Improved all-pairs approximate shortest paths in congested clique*

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 rounds in the Congested-Clique model. Before our work, the fastest algorithms achieving an O(1)-approximation for APSP in weighted undirected graphs required 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 rounds. By terminating our algorithm early, for any given parameter, we obtain an O(t)-round algorithm that guarantees an 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, an -approximation can be obtained in O(1) rounds. Previously, O(1)-round algorithms were only known for -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 -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.