Improved Bounds for Range Mode and Range Median Queries

Abstract

We investigate the following problem: Given a list of n items and a function defined over lists of these items, generate a bounded amount of auxiliary information such that range queries asking for the value of the function on sub-lists can be answered within a certain time bound.

For the function “mode” we improve the previously known time bound O(n ^εlogn) to O(n ^ε) with space O(n ^{2 − 2ε}), where 0 ≤ ε< 1/2. We improve the space bound O(n ²loglogn/logn) for an O(1) time bounded solution to O(n ²/logn).

For the function “median” the space bound O(n ²loglogn/logn) is improved to O(n ²log^(k) n/logn) for an O(1) time solution, where k is an arbitrary constant and log^(k) is the iterated logarithm.