Range Minimum Queries in Minimal Space

TL;DR

This paper introduces a space-efficient data structure for range minimum queries that operates in constant or near-constant time, suitable for both online and offline scenarios, with practical implementation showing competitive performance.

Contribution

The paper presents a novel minimal-space data structure for range minimum queries with optimal query times for online and offline processing, improving efficiency and space usage.

Findings

Achieves O(1) query time offline

Achieves O(α(n)) amortized online query time

Uses minimal space proportional to active positions

Abstract

We consider the problem of computing a sequence of range minimum queries. We assume a sequence of commands that contains values and queries. Our goal is to quickly determine the minimum value that exists between the current position and a previous position $i$. Range minimum queries are used as a sub-routine of several algorithms, namely related to string processing. We propose a data structure that can process these commands sequences. We obtain efficient results for several variations of the problem, in particular we obtain $O(1)$ time per command for the offline version and $O(\alpha(n))$ amortized time for the online version, where $\alpha(n)$ is the inverse Ackermann function and $n$ the number of values in the sequence. This data structure also has very small space requirements, namely $O(\ell)$ where $\ell$ is the maximum number active $i$ positions. We implemented our data structure and show that it is competitive against existing alternatives. We obtain comparable command processing time, in the nano second range, and much smaller space requirements.