SELECTION AND SORTING IN TOTALLY MONOTONE ARRAYS

A two-dimensional array A = {a[i,j]} is called totally monotone if, for all i1 < i2 and j1 < j2, a[i1, j1] < a[i1, j2] implies a[i2, j1] < a[i2, j2]. Totally monotone arrays were introduced in 1987 by Aggarwal, Klawe, Moran, Shor, and Wilber, who showed that several problems in computational geometry and VLSI river routing could be reduced to the problem of finding a maximum entry in each row of a totally monotone array. In this paper we consider several selection and sorting problems involving totally monotone arrays and give a number of applications of solutions for these problems. In particular, we obtain the following results for an m x n totally monotone array A: 1. The k largest (or k smallest) entries in each row of A can be computed in O(k(m + n)) time. This result allows us to determine the k farthest (or k nearest) neighbors of each vertex of a convex n-gon in O(kn) time. 2. Provided the transpose of A is also totally monotone, the k largest (or k smallest) entries overall in A can be computed in O(m + n + k lg(mn/k)) time. This result allows us to find the k farthest (or k nearest) pairs of vertices from a convex n-gon in O(n + k lg(n2/k)) time. 3. The rows of A can be sorted in O(mn) time when m greater-than-or-equal-to n and in O(mn(1 + lg(n/m))) time when m < n. This result allows us to solve the following problem in O(n2 lg l) time: given l convex polygons with a total of n vertices, for all vertices upsilon, sort the other vertices by distance from upsilon. 4. Sorting all the entries of A requires OMEGA(mn lg m) time.