Two-Dimensional Range Minimum Queries

We consider the two-dimensional Range Minimum Query problem: for a static (m × n)-matrix of size N = mn which may be preprocessed, answer on-line queries of the form “where is the position of a minimum element in an axis-parallel rectangle?”. Unlike the onedimensional version of this problem which can be solved in provably optimal time and space, the higher-dimensional case has received much less attention. The only result we are aware of is due to Gabow, Bentley and Tarjan [1], who solve the problem in O(N log N) preprocessing time and space and O(log N) query time. We present a class of algorithms which can solve the 2-dimensional RMQ-problem with O(kN) additional space, O(N log[k+1] N) preprocessing time and O(1) query time for any k > 1, where log[k+1] denotes the iterated application of k + 1 logarithms. The solution converges towards an algorithm with O(N log∗ N) preprocessing time and space and O(1) query time. All these algorithms are significant improvements over th...

• Type: Conference
• Year: 2007
• Where: CPM
• Authors: Amihood Amir, Johannes Fischer, Moshe Lewenstein
• Keywords: CPM 2007, Minimum Query Problem, Preprocessing, Query Time