Efficient Deterministic Distributed Coloring with Small Bandwidth

Bamberger, Philipp; Kühn, Fabian; Maus, Yannic

doi:10.1145/3382734.3404504

Cited by 33 publications

(44 citation statements)

References 44 publications

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“…Deterministic distributed algorithms for network decomposition can be applied to transform deterministic distributed algorithms with round complexity O(D) • poly(log n) into ones with round complexity poly(log n). This approach was used in designing poly(log n)-round deterministic distributed algorithms for fundamental graph problems such as maximal independent set [6,29] and (∆ + 1)-coloring [3] in the CONGEST model. Via other connections, these also led to improved randomized algorithms in the CONGEST model as well as the massively parallel computation model [7,14,22].…”

Section: Background: Model and Definitionsmentioning

confidence: 99%

Strong-Diameter Network Decomposition

Chang

Ghaffari

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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Network decomposition is a central concept in the study of distributed graph algorithms. We present the first polylogarithmicround deterministic distributed algorithm with small messages that constructs a strong-diameter network decomposition with polylogarithmic parameters.Concretely, a (C, D) strong-diameter network decomposition is a partitioning of the nodes of the graph into disjoint clusters, colored with C colors, such that neighboring clusters have different colors and the subgraph induced by each cluster has a diameter at most D. In the weak-diameter variant, the requirement is relaxed by measuring the diameter of each cluster in the original graph, instead of the subgraph induced by the cluster.A recent breakthrough of Rozhoň and Ghaffari [STOC 2020] presented the first poly(log n)-round deterministic algorithm for constructing a weak-diameter network decomposition where C and D are both in poly(log n). Their algorithm uses small O(log n)-bit messages. One can transform their algorithm to a strong-diameter network decomposition algorithm with similar parameters. However, that comes at the expense of requiring unbounded messages.The key remaining qualitative question in the study of network decompositions was whether one can achieve a similar result for strong-diameter network decompositions using small messages. We resolve this question by presenting a novel technique that can transform any black-box weak-diameter network decomposition algorithm to a strong-diameter one, using small messages and with only moderate loss in the parameters.

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Section: Background: Model and Definitionsmentioning

confidence: 99%

Strong-Diameter Network Decomposition

Chang

Ghaffari

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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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: 99%

“…Just the network decomposition algorithm of [45], on its own, solved several special cases: for instance, it implies a poly(log n) round algorithms for MISthus resolving Linial's question-and Δ + 1 coloring. These algorithms can be extended also to the CONGEST model, with O(log n)-bit messages [7,12,45].…”

Section: Motivationsmentioning

confidence: 99%

“…This is basically what we call network decomposition, as introduced first by Awerbuch, Goldberg, Luby, and Plotkin [3], and as we define formally next. Before that, it is worth commenting that, even though the above approach is in the LOCAL model and uses large messages for topology gathering, one can use network decomposition to solve MIS and Δ + 1 vertex coloring also in the CONGEST model [7,12,45].…”

Section: Warm Up: Mis and Network Decompositionmentioning

confidence: 99%

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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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“…Our method applies to all versions of PRAM, particularly the most powerful concurrent read concurrent write version of PRAM (CRCW PRAM), which allows for multiple processors to read and write to the same memory cell at the same time. 2 In contrast with prior work which mostly focused on obtaining randomness-efficient versions of specific parallel algorithms, a key feature of Theorem 2 is that it applies to general PRAM algorithms.…”

Section: This Work: Randomness-efficient Mpc Algorithm For Connectivitymentioning

confidence: 99%

Brief Announcement

Charikar

Tan

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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We give a randomness-efficient Massively Parallel Computation (MPC) algorithm for deciding whether an undirected graph is connected. For Connectivity on n-vertex, m-edge graphs whose components have diameter at most D = 2 o(log n/log log n) , our algorithm runs in R = O(log D + log log m/n n) rounds and uses a total of (log n) O (R) random bits, O(m) machines, and n 1−Ω(1) space per machine with good probability. 1 Our algorithm achieves a superpolynomial saving in randomness complexity as compared to the breakthrough algorithm of Andoni et al. (FOCS '18) and the subsequent improvement by Behnezhad et al. (FOCS '19). Our algorithm has the same round complexity as that of Behnezhad et al., but uses more total space.Our Connectivity algorithm is an instantiation of a general method we develop for converting randomized algorithms in the PRAM model to highly randomness-efficient MPC algorithms. We show that for k = o(log n/log log n) and p = n O (1) , any time-k pprocessor randomized PRAM algorithm computing a function on n input bits can be converted to an equivalent strongly sublinear MPC algorithm with O(k) rounds and only a total of (log n) O (k ) random bits. Our Connectivity algorithm follows from applying this method to the recent CRCW PRAM algorithm of Liu, Tarjan, and Zhong (SPAA '20).Our approach is based on the design of a pseudorandom generator for PRAM algorithms. The analysis of our generator is built on classic and influential results in circuit complexity (Håstad '86; Nisan and Wigderson '88), which we generalize from the setting of small-depth circuits to the more powerful setting of PRAM algorithms. The parameters that we achieve are optimal given the current state of the art in complexity theory, in the sense that further improvements will imply P NC 1 . CCS CONCEPTS• Mathematics of computing → Graph algorithms; • Theory of computation → Massively parallel algorithms; Pseudorandomness and derandomization.1 With good probability means with probability at least 1 − 1/poly((m log n)/n), which is the same as in Liu, Tarjan, and Zhong (SPAA '20).

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Efficient Deterministic Distributed Coloring with Small Bandwidth

Cited by 33 publications

References 44 publications

Strong-Diameter Network Decomposition

Strong-Diameter Network Decomposition

Network Decomposition and Distributed Derandomization (Invited Paper)

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