Counting Ones Without Broadword Operations

TL;DR

This paper establishes tight bounds on counting the number of ones in a binary word using basic operations, providing both lower and nearly matching upper bounds.

Contribution

It introduces a new lower bound for counting ones with minimal operations and presents an almost matching upper bound, advancing understanding of computational complexity for this problem.

Findings

Lower bound of Ω(min(ν(x), n-ν(x))) for counting ones.

Almost matching upper bound achieved.

Operates with increment, decrement, logical, and assignment operations.

Abstract

A lower time bound $\Omega(\min(\nu(x), n-\nu(x))$ for counting the number of ones in a binary input word $x$ of length $n$ is presented, where $\nu(x)$ is the number of ones. The operations available are increment, decrement, bit-wise logical operations, and assignment. The only constant available is zero. An almost matching upper bound is also obtained.