Submatrix maximum queries in Monge matrices and Monge partial matrices, and their applications

We describe a data structure for submatrix maximum queries in Monge matrices or Monge partial matrices, where a query specifies a contiguous submatrix of the given matrix, and its output is the maximum element of that submatrix. Our data structure for an n x n Monge matrix takes O(n log n) space, O(n log² n) preprocessing time, and can answer queries in O(log ² n) time. For a Monge partial matrix the space bound and the preprocessing time both grow by the small factor α(n), where α(n) is the inverse Ackermann function. Our design exploits an interpretation of the column maxima in a Monge matrix (resp., Monge partial matrix) as an upper envelope of pseudo-lines (resp., pseudosegments). We give two applications for this data structure: (1) For a set of n points in a rectangle B in the plane, we build a data structure that, given a query point p, returns the largest-area empty axis-parallel rectangle contained in B and containing p, in O(log ⁴ n) time. The preprocessing time is O(nα(n) log⁴ n), and the space required is O(nα(n) log³ n). This improves substantially a previous data structure of Augustine et al. [arXiv:1004.0558] that requires quadratic space. (2) Given an n-node arbitrarily weighted planar digraph, with possibly negative edge weights, we build, in O(n log² n/ log log n) time, a linear-size data structure that supports edge-weight updates and distance queries between arbitrary pairs of nodes (where the distance is minimum weight of a path in the graph between the pair of nodes), in O(n^2/3 log^5/3 n) time for each update and query. This improves the O(n^4/5 log^13/5 n)-time bound of Fakcharoenphol and Rao [JCSS 72, 2006]. Our data structure has already been applied in a recent maximum flow algorithm for planar graphs of Borradaile et al. [FOCS 2011], and we believe it will find additional applications.