Towards optimal range medians

Abstract

We consider the following problem: given an unsorted array of n elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which uses O(n) space and needs O(nlogk+klogn) time to answer k such median queries. This improves previous algorithms by a logarithmic factor and matches a lower bound for k=O(n). Since, in contrast to previous approaches, the algorithm decomposes the range of element values rather than the array, it has natural generalizations to higher-dimensional problems - it reduces a range median query to a logarithmic number of range counting queries. © 2009 Springer Berlin Heidelberg.