A SWAR Approach to Counting Ones

TL;DR

This paper explores the computational complexity of counting ones in binary representations using various operations, revealing new bounds for algorithms with different sets of allowed operations.

Contribution

It introduces improved complexity bounds for counting ones with addition, logical operations, and multiplication, extending previous results.

Findings

Counting ones with addition and logical operations takes O(log^2(n)) steps.

Parity can be computed in O(log(n)) steps, matching shift-based methods.

Multiplication enables counting ones in O(log^*(n)) time, better than previous bounds.

Abstract

We investigate the complexity of algorithms counting ones in different sets of operations. With addition and logical operations (but no shift) $O(\log^2(n))$ steps suffice to count ones. Parity can be computed with complexity $O(\log(n))$, which is the same bound as for methods using shift-operations. If multiplication is available, a solution of time complexity $O(\log^*(n))$ is possible improving the known bound $O(\log\log(n))$.