Sorting Short Integers

Abstract

We build boolean circuits of size $O(n{m}^2)$ 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 $O(nmk(1+{\log }^{\ast }(n)-{\log }^{\ast }(m)))$ 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.