Quantum data structure for range minimum query

Abstract

Given an array a[1..n], the Range Minimum Query (RMQ) problem is to maintain a data structure that supports RMQ queries: given a range [l,r], find the index of the minimum element among a[l..r], i.e., arg mini∈[l,r]a[i]. In this paper, we propose a quantum data structure that supports RMQ queries and range updates, with an optimal time complexity Θ˜(nq) for performing q=O(n) operations without preprocessing, compared to the classical Θ˜(n+q).1 As an application, we obtain a time-efficient quantum algorithm for k-minimum finding without the use of quantum random access memory.