Deterministic coloring algorithms in the LOCAL model

We study the problem of bi-chromatic coloring of hypergraphs in the LOCAL distributed model of computation. This problem can easily be solved by a randomized local algorithm with no communication. However, it is not known how to solve it deterministically with only a polylogarithmic number of communication rounds. In this paper we indeed design such a deterministic algorithm that solves this problem with polylogarithmic number of communication rounds. This is an almost exponential improvement on the previously known deterministic local algorithms for this problem. Because the bi-chromatic coloring of hypergraphs problem is known to be complete in the class of all locally checkable graph problems, our result implies deterministic local algorithms with polylogarithmic number of communication rounds for all such problems for which an efficient randomized algorithm exists. This solves one of the fundamental open problems in the area of local distributed graph algorithms. By reductions due to Ghaffari, Kuhn and Maus [STOC 2017] this implies such polylogarithmically efficient deterministic local algorithms for many graph problems.Comment: Version date: 10 July 2019; some typos corrected; added explanation p. 5; paper submitted to ACM-SIAM SODA 2020; 13 page

Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization

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.Comment: Extended version of an article that appears at the Symposium on Theory of Computing (STOC) 202