Geometric applications of a matrix-searching algorithm

Let A be a matrix with real entries and let j(i) be the index of the leftmost column containing the maximum value in row i of A. A is said to be monotone if i 1 >i 2 implies that j(i 1) ≥J(i 2). A is totally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitrary n x m monotone matrix requires Θ(m log n) time, whereas if the matrix is totally monotone the time is Θ(m) when m≥n and is Θ(m(1 + log(n/m))) when m<n. The problem of finding the maximum value within each row of a totally monotone matrix arises in several geometric algorithms such as the all-farthest-neighbors problem for the vertices of a convex polygon. Previously only the property of monotonicity, not total monotonicity, had been used within these algorithms. We use the Θ(m) bound on finding the maxima of wide totally monotone matrices to speed up these algorithms by a factor of log n. © 1987 Springer-Verlag New York Inc.