Dynamic Range Minimum Queries on the Ultra-Wide Word RAM

TL;DR

This paper introduces a novel data structure for dynamic range minimum queries on the ultra-wide word RAM model, achieving significantly faster query and update times by leveraging vectorized operations and recursive techniques.

Contribution

It presents the first linear space data structure with $O(\log \log \log n)$ query/update time for dynamic range minimum queries on the ultra-wide word RAM.

Findings

Supports range minimum queries and updates in $O(\log \log \log n)$ time.

Achieves exponential improvement over previous methods.

Utilizes a reduction to prefix minimum computations with a recursive parallel implementation.

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.