# Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization

## TL;DR

This paper introduces a polylogarithmic-time deterministic distributed algorithm for network decomposition, resolving longstanding open problems and enabling derandomization of many distributed algorithms, thus significantly advancing the field.

## Contribution

It presents the first polylogarithmic-time deterministic network decomposition algorithm, settling key open problems and establishing a derandomization theorem for distributed algorithms.

## Key findings

- Improves the network decomposition time from $2^{O(\sqrt{\log n})}$ to polylogarithmic.
- Enables deterministic algorithms for problems like maximal independent set and graph coloring.
- Shows that randomness is unnecessary for efficient distributed algorithms in the polylogarithmic-time regime.

## Abstract

We present a simple polylogarithmic-time deterministic distributed algorithm for network decomposition. This improves on a celebrated $2^{O(\sqrt{\log n})}$-time algorithm of Panconesi and Srinivasan [STOC'92] and settles a central and long-standing question in distributed graph algorithms. It also leads to the first polylogarithmic-time deterministic distributed algorithms for numerous other problems, hence resolving several well-known and decades-old open problems, including Linial's question about the deterministic complexity of maximal independent set [FOCS'87; SICOMP'92]---which had been called the most outstanding problem in the area. The main implication is a more general distributed derandomization theorem: Put together with the results of Ghaffari, Kuhn, and Maus [STOC'17] and Ghaffari, Harris, and Kuhn [FOCS'18], our network decomposition implies that $$\mathsf{P}\textit{-}\mathsf{RLOCAL} = \mathsf{P}\textit{-}\mathsf{LOCAL}.$$ That is, for any problem whose solution can be checked deterministically in polylogarithmic-time, any polylogarithmic-time randomized algorithm can be derandomized to a polylogarithmic-time deterministic algorithm. Informally, for the standard first-order interpretation of efficiency as polylogarithmic-time, distributed algorithms do not need randomness for efficiency. By known connections, our result leads also to substantially faster randomized distributed algorithms for a number of well-studied problems including $(\Delta+1)$-coloring, maximal independent set, and Lov\'{a}sz Local Lemma, as well as massively parallel algorithms for $(\Delta+1)$-coloring.