Linear-Space Data Structures for Range Mode Query in Arrays

Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, Bryan T. Wilkinson

Abstract: A mode of a multiset S is an element a∈S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1:n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i,j) for which a mode of A[i:j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (Proceedings of the International Symposium on Algorithms and Computation (ISAAC), pp. 517–526, 2003), requires $\varTheta (\sqrt{n}\log\log n)$ query time in the worst case. We improve their result and present an O(n)-space data structure that supports range mode queries in $O(\sqrt{n/\log n})$ worst-case time. In the external memory model, we give a linear-space data structure that requires $O(\sqrt{n/B})$ I/Os per query, where B denotes the block size. Furthermore, we present strong evidence that a query time significantly below $\sqrt{n}$ cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two $\sqrt{n} \times \sqrt{n}$ matrices reduces to n range mode queries in an array of size O(n).Additionally, we give linear-space data structures for the dynamic problem (queries and updates in near O(n 3/4) time), for orthogonal range mode in d dimensions (queries in near O(n 1−1/2d ) time) and for half-space range mode in d dimensions (queries in $O(n^{1-1/d^{2}})$ time). Finally, we complement our dynamic data structure with a reduction from the multiphase problem, again supporting that we cannot hope for much more efficient data structures.