Improved Distributed Network Decomposition, Hitting Sets, and Spanners, via Derandomization

TL;DR

This paper introduces improved deterministic algorithms for distributed graph problems like network decomposition, hitting sets, and spanners, using novel derandomization techniques to achieve near-optimal parameters efficiently.

Contribution

It develops new randomized algorithms analyzed with pairwise independence, enabling efficient derandomization and leading to the first near-optimal deterministic network decomposition.

Findings

Deterministic construction of $O(\log n)$-color network decomposition

Achieves $O(\log n imes \log\log\log n)$-strong diameter in $ ilde{O}(\log^3 n)$ rounds

Improves upon previous deterministic network decomposition results

Abstract

This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for $O(\log n)$-color $O(\log n \cdot \log\log\log n)$-strong diameter network decomposition in $\tilde{O}(\log^3 n)$ rounds. This is the first construction that achieves almost $\log n$ in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozho\v{n}, Ghaffari STOC'20; Ghaffari, Grunau, Rozho\v{n} SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozho\v{n}, Grunau FOCS'22].