短键的最佳排序电路
Optimal Sorting Circuits for Short Keys
摘要：
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最近的一项工作（SODA'21）表明，如果要排序的元素具有长度为$k = o(\log n)$的短键，那么通过利用非基于比较的技术，确实可以克服排序电路的$n\log n$障碍。更具体地说，Asharov等人。结果表明，在不考虑$poly\log^*$因素的情况下，对于$k$位密钥，存在$O(n) \cdot \min(k, \log n)$大小的排序电路.有趣的是，Farhadi等人（STOC'19）和Asharov等人（SODA'21）的最近工作也表明，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$k$基本上是最优的。还应注意，证明任何{ \it无条件}超线性电路的下界对于广泛的一类问题是超出当前技术的范围的。不幸的是，Asharov等人采取的方法。为了实现尺寸的最优性，在某种程度上关键地依赖于牺牲深度：具体地，即使对于1位密钥，他们的电路在深度上也是超{ \it poly}对数的。Asharov等人~将其表述为一个悬而未决的问题，即如何在规模和深度上实现最佳化。在本文中，我们将弥合我们理解中的这一重要差距。我们构造了一个大小为$O(n) \cdot \min(k, \log n)$（忽略$poly\log^*$项）和深度为$O(\log n)$的排序电路.为了实现这一点，我们的方法显着偏离以前的作品。我们的结果可以看作是Ajtai，Komlós和Szemerédi（STOC'83）的里程碑式结果的推广，同时在大小和深度方面。具体而言，对于$k = o(\log n)$，我们实现了渐进的AKS排序电路的大小的改进，同时保持深度的最优性。
英文摘要：
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, 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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