We design fast deterministic algorithms for distance computation in the Congested Clique model. Our key contributions include: A $$(2+\epsilon )$$ ( 2 + ϵ ) -approximation for all-pairs shortest paths in $$O(\log ^2{n} / \epsilon )$$ O ( log 2 n / ϵ ) rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model. A $$(1+\epsilon )$$ ( 1 + ϵ ) -approximation for multi-source shortest paths from $$O(\sqrt{n})$$ O ( n ) sources in $$O(\log ^2{n} / \epsilon )$$ O ( log 2 n / ϵ ) rounds on weighted undirected graphs. This is the first sub-polynomial algorithm obtaining this approximation for a set of sources of polynomial size. Our main techniques are new distance tools that are obtained via improved algorithms for sparse matrix multiplication, which we leverage to construct efficient hopsets and shortest paths. Furthermore, our techniques extend to additional distance problems for which we improve upon the state-of-the-art, including diameter approximation, and an exact single-source shortest paths algorithm for weighted undirected graphs in $$\tilde{O}(n^{1/6})$$ O ~ ( n 1 / 6 ) rounds.

中文翻译：

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拥塞团中的快速近似最短路径

我们为 Congested Clique 模型中的距离计算设计了快速确定性算法。我们的主要贡献包括： A $$(2+\epsilon )$$ ( 2 + ϵ ) - $O(\log ^2{n} / \epsilon )$$ O ( log 2 n / ϵ ) 在未加权的无向图上循环。使用一个小的附加附加因子，这也适用于加权图。这是该模型中 APSP 的第一个子多项式常数因子近似。A $$(1+\epsilon )$$ ( 1 + ϵ ) - 来自 $$O(\log ^ 中的 $$O(\sqrt{n})$$O ( n ) 源的多源最短路径的近似值2{n} / \epsilon )$$ O ( log 2 n / ϵ ) 在加权无向图上循环。这是第一个为一组多项式大小的源获得这种近似的子多项式算法。我们的主要技术是新的距离工具，这些工具是通过改进的稀疏矩阵乘法算法获得的，我们利用它来构建高效的跳跃集和最短路径。此外，我们的技术扩展到额外的距离问题，我们改进了最先进的技术，包括直径近似和精确的单源最短路径算法，用于 $$\tilde{O}(n ^{1/6})$$ O ~ ( n 1 / 6 ) 轮。