Sorting Short Keys in Circuits of Size o(n log n)

TL;DR

This paper presents a novel boolean circuit design that sorts arrays with short keys more efficiently than traditional methods, breaking the $n \log n$ barrier for certain key lengths, and introduces non-comparison-based techniques.

Contribution

It demonstrates the first non-trivial sorting circuit results using non-comparison-based methods for short keys, surpassing classical bounds for specific key sizes.

Findings

Achieves sorting with $(k + w) imes O(n k) imes ext{poly}( ext{log}^* n - ext{log}^* (w + k))$ gates.

Surpasses classical $n \log n$ bounds for short keys ($k < o( ext{log} n)$).

Shows the upper bound is optimal under the Li-Li network coding conjecture for $k=O( ext{log} n)$.

Abstract

We consider the classical problem of sorting an input array containing $n$ elements, where each element is described with a $k$-bit comparison-key and a $w$-bit payload. A long-standing open problem is whether there exist $(k + w) \cdot o(n \log n)$-sized boolean circuits for sorting. We show that one can overcome the $n\log n$ barrier when the keys to be sorted are short. Specifically, we prove that there is a circuit with $(k + w) \cdot O(n k) \cdot \poly(\log^*n - \log^* (w + k))$ boolean gates capable of sorting any input array containing $n$ elements, each described with a $k$-bit key and a $w$-bit payload. Therefore, if the keys to be sorted are short, say, $k < o(\log n)$, our result is asymptotically better than the classical AKS sorting network (ignoring $\poly\log^*$ terms); and we also overcome the $n \log n$ barrier in such cases. Such a result might be surprising initially because it is long known that comparator-based techniques must incur $\Omega(n \log n)$ comparator gates even when the keys to be sorted are only $1$-bit long (e.g., see Knuth's "Art of Programming" textbook). To the best of our knowledge, we are the first to achieve non-trivial results for sorting circuits using non-comparison-based techniques. We also show that if the Li-Li network coding conjecture is true, our upper bound is optimal, barring $\poly\log^*$ terms, for every $k$ as long as $k = O(\log n)$.