The Complexity of (Δ+1) Coloring in Congested Clique, Massively Parallel Computation, and Centralized Local Computation

Chang, Yi‐Jun; Fischer, Manuela; Ghaffari, Mohsen; Uitto, Jara; Zheng, Yufan

doi:10.1145/3293611.3331607

Cited by 50 publications

(68 citation statements)

References 46 publications

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“…Subsequent to our work, Chang et al [15] further studied the (∆ + 1) coloring problem and among other results, gave an O( √ log log n) round MPC algorithm for this problem on machines with memory as small as n Ω(1) .…”

Section: Recent Related Workmentioning

confidence: 78%

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Assadi¹,

Chen²,

Khanna³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

195

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Any graph with maximum degree ∆ admits a proper vertex coloring with ∆ + 1 colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm?We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design:

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Section: Recent Related Workmentioning

confidence: 78%

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Assadi¹,

Chen²,

Khanna³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

195

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“…Recent Developments. After the initial publication of this work [13], our algorithm was adapted to solve (∆ + 1)-coloring in several other models of computation, namely the congested clique, the MPC 4 model, and the centralized local computation model [4,35,36,11]. Chang, Fischer, Ghaffari, Uitto, and Zheng [11], improving [35,36], showed that (∆ + 1)-coloring can be solved in the congested clique in O(1) rounds, w.h.p.…”

Section: New Resultsmentioning

confidence: 99%

“…Using the previous probability calculations, for each cluster S j the invariant U (11) holds with probability at least 1 − exp(−Ω(∆ 1/20 poly log ∆)), and the invariant L (11) holds with certainty. We will show that for a given cluster S j , the probability that D (11) is a valid degree bound (i.e., D (11) holds) is at least 1 − ∆ −Ω(c) . If a cluster S j does not meet at least one of U (11) , L (11) , or D (11) , then all vertices in S j halt and join V bad .…”

Section: Degree Upper Bounds By Lemma 4 Dmentioning

confidence: 97%

“…In this paper we improve the number of stages to O(1). This improvement does not affect the overall asymptotic time in the LOCAL model but it simplifies the algorithm, and is critical to adaptations of our algorithm to models in which Linial's lower bound [30] does not apply, e.g., the congested clique [4,11].…”

Section: Block Sizes and Excess Colorsmentioning

confidence: 99%

“…However, we still define U (11) = βδ (10) · U (10) = Θ(∆ 1/20 log 18 ∆) and L (11) = δ (10) · L (10) = Θ(∆ 1/20 log 17 ∆)…”

Section: Degree Upper Bounds By Lemma 4 Dmentioning

confidence: 99%

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An optimal distributed (Δ+1)-coloring algorithm?

Chang

Pettie

2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

Self Cite

111

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Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for (∆ + 1)-list coloring in the randomized LOCAL model running in O(Detd(poly log n)) time, where Detd(n ) is the deterministic complexity of (deg +1)-list coloring on n -vertex graphs. (In this problem, each v has a palette of size deg(v) + 1.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC 2016, J. ACM 2018] with complexity O( √ log ∆ + log log n + Detd(poly log n)), and, for some range of ∆, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS 2016] and Barenboim, Elkin, and Goldenberg [PODC 2018], with complexity O( √ ∆ log ∆ log * ∆ + log * n). Our algorithm appears to be optimal, in view of the Ω(Det(poly log n)) randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS 2016, SIAM J. Comput. 2019], where Det is the deterministic complexity of (∆ + 1)-list coloring. At present, the best upper bounds on Detd(n ) and Det(n ) are both 2 O √ log n and use a black box application of network decompositions due to Panconesi and Srinivasan [J.Algorithms 1996]. It is quite possible that the true deterministic complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our (∆ + 1)-list coloring algorithm.1 In the case of MIS, the subproblems actually have size poly(∆) log n, but satisfy the additional property that they contain distance-5 dominating sets of size O(log n), which is often just as good as having poly log(n) size. See [9, §3] or [21, §4] for more discussion of this.2 See [33,14,12] for the formal definition of the class of locally checkable labeling (LCL) problems.

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Breaking the Linear-Memory Barrier in $$\mathsf {MPC}$$: Fast $$\mathsf {MIS}$$ on Trees with Strongly Sublinear Memory

Brandt

Fischer

Uitto

2019

Structural Information and Communication Complexity

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Recently, studying fundamental graph problems in the Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow Ω(n) memory per machine, where n is the number of nodes in the graph and Ω hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory.In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality.In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in O(log 3 log n) rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in O( √ log n)-algorithm in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from Ω(n) required by the recent O(log log n)-round MIS algorithm of Ghaffari et al. [PODC'18] to n ε for any ε > 0, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the LOCAL and PRAM models by Lenzen and Wattenhofer [PODC'11].

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The Complexity of (Δ+1) Coloring in Congested Clique, Massively Parallel Computation, and Centralized Local Computation

Cited by 50 publications

References 46 publications

Sublinear Algorithms for (Δ + 1) Vertex Coloring

Sublinear Algorithms for (Δ + 1) Vertex Coloring

An optimal distributed (Δ+1)-coloring algorithm?

Breaking the Linear-Memory Barrier in $$\mathsf {MPC}$$: Fast $$\mathsf {MIS}$$ on Trees with Strongly Sublinear Memory

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