Selection in Monotone Matrices and Computing kth Nearest Neighbors

' ŽŽ . . We present an O m q n n log n time algorithm to select the k th smallest item from an m = n totally monotone matrix for any k F mn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for a generalized 4 row-selection problem in monotone matrices. Given a set S s p , . . . , p of n 1 n 4 points in convex position and a vector k s k , . . . , k , we also present an 1 n Ž 4r3 c . O n log n algorithm to compute the k th nearest neighbor of p for every i i i F n; here c is an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points of S are arbitrary, then the k th nearest neighbor of p , for all i F n, can be i i Ž 7r5 c . computed in time O n log n , which also improves upon the previously bestknown result. Q 1996 Academic Press, Inc.