Submatrix Maximum Queries in Monge Matrices and Partial Monge Matrices, and Their Applications

Abstract

We describe a data structure for submatrix maximum queries in Monge matrices or partial Monge matrices, where a query seeks the maximum element in a contiguous submatrix of the given matrix. The structure, for an n × n Monge matrix, takes O ( n log n ) space and O ( n log n ) preprocessing time, and answers queries in O (log 2 n ) time. For partial Monge matrices, the space grows by α( n ), the preprocessing grows by α( n )log n (α( n ) is the inverse Ackermann function), and the query remains O (log 2 n ). Our design exploits an interpretation of the column maxima in a Monge (partial Monge, respectively) matrix as an upper envelope of pseudo-lines (pseudo-segments, respectively). We give two applications: (1) For a planar set of n points in an axis-parallel rectangle B , we build a data structure, in O ( n α( n )log 4 n ) time and O ( n α( n )log 3 n ) space, that returns, for a query point p , the largest-area empty axis-parallel rectangle contained in B and containing p , in O (log 4 n ) time. This improves substantially the nearly quadratic storage and preprocessing obtained by Augustine et al. [2010]. (2) Given an n -node arbitrarily weighted planar digraph, with possibly negative edge weights, we build, in O ( n log 2 n /log log n ) time, a linear-size data structure that supports edge-weight updates and graph-distance queries between arbitrary pairs of nodes in O ( n 2/3 log 5/3 n ) time per operation. This improves a previous algorithm of Fakcharoenphol and Rao [2006]. Our data structure has already been applied in a recent maximum flow algorithm for planar graphs in Borradaile et al. [2011].