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Abstract:
Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straightline distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less that Ã10 times [ab], where [ab] is the straightline Euclidean distance between a and b. The triangulation that has this property is the L1 metric Delauney triangulation for the set S. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O (n log n) time.
Bibliographic citation for this report: [plain text] [BIB] [BibTeX] [Refer]
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L. Paul Chew,
"There is a Planar Graph Almost as Good as the Complete Graph."
Dartmouth Computer Science Technical Report PCSTR90148,
1990.
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