weighted Euclidean metric: The length of a path is defined to be the weighted sum of (Euclidean) lengths of the subpaths within each region. An algorithm that constructs a (restricted) “shortest path map” with respect to a given source point is presented. The output is a partitioning of each edge of the subdivion into intervals of &egr;-optimality, allowing an &egr;-optimal path to be traced from the source to any query point along any edge. The algorithm runs in worst-case time O(ES) and requires O(E) space, where E is the number of “events” in our algorithm and S is the time it takes to run a numerical search procedure. In the worst case, E is bounded above by O(n4) (and we give an &OHgr;(n4) lower bound), but it is likeky that E will be much smaller in practice. We also show that S is bounded by O(n4L), where L is the precision of the problem instance (including the number of bits in the user-specified tolerance &egr;). Again, the value of S should be smaller in practice. The algorithm applies the “continuous Dijkstra” paradigm and exploits the fact that shortest paths obey Snell's Law of Refraction at region boundaries, a local optimaly property of shortest paths that is well known from the analogous optics model. The algorithm generalizes to the multi-source case to compute Voronoi diagrams.