An Optimal Algorithm for Sliding Window Order Statistics

Assume there is a data stream of elements and a window of size m . Sliding window algorithms compute various statistic functions over the last m elements of the data stream seen so far. The time complexity of a sliding window algorithm is measured as the time required to output an updated statistic function value every time a new element is read. For example, it is well known that computing the sliding window maximum/minimum has time complexity O (1) while computing the sliding window median has time complexity O (log m ). In this paper we close the gap between these two cases by (1) presenting an algorithm for computing the sliding window k -th smallest element in O (log k ) time and (2) prove that this time complexity is optimal.