Sorting short keys in circuits of size O(n log n)

Abstract

We consider the classical problem of sorting 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) · o(n log n)sized boolean circuits for sorting. Ajtai, Komlós, and Szemerédi (STOC'83) constructed the famous AKS sorting network with (k + w) · O(n log n) boolean gates. Recently, Farhadi et al. (STOC'19) showed that if the famous Li-Li network coding conjecture is true, then sorting circuits of size w · o(n log n) do not exist for general k (while unconditional circuit lower bound is out of the reach of existing techniques). In this paper, we show that one can overcome the n log n barrier when the comparison-keys are short. Specifically, we construct a sorting circuit with (k + w) · O(nk) · poly(log∗ n− log∗(w + k)) boolean gates, asymptotically better than AKS sorting network if the keys are short, say, k = o(log n) (ignoring poly log∗ terms). Such a result might be surprising since comparator-based techniques must incur Ω(n log n) comparators even when the keys are only 1-bit long (e.g., see Knuth's “Art of Programming” textbook). To the best of our knowledge, this is also the first non-trivial result on non-comparison-based sorting circuits. 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 = O(log n).