Encodings for Range Minimum Queries over Bounded Alphabets

2026-04-14

AI Summary

This work addresses the lack of efficient encoding schemes for range minimum queries (RMQ) on one- and two-dimensional arrays over bounded alphabets. It establishes, for the first time, tight information-theoretic space bounds for this setting. By combining lower-bound analysis with combinatorial encoding techniques and carefully designed data structures, the study achieves constant-time queries with near-optimal space in the one-dimensional case under constant-sized alphabets. For two-dimensional RMQ restricted to 1- to 4-sided queries, it presents succinct encodings whose space usage matches the derived lower bounds while supporting efficient query operations. These results substantially advance both the theoretical understanding and practical feasibility of RMQ over bounded alphabets.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Applications of cryptography

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.

Problem

• Range Minimum Queries
• Bounded Alphabets
• Encoding Complexity
• Space Efficiency
• Two-dimensional Arrays

Innovation

• Range Minimum Queries
• Bounded Alphabet
• Space-Efficient Encoding
• Two-Dimensional RMQ
• Query Complexity