Simple, Deterministic, Constant-Round Coloring in the Congested Clique

2021

ACM Trans. Algorithms

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The Massively Parallel Computation (MPC) model is an emerging model that distills core aspects of distributed and parallel computation, developed as a tool to solve combinatorial (typically graph) problems in systems of many machines with limited space. Recent work has focused on the regime in which machines have sublinear (in n , the number of nodes in the input graph) space, with randomized algorithms presented for the fundamental problems of Maximal Matching and Maximal Independent Set. However, there have been no prior corresponding deterministic algorithms. A major challenge underlying the sublinear space setting is that the local space of each machine might be too small to store all edges incident to a single node. This poses a considerable obstacle compared to classical models in which each node is assumed to know and have easy access to its incident edges. To overcome this barrier, we introduce a new graph sparsification technique that deterministically computes a low-degree subgraph, with the additional property that solving the problem on this subgraph provides significant progress towards solving the problem for the original input graph. Using this framework to derandomize the well-known algorithm of Luby [SICOMP’86], we obtain O (log Δ + log log n )-round deterministic MPC algorithms for solving the problems of Maximal Matching and Maximal Independent Set with O ( n ɛ ) space on each machine for any constant ɛ > 0. These algorithms also run in O (log Δ) rounds in the closely related model of CONGESTED CLIQUE, improving upon the state-of-the-art bound of O (log 2 Δ) rounds by Censor-Hillel et al. [DISC’17].

Section: The Mpc Modelmentioning

confidence: 99%

Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space

2021

ACM Trans. Algorithms

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“…We then turn to consider the impact of component-stability on deterministic low-space MPC algorithms. Since the recent derandomization technique of [11,12] leads to inherently componentunstable algorithms, we ask: Question 3. Does component-instability help for obtaining improved deterministic low-space MPC algorithms for graph problems?…”

Section: Our Aimsmentioning

confidence: 99%

“…The first indication that component-stability might actually matter was provided by recent works [11,12], which present deterministic low-space component-unstable MPC algorithms for several classic graph problems, even though the validity of solutions to these problems depends only on local information. Specifically, by derandomizing a basic graph sparsification technique, one can obtain (log Δ + log log )-round deterministic low-space componentunstable MPC algorithms for MIS, maximal matching, and (Δ + 1)coloring.…”

Section: Introductionmentioning

confidence: 99%

Component Stability in Low-Space Massively Parallel Computation

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

2021

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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.

“…Their algorithm also supports the list variant of the problem, by employing a new randomized partitioning of both the nodes and their colors. Recently, Czumaj, Davies and Parter [14] provided a simplified (1)-round deterministic algorithm for the problem. In contrast to prior works, their algorithm is not based on the CLP algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…The deterministic complexity of the (Δ + 1) coloring in lowspace MPC has been studied independently by Bamberger, Kuhn and Maus [3] and by Czumaj, Davies and Parter [14]: [3] presented an (log 2 Δ + log ) round solution for the (deg +1) list coloring problem; [14] presented an (log Δ + log log )-round algorithm for the (Δ + 1) list coloring problem. No sublogarithmic bounds are currently known.…”

Section: Introductionmentioning

confidence: 99%

Improved Deterministic (Δ+1) Coloring in Low-Space MPC

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

2021

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We present a deterministic (log log log )-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of (Δ + 1)-coloring on -vertex graphs. In this model, every machine has sublinear local space of size for any arbitrary constant ∈ (0, 1). Our algorithm works under the relaxed setting where each machine is allowed to perform exponential local computations, while respecting the space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious (Δ + 1)-coloring local algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in = (log log ) rounds, which sets the state-ofthe-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, notably pseudorandom generators (PRG) and bounded-independence hash functions.The achieved round complexity of (log log log ) rounds matches the bound of log(), which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the (Δ + 1)-coloring problem have been known before. CCS CONCEPTS• Computing methodologies → Distributed algorithms; • Mathematics of computing → Graph algorithms; • Theory of computation → Pseudorandomness and derandomization.