Dynamic Range Minimum Queries on the Ultra-Wide Word RAM

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. The goal in the dynamic range minimum problem is to maintain an array $A$ of $n$ $w$-bit integers subject to range minimum queries (given indices $i$ and $j$ return a smallest integer in the subarray $A[i..j]$) and updates (given index $i$ and integer $α$ set $A[i] \leftarrow α$). Our main result is a data structure that supports range minimum queries and updates in $O(\log \log \log n)$ time and uses $O(n/\log n)$ space in addition to the input array. 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.