Abstract

An O(n log n) algorithm is presented for computing the maximum euclidean distance between two finite planar sets of n points. When the n points form the vertices of simple polygons this complexity can be reduced to O(n). The algorithm is empirically compared to the brute-force method as well as an alternate O(n 2 ) algorithm. Both the O(n log n) and O(n 2 ) algorithms run in O(n) expected time for many underlying distributions of the points. An e-approximate algorithm can be obtained that runs in O(n + 1/e) worst-case time. 1. Introduction Let S 1 = p 1 , p 2 ,..., p n and S 2 = q 1 , q 2 ,..., q n be two planar sets of n points, and let S = S 1 È S 2 . (The sets need not have equal cardinality, but this assumption simplifies notation.) A point p i is given by the cartesian coordinates x i and y i . The maximum distance between S 1 and S 2 , denoted by d max (S 1 , S 2 ), is defined as d max (S 1 , S 2 ) = {d(p i ,q j )}, i, j = 1,2,...,n, where d(p i ,q j ) is the euclidean dista...

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