Linear-Space Data Structures for Range Mode Query in Arrays

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 (ISAAC 2003), requires O( √ n log log n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O( p n/ log n) worst-case time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two √ n × √ n matrices reduces to n range mode queries in an array of size O(n). Additionally, we give linear-space data structures for orthogona...

• Type: Journal
• Year: 2012
• Where: STACS
• Authors: Timothy M. Chan, Stephane Durocher, Kasper Green Larsen, Jason Morrison, Bryan T. Wilkinson
• Keywords: Combinatorial Techniques, Linear Space, Matrix Multiplication, STACS 2012, Theoretical Computer Science