An efficient algorithm for row minima computations in monotone matrices

A matrix A of size m×n containing items from a totally ordered universe is termed monotone if for every i, j, 1⩽i<j⩽m, the minimum value in row j lies below or to the right of the minimum in row i. Monotone matrices and variations thereof are known to have many important applications. In particular, the problem of computing the row minima of a monotone matrix is of importance in image processing, pattern recognition, text editing, facility location, optimization, and VLSI. Our first main contribution is to show that the task of computing the row minima of an m×n monotone matrix, 1⩽m⩽n, pretiled onto a basic reconfigurable mesh of the same size can be performed in O(log n) time if m=1,2 and in O(log n/log m log log m) time if m>2. Our second contribution is to exhibit a number of non-trivial lower bounds for matrix search problems. These lower bound results hold for arbitrary infinite two-dimensional reconfigurable meshes as long as the input is pretiled onto an n×n submesh thereof. Specifically, in this context we show that every algorithm that solves the problem of computing the minimum of an n×n matrix must take Ω(log log n) time. The same lower bound is shown to hold for the problem of computing the minima in each row of all arbitrary n×n matrix. As a by product we obtain an Ω(log log n) time lower bound for the problem of selecting the k-th smallest item in a monotone matrix, thus extending the previously best known lower bound for selection of the reconfigurable mesh. Finally, we show an almost tight Ω(√(log log n)) time lower bound for the task of computing the row minima of a monotone n×n matrix