Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization (1907.10937v2)

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.

Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization

The paper presents significant advancements in the field of distributed graph algorithms, notably addressing a long-standing question regarding deterministic network decomposition with polylogarithmic-time algorithms. The authors propose a novel deterministic approach that surpasses the complexity of the previously celebrated algorithm by Panconesi and Srinivasan, settling a central problem in the discipline and resolving multiple decades-old open questions. Among these questions is Linial's inquiry about the deterministic complexity of the maximal independent set, which was considered an outstanding problem in distributed graph algorithms.

Key Contributions

The primary contribution is the development of a simple polylogarithmic-time deterministic algorithm for network decomposition. This core innovation paves the way for deterministic polylogarithmic-time solutions for various other distributed problems. Specifically, the research establishes a more generalized distributed derandomization theorem, demonstrating that for any problem whose solution can be verified deterministically in polylogarithmic-time, a polylogarithmic-time randomized algorithm can be effectively transformed into a deterministic counterpart. In essence, this equates the efficiency standards of randomized local algorithms to their deterministic equivalents, indicating that distributed algorithms can achieve efficiency without relying on randomness.

Implications and Future Directions

This advancement has profound implications, both theoretically and practically. From a theoretical perspective, the research offers fresh insights into the derandomization of distributed algorithms, potentially redrawing the boundaries between deterministic and randomized algorithmic paradigms in distributed computing. Practically, the implications extend to a variety of problems with distributed algorithms, such as $(\Delta+1)$-coloring, maximal independent set, and applications involving Lovász Local Lemma, which traditionally exploited randomization for efficiency.

Furthermore, the paper presents a promising outlook on the improvements in the landscape of massively parallel computation, providing faster deterministic algorithms grounded in the new network decomposition framework. The deterministic approaches described not only enhance the complexity landscape of distributed algorithms but also guide future developments in distributed derandomization. Such developments could, in turn, influence efficient solutions in computer networks, distributed systems, and relevant scenarios in parallel processing.

Conclusion

In summary, the paper successfully bridges a substantial gap between deterministic and randomized distributed algorithms, revealing that randomness is not a prerequisite for achieving polylogarithmic-time efficiency in distributed graph computations. This alignment through a deterministic approach heralds new opportunities for research in distributed computing, promising further exploration of algorithmic efficiencies and their subsequent applications across a multitude of distributed and parallel systems.