Abstract
We study the Euclidean bottleneck Steiner tree problem: given a set P of n points in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edge in the tree is minimized. This problem is known to be NP-hard even to approximate within ratio 2 and there was no known exact algorithm even for k = 1 prior to this work. In this paper, we focus on finding exact solutions to the problem for a small constant k. Based on geometric properties of optimal location of Steiner points, we present an optimal Θ ( n log n ) -time exact algorithm for k = 1 and an O ( n 2 ) -time algorithm for k = 2 . Also, we present an optimal Θ ( n log n ) -time exact algorithm for any constant k for a special case where there is no edge between Steiner points.
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