On Derandomizing Local Distributed Algorithms

Abstract

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we combine the method of conditional expectation with network decompositions to obtain a generic and clean recipe for derandomizing randomized LOCAL algorithms and transforming them into efficient deterministic LOCAL algorithms. This simple recipe leads to significant improvements on a number of problems, in cases resolving known open problems. Two main results are:• An improved deterministic distributed algorithm for hypergraph maximal matching, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The last result gives the first positive resolution in the Open Problem 11.10 in the book of Barenboim and Elkin. • Improved randomized and deterministic distributed algorithms for the Lovász Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and k-SAT. * Supported by ERC Grant No. 336495 (ACDC).SLOCAL is similar to the LOCAL model, in that each node can read its r-hop neighborhood in the graph G, for some parameter r. However, in the SLOCAL model, the neighborhoods are read sequentially. Formally, the nodes are processed in an arbitrary (adversarially chosen) order. When node v is processed, v can read its r-hop neighborhood and it computes and locally stores its output y v and potentially additional information. When reading the r-hop neighborhood, v also reads all the information that has been locally stored by the previously processed nodes there. We call the parameter r the locality of an SLOCAL algorithm.The SLOCAL model can be seen as a natural extension of sequential greedy algorithms. In fact, the classic distributed graph problems such as MIS or (∆ + 1)-coloring have simple SLOCAL algorithms with locality 1: in order to determine whether a node v is in the MIS or which color v gets in a (∆ + 1)-coloring, it suffices to know the decisions of all the neighbors that have been processed before.The SLOCAL model is inherently sequential. The main reason it is useful is that there are transformations from SLOCAL algorithms to LOCAL algorithms, which handles symmetry breaking in a "black-box" or generic way. By developing and analyzing SLOCAL algorithms, we are therefore able to treat a number of diverse LOCAL problems in a unified and abstract fashion. We are also able to adapt multiple types of algorithms to take advantage of special structure in the graphs (for example, bounds on its maximum degree). This two-part method of algorithm analysis -constructing SLOCAL algorithms, and transforming them to LOCAL algorithms generically -will be a key technical tool. Our Contributions, Part I: DerandomizationIn the first part of this paper, we present a simple and clean recipe for derandomizing...

A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

“…There is an O(d 2 + log * n) round deterministic distributed algorithm that solves the LLL problem. This deterministic algorithm improves on the previously best randomized algorithm which has a runtime of ω(poly log log n) [GHK18] (see related work section for more details).…”

“…All the related work that we surveyed so far are for randomized algorithms and there are very few deterministic results for the distributed LLL problem. A λ · n 1/λ · 2 √ log n time algorithm under the criterion p(ed) λ < 1 is known [FG17] and the state-of-the-art runtime of exp (i) O (log (i) n) 0.5 is by Ghaffari, Harris and Kuhn [GHK18].…”

On the Use of Randomness in Local Distributed Graph Algorithms

“…the respective discussion in Section 1), we can boost the success probability when computing a network decomposition significantly beyond this. By the reductions in [GKM17,GHK18], we then also immediately get the same improvement in the success probability for all poly(log n)-locally checkable graph problems that have poly(log n)-time randomized distributed algorithms with only polynomially small error probability.…”

“…Given this, Ghaffari et al [GKM17] studied the question of P-SLOCAL vs. P-LOCAL: whether any locally checkable problem that admits a deterministic SLOCAL algorithm with locality poly(log n) can be solved using a poly(log n)-round deterministic LOCAL algorithm. A number of problems were shown to be complete with respect to P-SLOCAL [GKM17,GHK18], in the following sense: they admit deterministic SLOCAL algorithms with locality poly(log n) and if one can provide a poly(log n)-round deterministic LOCAL for any of them, one has proven that P-SLOCAL = P-LOCAL. Example complete problems include network decompositions, splitting [GKM17], and certain locally verifiable versions of approximating dominating set or set cover [GHK18].…”

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

“…Whether an exponential gap between the best randomized and deterministic algorithms for these problems is really necessary is considered to be one of the key open questions in the area of distributed graph algorithms [BE13,GKM17,Lin92,PS95,Mau18]. In the context of this more general question, Ghaffari, Harris, and Kuhn recently showed in [GHK18] that also the MDS problem has a key role in this more general question. By using (and extending) a frame-work developed in [GKM17], they showed the MDS problem is P − SLOCAL complete, meaning: If there is a polylogarithmic-time distributed deterministic algorithm that computes any polylogarithmic approximation for the MDS problem, there are polylogarithmic-time deterministic distributed algorithms for essentially all 4 problems for which there are efficient randomized algorithms.…”