On Space Efficient Two Dimensional Range Minimum Data Structures

The two dimensional range minimum query problem is to preprocess a static two dimensional m by n array A of size N = m . n, such that subsequent queries, asking for the position of the minimum element in a rectangular range within A, can be answered efficiently. We study the trade-off between the space and query time of the problem. We show that every algorithm enabled to access A during the query and using O(N/c) bits additional space requires, Omega(c) query time, for any c where 1 <= c <= N. This :lower bound holds for any dimension. In particular, for the one dimensional version of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits additional space which can be preprocessed in O(N) time and achieves O(c log(2) c) query time. For c = O(1), this is the first O(1) query time algorithm using optimal O(N) bits additional space. For the case where queries can not probe A, we give a data structure of size O(N . min{m, log n}) bits with O(1) query time, assuming m <= n. This leaves a gap to the lower bound of Omega(N log m) bits for this version of the problem.