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 × n Monge matrix takes O(n log n) space, O(n log2 n) preprocessing time, and can answer queries in O(log2 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(log4 n) time. The preprocessing time is O(nα(n...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Haim Kaplan, Shay Mozes, Yahav Nussbaum, Micha Sharir
• Keywords: Algorithms, Column Maxima, Monge Matrix, Parallel Rectangle, SODA 2012