# Fast Approximate Shortest Paths in the Congested Clique

## TL;DR

This paper introduces fast deterministic algorithms for approximate shortest path computations in the congested clique model, achieving sub-polynomial round complexities for various approximation guarantees and extending to other distance problems.

## Contribution

It presents the first sub-polynomial constant-factor approximation for APSP and the first sub-polynomial approximation for multiple sources, using new distance tools and improved matrix multiplication algorithms.

## Key findings

- Achieves a (2+ε)-approximation for APSP in O(log^2 n / ε) rounds.
- Provides a (1+ε)-approximation for multi-source shortest paths from O(√n) sources.
- Improves algorithms for diameter approximation and exact single-source shortest paths.

## Abstract

We design fast deterministic algorithms for distance computation in the congested clique model. Our key contributions include: -- A $(2+\epsilon)$-approximation for all-pairs shortest paths in $O(\log^2{n} / \epsilon)$ 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 $(1+\epsilon)$-approximation for multi-source shortest paths from $O(\sqrt{n})$ sources in $O(\log^2{n} / \epsilon)$ 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 $\tilde{O}(n^{1/6})$ rounds.