Two Dimensional Range Minimum Queries and Fibonacci Lattices

Given a matrix of size N , two dimensional range minimum queries (2D-RMQs) ask for the position of the minimum element in a rectangular range within the matrix. We study trade-offs between the query time and the additional space used by indexing data structures that support 2D-RMQs. Using a novel technique—the discrepancy properties of Fibonacci lattices—we give an indexing data structure for 2D-RMQs that uses O(N/c) bits additional space with O(clog c(log log c) ²) query time, for any parameter c , 4≤c≤N. Also, when the entries of the input matrix are from {0,1}, we show that the query time can be improved to O(clog c) with the same space usage.