Derandomizing local distributed algorithms under bandwidth restrictions

We present O(log log n)-round algorithms in the Massively Parallel Computation (MPC) model, withÕ(n) memory per machine, that compute a maximal independent set, a 1 + ε approximation of maximum matching, and a 2 + ε approximation of minimum vertex cover, for any n-vertex graph and any constant ε > 0. These improve the state of the art as follows:• Our MIS algorithm leads to a simple O(log log ∆)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on theÕ( log ∆)-round algorithm of Ghaffari [PODC'17].• Our O(log log n)-round (1 + ε)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log 2 log n)-round (1 + ε)-approximation algorithm of Czumaj et al. [STOC'18] and O(log log n)-round (1 + ε)-approximation algorithm of Assadi et al. [SODA'19]. • Our O(log log n)-round (2 + ε)-approximate minimum vertex cover algorithm improves on an O(log log n)-round O(1)-approximation of Assadi et al. [arXiv'17]. The ModelsWe consider two closely related models: Massively Parallel Computation (MPC), and the CONGESTED-CLIQUE model of distributed computing. Indeed, we consider it as a conceptual contribution of this paper to (further) exhibit the proximity of these two models. We next review these models. The MPC modelThe MPC model was first introduced in [KSV10] and later refined in [GSZ11, BKS13, ANOY14].The computation in this model proceeds in synchronous rounds carried out by m machines. At the beginning of every round, the data (e.g. vertices and edges) is distributed across the machines. During a round, each machine performs computation locally without communicating to other machines. At the end of the round, the machines exchange messages which are used to guide the computation in the next round. In every round, each machine receives and outputs messages that fit into its local memory.Space: In this model, each machine has S words of space. If N is the total size of the data and each machine has S words of space, the typical settings that are of interest are when S is sublinear in N and S · m = Θ(N ). That is, the total memory across all the machines suffices to fit all the data, but is not much larger than that. If we are given a graph on n vertices, in our work we consider the regimes in which S ∈ Θ(n/ polylog n) or S ∈ Θ(n). Communication vs. computational complexity:Our main focus is the number of rounds required to finish the computation, which is essentially the complexity of the communication needed to solve the problem. Although we do not explicitly state the computational complexity in our results, it will be apparent from the description of our algorithms that the total computation time across all the machines is nearly-linear in the input size. CONGESTED-CLIQUEA second model that we consider is the CONGESTED-CLIQUE model of distributed computing, which was introduced by Lotker, Pavlov, Patt-Shamir, and Peleg [LPPSP03] and has been stud-ied extensively since then, see e.g.]. In this model, we have n players which can communicate in sync...

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“…deterministic O(log n log ∆)-round CONGESTED-CLIQUE algorithm was given by Censor-Hillel et al [CHPS17].…”

Section: Introductionmentioning

confidence: 99%

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

Ghaffari

Gouleakis

Konrad

et al. 2018

Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing

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“…Running Time Forest-Decomposition O(log a) Forest-Decomposition [2] O(log n) O(a 2+ε )-coloring O(log * n) O(a 2+ε )-coloring [2] O(log n) O(a 2 )−coloring O(log a) + log * n O(a 2 )−coloring [2] O(log n) O(a 1+ε )-coloring O(log 2 a) O(a 1+ε )-coloring [3] O(log a log n) O(a)-coloring O(a ε ) O(a)-coloring [3] O(min(a ε log n, a ε + log 1+ε n)) MIS O( √ a) MIS [2] O(a + log n) MIS [6] O(log ∆ log n) MIS (rand.) [9]…”

Section: Our Results (Deterministic)mentioning

confidence: 99%

“…Thus obtaining deterministic algorithms that never fail seems to be a more challenging task in this setting. Since the publication of the result of [21] many additional problems have been studied in the Congested Clique setting [5,6,8,9,14]. In particular, several symmetry-breaking problems were investigated.…”

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confidence: 99%

Distributed Symmetry-Breaking Algorithms for Congested Cliques

Barenboim

Khazanov

2018

Computer Science – Theory and Applications

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The Congested Clique is a distributed-computing model for single-hop networks with restricted bandwidth that has been very intensively studied recently. It models a network by an n-vertex graph in which any pair of vertices can communicate one with another by transmitting O(log n) bits in each round. Various problems have been studied in this setting, but for some of them the best-known results are those for general networks. For other problems, the results for Congested Cliques are better than on general networks, but still incure significant dependency on the number of vertices n. Hence the performance of these algorithms may become poor on large cliques, even though their diameter is just 1. In this paper we devise significantly improved algorithms for various symmetry-breaking problems, such as forests-decompositions, vertex-colorings, and maximal independent set.We analyze the running time of our algorithms as a function of the arboricity a of a clique subgraph that is given as input. The arboricity is always smaller than the number of vertices n in the subgraph, and for many families of graphs it is significantly smaller. In particular, trees, planar graphs, graphs with constant genus, and many other graphs have bounded arboricity, but unbounded size. We obtain O(a)-forest-decomposition algorithm with O(log a) time that improves the previously-known O(log n) time, O(a 2+ǫ )-coloring in O(log * n) time that improves upon an O(log n)-time algorithm, O(a)-coloring in O(a ǫ )-time that improves upon several previous algorithms, and a maximal independent set algorithm with O( √ a) time that improves at least quadratically upon the state-of-the-art for small and moderate values of a.Those results are achieved using several techniques. First, we produce a forest decomposition with a helpful structure called H-partition within O(log a) rounds. In general graphs this structure requires Θ(log n) time, but in Congested Cliques we are able to compute it faster. We employ this structure in conjunction with partitioning techniques that allow us to solve various symmetry-breaking problems efficiently. . This research has been supported by ISF grant 724/15 and Open University of Israel research fund.networks, a more realistic model has been studied. This is the CONGEST model that is similar to the LOCAL model, except that each edge is only allowed to transmit O(log n) bits per round. An important type of CONGEST networks that has been intensively studied recently is the Congested Clique model. It represents single-hop networks with limited bandwidth. Although the diameter of such networks is 1, which would make any problem on such graphs trivial in the LOCAL model, in the Congested Cliques various tasks become very challenging. Note that the Congested Clique is equivalent to a general n-vertex graph in which any pair of vertices (not necessarily neighbors) can exchange messages of size O(log n) in each round. Such a general graph corresponds to a subgraph of an n-clique. The subgraph constitutes the input, while the clique...

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“…Unlike in the LOCAL model, where partial solutions of clusters of small diameter can be computed in time linear in the cluster diameter by simply collecting the whole cluster topology at one node, in the CONGEST model, we have to be much more careful. Inspired by recent work on other local problems [CPS17,GK18], we design a randomized CONGEST algorithm that only requires poly log n-wise independent randomness. Then we use the method of conditional expectations to efficiently derandomize this algorithm.…”

Section: Contributionsmentioning

confidence: 99%

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

Deurer

Kühn

Maus

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For ε > 1/ poly log ∆ we obtain two algorithms with approximation factor (1 + ε)(1 + ln(∆ + 1)) and with runtimes 2 O( √ log n log log n) and O(∆ poly log ∆ + poly log ∆ log * n), respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic O(log ∆)-approximation algorithm for the minimum connected dominating set with time complexity 2 O( √ log n log log n) .

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Derandomizing local distributed algorithms under bandwidth restrictions

Cited by 44 publications

References 60 publications

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

Distributed Symmetry-Breaking Algorithms for Congested Cliques

Deterministic Distributed Dominating Set Approximation in the CONGEST Model

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