Improved All-Pairs Approximate Shortest Paths in Congested Clique

TL;DR

This work advances APSP in the Congested-Clique by delivering a randomized $O(1)$-approximation in $O(\log\log\log n)$ rounds for weighted undirected graphs, a first sub-logarithmic-round result for this problem. The authors introduce a modular framework that iteratively sharpens an initial $O(\log n)$-approximation via a novel $a\Rightarrow O(\sqrt{a})$-approximation reduction, aided by new constructs such as $k$-nearest $\beta$-hopsets and skeleton graphs to compress the global problem. By combining fast hopset construction, fast $k$-nearest computations, and skeleton-based reductions, they obtain a spectrum of tradeoffs: early termination yields $O(t)$ rounds with $O(\log^{1/2^t} n)$-approximation, and, in particular, $O(1)$ rounds with $O(\log^{\varepsilon} n)$-approximation for any fixed $\varepsilon>0$. The core techniques—hopsets tuned to nearest-neighbor distances, efficient matrix-exponentiation-like rounds, and skeleton-based problem refactoring—offer potential extensions to general graphs, weighted.regular bandwidth regimes, and related distributed models, and stimulate several open questions on reducing constants and extending to MPC-like settings.

Abstract

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.

Figures (2)

• Figure 1: Illustration of the selection of $t_i$.
• Figure 2: Illustration of the construction. Red nodes are skeleton nodes.

Theorems & Definitions (69)

• Theorem 1.1: name=APSP approximation,restate=main
• Theorem 1.2: name=Round-approximation tradeoff, restate=truncatedalgorithm
• Theorem 2.1
• Lemma 2.1: name=Lenzen-routing
• Lemma 2.2: CensorHillel2021FastDA
• Lemma 3.1: name=Approximation factor reduction,restate=approximatereduction
• Lemma 3.2: name=Computing $\sqrt{n}$-nearest $\beta$-hopsets, restate=ahop
• Lemma 3.3: name=Computing $k$-nearest nodes,restate=matrixexp
• Lemma 3.4: name=Skeleton graphs -- simplified version, restate=skel
• Claim 4.1