The Weighted Euclidean 1-Center Problem

The special case where wi = 1, i = 1, . . . , n, was proposed by J. Sylvester in 1857 and amounts to finding the smallest circle that contains all the given points. The general case was introduced by Francis [3]. The most efficient algorithm known to date for the unweighted case is an O(n logn) algorithm by Shamos and Hoey [lo].' This algorithm utilizes the data structure so-called "Farthest point Voronoi diagram." It is not clear whether the generalization of this concept into weighted Voronoi diagrams may yield equally efficient algorithms for problems such as the weighted 1-center problem. Previous algorithms and analysis of the unweighted case appeared in Rademacher and Toeplitz [8], Courant and Robbins [I], Smallwood [I 11, Nair and Chandrasekaran [6] and Elzinga and Hearn [2]. The weighted case is solvable in 0(n3) time assuming that we can solve quadratic equations in constant time. The 0(n3) solution is achieved as follows. The problem can be posed as of finding the minimal value r* of the variable r such that the balls Bi(r) = {(x, y) : (x xi)' + ( y yi)2 < ( r / ~ ; ) ~ ) have a nonempty intersection. By Helly's Theorem [9] this intersection is nonempty if and only if every three of the balls intersect. We can thus consider triples of balls as follows. Let rgk = min{r : Bi(r) n Bj(r) f l Bk(r) f 0). Then r* = maxrvk and the point (x, y ) is selected as the unique point in Bi(r*) fl Bj(r*) n Bk(r*) where i, j , k are such that rijk = r*. The goal of the present paper is to develop an algorithm for this weighted 1-center problem which runs in 0 (n (log r~)~(log log n)2) time.