Sorted Range Reporting and Range Minima Queries

TL;DR

This paper introduces a new optimal trade-off data structure for sorted range reporting that uses linear space and answers queries in O(k log k) time, along with a simple method for range minima queries.

Contribution

It presents a linear space data structure for sorted range reporting with O(k log k) query time, offering an alternative to the existing optimal solution, and introduces a simple approach for range minima queries.

Findings

Achieves linear space complexity.

Provides O(k log k) query time for range selection.

Introduces a simple, practical method for range minima queries.

Abstract

Given an array A[1: n] of n elements drawn from an ordered set, the sorted range selection problem is to build a data structure that can be used to answer the following type of queries efficiently: Given a pair of indices i, j $ (1\le i\le j \le n)$, and a positive integer k, report the k smallest elements from the sub-array A[i: j] in order. Brodal et al. (Brodal, G.S., Fagerberg, R., Greve, M., and L{\'o}pez-Ortiz, A., Online sorted range reporting. Algorithms and Computation (2009) pp. 173--182) introduced the problem and gave an optimal solution. After O(n log n) time for preprocessing, the query time is O(k). The space used is O(n). In this paper, we propose the only other possible optimal trade-off for the problem. We present a linear space solution to the problem that takes O(k log k) time to answer a range selection query. The preprocessing time is O(n). Moreover, the proposed algorithm reports the output elements one by one in non-decreasing order. Our solution is simple and practical. We also describe an extremely simple method for range minima queries (most of whose parts are known) which takes al most (but not exactly) linear time. We believe that this method may be, in practice, faster and easier to implement in most cases.