Encodings for Range Minimum Queries over Bounded Alphabets

TL;DR

This paper studies space-efficient encodings for range minimum queries on arrays with small alphabets, providing near-optimal solutions for 1D and analyzing bounds for 2D cases.

Contribution

It introduces near-optimal encoding schemes for 1D RMQ with small alphabets and analyzes bounds for 2D RMQ queries, filling a gap in existing research.

Findings

Near-optimal space encoding for 1D RMQ with constant-sized alphabets

Derived lower bounds for 2D RMQ encoding space

Matching upper bounds for 2D RMQ in most cases

Abstract

Range minimum queries (RMQs) are fundamental operations with widespread applications in database management, text indexing and computational biology. While many space-efficient data structures have been designed for RMQs on arrays with arbitrary elements, there has not been any results developed for the case when the alphabet size is small, which is the case in many practical scenarios where RMQ structures are used. In this paper, we investigate the encoding complexity of RMQs on arrays over bounded alphabet. We consider both one-dimensional (1D) and two-dimensional (2D) arrays. For the 1D case, we present a near-optimal space encoding. For constant-sized alphabets, this also supports the queries in constant time. For the 2D case, we systematically analyze the 1-sided, 2-sided, 3-sided and 4-sided queries and derive lower bounds for encoding space, and also matching upper bounds that support efficient queries in most cases. Our results demonstrate that, even with the bounded alphabet restriction, the space requirements remain close to those for the general alphabet case.