Cell Probe Lower Bounds and Approximations for Range Mode

Abstract

The mode of a multiset of labels, is a label that occurs at least as often as any other label. The input to the range mode problem is an array A of size n. A range query [i, j] must return the mode of the subarray A[i], A[i+1],...,A[j]. We prove that any data structure that uses S memory cells of w bits needs Ω(log n/log(Sw/n)) time to answer a range mode query. Secondly, we consider the related range k-frequency problem. The input to this problem is an array A of size n, and a query [i, j] must return whether there exists a label that occurs precisely k times in the subarray A[i], A[i+1],...,A[j]. We show that for any constant k > 1, this problem is equivalent to 2D orthogonal rectangle stabbing, and that for k = 1 this is no harder than four-sided 3D orthogonal range emptiness. Finally, we consider approximate range mode queries. A c-approximate range mode query must return a label that occurs at least 1/c times that of the mode. We describe a linear space data structure that supports 3-approximate range mode queries in constant time, and a data structure that uses O(n/e) space and supports (1 + e)-approximation queries in O(log 1/e) time.