In this paper, we present new algorithms for approximating All-Pairs Shortest Paths (APSP) in the Congested Clique model. We present randomized algorithms for weighted undirected graphs. Our first contribution is an $O(1)$-approximate APSP algorithm taking just $O(\log \log \log n)$ rounds. Prior to our work, the fastest algorithms that give an $O(1)$-approximation for APSP take $\operatorname{poly}(\log{n})$ rounds in weighted undirected graphs, and $\operatorname{poly}(\log \log n)$ rounds in unweighted undirected graphs. If we terminate the execution of the algorithm early, we obtain an $O(t)$-round algorithm that yields an $O \big( (\log n)^{1/2^t} \big) $ distance approximation for a parameter $t$. The trade-off between $t$ and the approximation quality provides flexibility for different scenarios, allowing the algorithm to adapt to specific requirements. In particular, we can get an $O \big( (\log n)^{1/2^t} \big) $-approximation for any constant $t$ in $O(1)$-rounds. Such result was previously known only for the special case that $t=0$. A key ingredient in our algorithm is a lemma that allows to improve an $O(a)$-approximation for APSP to an $O(\sqrt{a})$-approximation for APSP in $O(1)$ rounds. To prove the lemma, we develop several new tools, including $O(1)$-round algorithms for computing the $k$ closest nodes, a certain type of hopset, and skeleton graphs.

翻译：本文提出了在拥塞团簇（Congested Clique）模型下近似计算全对最短路径（APSP）的新算法。针对加权无向图，我们提出了随机算法。首要贡献是一种仅需$O(\log \log \log n)$轮通信的$O(1)$近似APSP算法。在此之前，对于加权无向图中APSP的$O(1)$近似，最快算法需要$\operatorname{poly}(\log{n})$轮，而在无权无向图中需要$\operatorname{poly}(\log \log n)$轮。如果提前终止算法执行，可得到一种$O(t)$轮算法，对于参数$t$可提供$O \big( (\log n)^{1/2^t} \big) $距离近似。该参数$t$与近似质量之间的权衡为不同场景提供了灵活性，使算法能适应特定需求。特别地，对于任意常数$t$，我们能在$O(1)$轮内获得$O \big( (\log n)^{1/2^t} \big)$近似。此前这种结果仅对$t=0$特例已知。算法核心是一种引理，可将APSP的$O(a)$近似在$O(1)$轮内改进为$O(\sqrt{a})$近似。为证明该引理，我们开发了多项新工具，包括计算$k$最近邻节点的$O(1)$轮算法、特定类型的跳表（hopset）以及骨架图（skeleton graphs）。