JOURNAL “P ALGORITHMS 12, 38-56 (1991) Finding k Points with Minimum Diameter and Related Problems ALOK AGCARWAL IBM T J. Watson Research Center, Yorktown Heights, New York 10598 HI ROSHI IMAI Kyushu University, Japan NAOKIKATOH Kobe UniL,ersity of Commerce, Japan AND Su~HAsf3 Sum Bell Communications Research, Morristown, New Jersey 07960 Received April 27, 1989 Let S be a set consisting of n points in the plane. We consider the problem of finding k points of S that form a “small” set under some given measure, and present efficient algorithms for several natural measures including the diameter and the variance. ej 1991 Academic Press, 1nc. 1. INTRODUCTION We consider the problem of selectinga specified number of points, k, from a given set S, subject to some optimization criterion. Problems of this type often arise in statistical clustering and pattern recognition (see Andrews [3] and Hartigan [7]). From an algorithmic standpoint, these problems usually can be solved in time O(nk+c), where 12 is the number of points in S and c is a small constant; for arbitrary k, this time complexity is exponential in the size of theinput. Finding general methods to solve this problem more efficiently for a wide variety of optimization criteria is a challenging and elusive goal, and most researchers have concentrated on fixed values of k. A notable exception is thepaper by Dobkin, Drysdale, and Guibas [6], where some general methods for finding smallest polygons are developed. Among the results presented in [6] are algorithms for 38 0196-6774/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.