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Dilationoptimal edge deletion in polygonal cycles
 In Proc. 18th International Symposium on Algorithms and Computation, LNCS 4835
, 2007
"... Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pairs of vertices in G is called the dilation of G. In this paper, giv ..."
Abstract

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Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pairs of vertices in G is called the dilation of G. In this paper, given a polygonal cycle C on n vertices in the plane, a randomized algorithm is presented which computes in O(n log 3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that for each edge e of C, a (1 − ɛ)approximation to the dilation of the path C \ {e} can be computed in O(n log n) total time. 1