Towards Optimal Range Medians

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 needs O(n log k + k logn) time to answer k such median queries. This improves previous algorithms by a logarithmic factor and matches a comparison lower bound for k = O(n). The space complexity of our simple algorithm is O(n logn) in the pointer-machine model, and O(n) in the RAM model. In the latter model, a more involved O(n) space data structure can be constructed in O(n logn) time where the time per query is reduced to O(logn/ log log n). We also give efficient dynamic variants of both data structures, achieving O(log n) query time using O(n logn) space in the comparison model and O((logn/ log logn)) query time using O(n logn/ log log n) space in the RAM model, and show that in the cell-probe model, any data structure which supports updates in O(log n) time must have Ω(logn/ log log n) query time. Our approach naturally generalizes to higher-dimensional range median problems, where element positions and query ranges are multidimensional — it reduces a range median query to a logarithmic number of range counting queries.