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There is a Planar Graph Almost as Good as the Complete Graph
L. Paul Chew
Dartmouth PCS-TR90-148


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 straight-line 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 10 times [ab], where [ab] is the straight-line 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.


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   L. Paul Chew, "There is a Planar Graph Almost as Good as the Complete Graph." Dartmouth Computer Science Technical Report PCS-TR90-148, 1990.

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