Affiliation:
1. Cornell Univ., Ithaca, NY
2. Univ. of California at San Diego, La Jolla
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 ε-optimality, allowing an ε-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
(
n
4
) (and we give an Ω(
n
4
) lower bound), but it is likeky that
E
will be much smaller in practice. We also show that
S
is bounded by
O
(
n
4
L
), where
L
is the precision of the problem instance (including the number of bits in the user-specified tolerance ε). 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.
Publisher
Association for Computing Machinery (ACM)
Subject
Artificial Intelligence,Hardware and Architecture,Information Systems,Control and Systems Engineering,Software
Reference29 articles.
1. ~ALEXANDER~ R. Construction of optimal-path maps for homogeneous-cost-region path-planning ~problems. Ph.D. dissertaUon. Dept. Comput. Sci. U.S. Naval Postgraduate School Monterey ~Cahf. Sept. 1989. ~ALEXANDER~ R. Construction of optimal-path maps for homogeneous-cost-region path-planning ~problems. Ph.D. dissertaUon. Dept. Comput. Sci. U.S. Naval Postgraduate School Monterey ~Cahf. Sept. 1989.
2. Visibility of disjoint polygons
3. Generalized Unfoldings for Shortest Paths
4. A note on two problems in connexion with graphs
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