On the Use of Randomness in Local Distributed Graph Algorithms

Ghaffari, Mohsen; Kühn, Fabian

doi:10.1145/3293611.3331610

Cited by 12 publications

(12 citation statements)

References 38 publications

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“…A somewhat related note: one can see, via a simple application of the probabilistic method that generalizes a classic argument of Newman [42] for two-party protocols, that we need at most O(log n) bits, if any, of shared randomness in distributed algorithms 7 . See also [27] for other related observations.…”

Section: Open Problem 3 What Is the Largest Gap Possible Between Thementioning

confidence: 87%

Network Decomposition and Distributed Derandomization (Invited Paper)

Ghaffari

2020

Structural Information and Communication Complexity

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We overview a recent line of work [Rozhoň and Ghaffari at STOC 2020; Ghaffari, Harris, and Kuhn at FOCS 2018; and Ghaffari, Kuhn, and Maus at STOC 2017], which proved that any (locallycheckable) graph problem that admits an efficient randomized distributed algorithm also admits an efficient deterministic distributed algorithm, thereby resolving several central and decades-old open problems in distributed graph algorithms. We present a short and self-contained version of the proofs, and conclude by discussing several related questions that remain open.This article accompanies a keynote talk of the author at the International Colloquium on Structural Information and Communication Complexity (SIROCCO) 2020. The writing is based on [24,28,45] and primarily targets non-experts.

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Section: Open Problem 3 What Is the Largest Gap Possible Between Thementioning

confidence: 87%

Network Decomposition and Distributed Derandomization (Invited Paper)

Ghaffari

2020

Structural Information and Communication Complexity

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“…Finally, we consider the interesting gap between randomized and deterministic algorithms in the low-space MPC setting. As observed by [9] and [17], randomized algorithms that succeed with probability of 1 − 1/2 2 can be turned into non-uniform deterministic algorithms. This result can also be extended to the low-space MPC setting, with some caveat.…”

Section: Our Contributionsmentioning

confidence: 98%

“…This result can also be extended to the low-space MPC setting, with some caveat. In contrast to the LOCAL model where the space of the nodes is unlimited, in the low-space MPC setting, the transformation implied by [9] and [17] yields a non-uniform, non-explicit algorithm. By using success probability amplification with poly( ) machines, one can boost the success guarantee of any randomized MPC algorithm from 1−1/poly( ) to 1−1/2 2 without any slowdown in the round complexity.…”

Section: Our Contributionsmentioning

confidence: 99%

Component Stability in Low-Space Massively Parallel Computation

Czumaj

Davies

Parter

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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In this paper, we study the power and limitations of componentstable algorithms in the low-space model of Massively Parallel Computation (MPC). Recently Ghaffari, Kuhn and Uitto (FOCS 2019) introduced the class of component-stable low-space MPC algorithms, which are, informally, defined as algorithms for which the outputs reported by the nodes in different connected components are required to be independent. This very natural notion was introduced to capture most (if not all) of the known efficient MPC algorithms to date, and it was the first general class of MPC algorithms for which one can show non-trivial conditional lower bounds. In this paper we enhance the framework of component-stable algorithms and investigate its effect on the complexity of randomized and deterministic low-space MPC. Our key contributions include:• We revise and formalize the lifting approach of Ghaffari, Kuhn and Uitto. This requires a very delicate amendment of the notion of component stability, which allows us to fill in gaps in the earlier arguments. • We also extend the framework to obtain conditional lower bounds for deterministic algorithms and fine-grained lower bounds that depend on the maximum degree Δ. • We demonstrate a collection of natural graph problems for which non-component-stable algorithms break the conditional lower bound obtained for component-stable algorithms. This implies that, for both deterministic and randomized algorithms, component-stable algorithms are conditionally weaker than the non-component-stable ones.Altogether our results imply that component-stability might limit the computational power of the low-space MPC model, paving the way for improved upper bounds that escape the conditional lower bound setting of Ghaffari, Kuhn, and Uitto. CCS CONCEPTS• Computing methodologies → Distributed algorithms; • Mathematics of computing → Graph algorithms; • Theory of computation → Pseudorandomness and derandomization.

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“…We therefore introduce a slightly more general definition of a network decomposition, which also includes a congestion parameter κ. A similar definition was for example used previously in [GK19]. Definition 3.1 (Network decomposition with congestion).…”

Section: Efficient (Degree + 1)-list Coloring In Congestmentioning

confidence: 99%

Efficient Deterministic Distributed Coloring with Small Bandwidth

Bamberger¹,

Kühn²,

Maus³

2019

Preprint

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We show that the (degree + 1)-list coloring problem can be solved deterministically in O(D • log n • log 2 ∆) rounds in the CONGEST model, where D is the diameter of the graph, n the number of nodes, and ∆ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., poly log n-time) deterministic CONGEST algorithm for the (∆+1)coloring and the (degree+1)-list coloring problem. Previously the best known algorithm required 2 O( √ log n) rounds and was not based on network decompositions. Our techniques also lead to deterministic (degree + 1)-list coloring algorithms for the congested clique and the massively parallel computation (MPC) model. For the congested clique, we obtain an algorithm with time complexity O(log ∆•log log ∆), for the MPC model, we obtain algorithms with round complexity O(log 2 ∆) for the linear-memory regime and O(log 2 ∆+log n) for the sublinear memory regime.Distributed Graph Coloring. Computing a coloring with the optimal number of colors χ(G) was one of the first problems known to be NP-complete [Kar72]. In the distributed setting, one therefore aims for a more relaxed goal and the usual objective is to color a given graph G with ∆ + 1 colors, where ∆ is the maximum degree of G [BE13]. Note that any graph admits such a coloring and it can be computed by a simple sequential greedy algorithm. Despite the simplicity of the sequential algorithm, the question of determining the distributed complexity of computing a (∆ + 1)-coloring has been an extremely challenging question. In particular, while O(log n)-time randomized distributed (∆+1)-coloring algorithms have been known for more than 30 years, the question whether a similarly efficient (i.e., polylogarithmic time) deterministic distributed coloring algorithm exists remained one of the most important open problems in the area [BE13, GKM17] until it was resolved very recently by Rozhoň and Ghaffari [RG19]. In [RG19] the question was answered in the affirmative for the LOCAL model by designing an efficient deterministic distributed method to decompose the communication graph into a logarithmic number of subgraphs consisting of connected components (clusters) of polylogarithmic (weak) diameter, a structure known as a network decomposition [AGLP89]. It was known before that an efficient deterministic algorithm for network decomposition would essentially imply efficient determinstic LOCAL algorithms for all problems for which efficient randomized algorithms exist [Lin92, AGLP89, BE13, GKM17, GHK18].Note that due to the unbounded message size any (solvable) problem can be solved in diameter time in the LOCAL model by simply collecting the whole graph topology at one node, solving the problem locally (potentially using unbounded computational power), and redistributing the solution to the nodes. Thus the small diameter components of a network decomposition almost immediately give rise to an efficient (∆ + 1)coloring algorithm in the...

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On the Use of Randomness in Local Distributed Graph Algorithms

Cited by 12 publications

References 38 publications

Network Decomposition and Distributed Derandomization (Invited Paper)

Network Decomposition and Distributed Derandomization (Invited Paper)

Component Stability in Low-Space Massively Parallel Computation

Efficient Deterministic Distributed Coloring with Small Bandwidth

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