# Geometric applications of a matrix-searching algorithm

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## Abstract

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*.

## Key words

All-farthest neighbors Monotone matrix Convex polygon Wire routing Inscribed polygons Circumscribed polygons## Preview

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