Optimal Sorting Circuits for Short Keys
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A long-standing open question in the algorithms and complexity literature is whether there exist sorting circuits of size o(n log n). A recent work by Asharov, Lin, and Shi (SODA'21) showed that if the elements to be sorted have short keys whose length k = o(log n), then one can indeed overcome the nlog n barrier for sorting circuits, by leveraging non-comparison-based techniques. More specifically, Asharov et al. showed that there exist O(n) ·min(k, log n)-sized sorting circuits for k-bit keys, ignoring polylog^* factors. Interestingly, the recent works by Farhadi et al. (STOC'19) and Asharov et al. (SODA'21) also showed that the above result is essentially optimal for every key length k, assuming that the famous Li-Li network coding conjecture holds. Note also that proving any unconditional super-linear circuit lower bound for a wide class of problems is beyond the reach of current techniques. Unfortunately, the approach taken by Asharov et al. to achieve optimality in size somewhat crucially relies on sacrificing the depth: specifically, their circuit is super-polylogarithmic in depth even for 1-bit keys. Asharov et al. phrase it as an open question how to achieve optimality both in size and depth. In this paper, we close this important gap in our understanding. We construct a sorting circuit of size O(n) ·min(k, log n) (ignoring polylog^* terms) and depth O(log n). To achieve this, our approach departs significantly from the prior works. Our result can be viewed as a generalization of the landmark result by Ajtai, Komlós, and Szemerédi (STOC'83), simultaneously in terms of size and depth. Specifically, for k = o(log n), we achieve asymptotical improvements in size over the AKS sorting circuit, while preserving optimality in depth.
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