Sorting Short Keys in Circuits of Size ${o(n \log n)}$

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 $w$-bit payload. A long-standing open whether there exist $(k + w) \cdot o(n \log n)$-sized Boolean circuits for sorting. landmark result in this area work by Ajtai, Komlós, Szemerédi (An $O(n n)$ network, STOC'83), they showed how to achieve O(n gates. The recent Farhadi et al. (Lower bounds external memory integer via network coding, STOC'19) that if famous Li-Li coding conjecture true, then size $w do not general $k$; however, no unconditional lower bound known (in fact proving superlinear circuit out reach existing techniques). In paper, we show one can overcome $n\log n$ barrier when keys be sorted are short. Specifically, prove k) (\log^*n - \log^* (w k))^{2+\epsilon}$ gates capable any Therefore, short, say, $k < o(\log n)$, our asymptotically better than (ignoring ${\sf poly}\log^*$ terms); also $n such cases. Such might surprising initially because it long comparator-based techniques must incur $\Omega(n comparator even only 1-bit (e.g., see Knuth's “Art Programming” textbook). To best knowledge, first nontrivial results using non-comparison-based techniques. upper optimal, barring terms, every $k$ as = O(\log n)$.