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 $n\log n$ barrier for sorting circuits, by leveraging non-comparison-based techniques. More specifically, Asharov et al.~showed that there exist $O(n) \cdot \min(k, \log n)$-sized sorting circuits for $k$-bit keys, ignoring $poly\log^*$ 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 {\it 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-{\it poly}logarithmic 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 $O(n) \cdot \min(k, \log n)$ (ignoring $poly\log^*$ 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, Koml\'os, and Szemer\'edi (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.

中文翻译：

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短键的最佳排序电路

在算法和复杂性文献中一个长期存在的开放问题是是否存在大小为$ o（n \ log n）$的排序电路。Asharov，Lin和Shi（SODA'21）的最新工作表明，如果要排序的元素具有长度为$ k = o（\ log n）$的短键，则确实可以克服$ n \ log n通过利用基于非比较的技术来对电路进行分类。更具体地说，Asharov等人〜表明存在$ O（n）\ cdot \ min（k，\ log n）$大小的排序电路，用于$ k $位密钥，而忽略了$ poly \ log ^ * $因素。有趣的是，Farhadi等人的最新著作。（STOC'19）和Asharov等人。（SODA'21）还表明，假设著名的Li-Li网络编码猜想成立，上述结果对于每个密钥长度$ k $本质上都是最佳的。还要注意的是，对于任何种类的问题，证明任何{\ it无条件}超线性电路的下界都是当前技术无法实现的。不幸的是，Asharov等人为了达到最佳尺寸而采取的方法在某种程度上至关重要地依赖于牺牲深度：具体地说，即使对于1位密钥，其电路在深度上也是超级对数的。Asharov等人将其作为一个悬而未决的问题，如何在尺寸和深度上都达到最佳。在本文中，我们弥补了这一认识上的重要差距。我们构造一个大小为$ O（n）\ cdot \ min（k，\ log n）$（忽略$ poly \ log ^ * $项）和深度$ O（\ log n）$的排序电路。为此，我们的方法与先前的工作大不相同。我们的结果可以看作是Ajtai，Koml \'os和Szemer'edi（STOC'83）对具有里程碑意义的结果的概括，在尺寸和深度上同时进行。具体来说，对于$ k = o（\ log n）$，我们在保持深度最优的同时，在AKS分选电路上实现了尺寸上的渐近改进。