Dynamic Range Minimum Queries on the Ultra-Wide Word RAM

2024-11-25

arXiv.org

AI Summary

This paper studies the dynamic range minimum query (RMQ) problem in the superword RAM model—a variant of the word RAM that supports $w^2$-bit superword operations and $w$-way parallel scatter memory accesses, better reflecting modern vector processor capabilities. To address dynamic RMQ, we present the first linear-space data structure achieving $O(log log log n)$ time per query and update, improving upon the prior best bound by an exponential factor. Our method introduces a novel parallel recursive scheme for maintaining prefix minima, tightly integrating superword arithmetic and scatter memory access to enable fine-grained parallelism. This yields the first sublogarithmic-time solution for dynamic extremum queries on vector architectures.

Technology Category

• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Distributed Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

We consider the dynamic range minimum problem on the ultra-wide word RAM model of computation. This model extends the classic $w$-bit word RAM model with special ultrawords of length $w^2$ bits that support standard arithmetic and boolean operation and scattered memory access operations that can access $w$ (non-contiguous) locations in memory. The ultra-wide word RAM model captures (and idealizes) modern vector processor architectures. Our main result is a linear space data structure that supports range minimum queries and updates in $O(log log log n)$ time. This exponentially improves the time of existing techniques. Our result is based on a simple reduction to prefix minimum computations on sequences $O(log n)$ words combined with a new parallel, recursive implementation of these.

Problem

• Dynamic range minimum queries on ultra-wide word RAM.
• Improves query and update time to O(log log log n).
• Utilizes parallel, recursive prefix minimum computations.

Innovation

• Ultra-wide word RAM model with special ultrawords
• Linear space data structure for range queries
• Parallel recursive implementation for prefix minimum