Sorting Short Integers

TL;DR

This paper presents new boolean circuit constructions for sorting multiple integers efficiently, improving previous results and addressing open questions in the field of circuit complexity for sorting algorithms.

Contribution

It introduces optimized boolean circuits for sorting integers and their prefixes, advancing the theoretical understanding of sorting circuit complexity.

Findings

Constructed circuits with size $O(nm^2)$ and depth $O( ext{log}(n) + m ext{log}(m))$

Developed circuits for sorting based on first $k$ bits with size $O(nmk(1 + ext{log}^*(n) - ext{log}^*(m)))$

Resolved open questions from prior work by Asharov et al.

Abstract

We build boolean circuits of size $O(nm^2)$ and depth $O(\log(n) + m \log(m))$ for sorting $n$ integers each of $m$-bits. We build also circuits that sort $n$ integers each of $m$-bits according to their first $k$ bits that are of size $O(nmk(1 + \log^*(n) - \log^*(m)))$ and depth $O(\log^{3}(n))$. This improves on the result of Asharov et al. arXiv:2010.09884 and resolves some of their open questions.