In this paper, we present a new randomized $O(1)$-approximation algorithm for the All-Pairs Shortest Paths (APSP) problem in weighted undirected graphs that runs in just $O(\log \log \log n)$ rounds in the Congested-Clique model. Before our work, the fastest algorithms achieving an $O(1)$-approximation for APSP in weighted undirected graphs required $\operatorname{poly}(\log n)$ rounds, as shown by Censor-Hillel, Dory, Korhonen, and Leitersdorf (PODC 2019 & Distributed Computing 2021). In the unweighted undirected setting, Dory and Parter (PODC 2020 & Journal of the ACM 2022) obtained $O(1)$-approximation in $\operatorname{poly}(\log \log n)$ rounds. By terminating our algorithm early, for any given parameter $t \geq 1$, we obtain an $O(t)$-round algorithm that guarantees an $O\left(\log^{1/2^t} n\right)$ approximation in weighted undirected graphs. This tradeoff between round complexity and approximation factor offers flexibility, allowing the algorithm to adapt to different requirements. In particular, for any constant $\varepsilon > 0$, an $O\left(\log^\varepsilon n\right)$-approximation can be obtained in $O(1)$ rounds. Previously, $O(1)$-round algorithms were only known for $O(\log n)$-approximation, as shown by Chechik and Zhang (PODC 2022). A key ingredient in our algorithm is a lemma that, under certain conditions, allows us to improve an $a$-approximation for APSP to an $O(\sqrt{a})$-approximation in $O(1)$ rounds. To prove this lemma, we develop several new techniques, including an $O(1)$-round algorithm for computing the $k$-nearest nodes, as well as new types of hopsets and skeleton graphs based on the notion of $k$-nearest nodes.

翻译：本文针对加权无向图中的全对最短路径问题，提出了一种新的随机化$O(1)$近似算法，该算法在拥塞团模型中仅需$O(\log \log \log n)$轮即可完成。在我们工作之前，如Censor-Hillel、Dory、Korhonen和Leitersdorf（PODC 2019与分布式计算2021）所示，在加权无向图中实现$O(1)$近似的APSP最快算法需要$\operatorname{poly}(\log n)$轮。在无权无向图设置中，Dory和Parter（PODC 2020与ACM期刊2022）在$\operatorname{poly}(\log \log n)$轮内获得了$O(1)$近似。通过提前终止我们的算法，对于任意给定参数$t \geq 1$，我们获得了一个$O(t)$轮算法，该算法在加权无向图中保证$O\left(\log^{1/2^t} n\right)$的近似比。这种轮复杂度与近似因子之间的权衡提供了灵活性，使算法能够适应不同的需求。特别地，对于任意常数$\varepsilon > 0$，可以在$O(1)$轮内获得$O\left(\log^\varepsilon n\right)$近似。此前，如Chechik和Zhang（PODC 2022）所示，仅已知$O(1)$轮算法可实现$O(\log n)$近似。我们算法的一个关键组成部分是一个引理，该引理在特定条件下允许我们在$O(1)$轮内将APSP的$a$近似改进为$O(\sqrt{a})$近似。为了证明这一引理，我们开发了若干新技术，包括计算$k$最近邻节点的$O(1)$轮算法，以及基于$k$最近邻节点概念的新型跳集和骨架图。