Partial Sums on the Ultra-Wide Word RAM

TL;DR

This paper introduces a new in-place data structure for the partial sums problem on the ultra-wide word RAM model, achieving doubly logarithmic time with constant space overhead, inspired by Fenwick trees.

Contribution

It presents the first in-place ultra-wide word RAM data structure for partial sums with optimal time bounds and constant ultraword space, improving previous superlinear space solutions.

Findings

Supports partial sum operations in doubly logarithmic time.

Uses only a constant number of ultrawords of space.

Based on an efficient parallel implementation of Fenwick trees.

Abstract

We consider the classic partial sums 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 new in-place data structure for the partial sum problem that only stores a constant number of ultraword in addition to the input and supports operations in doubly logarithmic time. This matches the best known time bounds for the problem (among polynomial space data structures) while improving the space from superlinear to a constant number of ultrawords. Our results are based on a simple and elegant in-place word RAM data structure, known as the Fenwick tree. Our main technical contribution is a new efficient parallel ultra-wide word RAM implementation of the Fenwick tree, which is likely of independent interest.