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 $\Omega(\frac{{\rm log} n}{\log (Sw/n)})$ 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 $O(\frac{n}{\varepsilon})$ space and supports (1 + ε)-approximation queries in $O({\rm log} {\frac {1}{\varepsilon}})$ time.