(2Δ — l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting

Elkin, Michael; Pettie, Seth; Su, Hsin-Hao

doi:10.1137/1.9781611973730.26

Cited by 51 publications

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“…Theorem 3 works equally well when t and r are functions of n, ∆, and possibly other quantitative global graph parameters. For example, the time may depend on measures of local sparsity (as in [14]), arboricity/degeneracy (as in [3,5]), or neighborhood growth (as in [31]). …”

Section: The Necessity Of Graph Shatteringmentioning

confidence: 99%

“…This second phenomenon is no coincidence! It is a direct result of the graph shattering approach to symmetry breaking used in [5] and further in [12,14,16,18,7,21,29]. The idea is to apply some randomized procedure that fixes some fragment of the output (e.g., part of the MIS is fixed, part of the coloring is fixed, etc.…”

Section: Introductionmentioning

confidence: 99%

“…This result has a very clear take-away message: the graph shattering technique applied by recent randomized symmetry breaking algorithms [5,16] is inherent to the RandLOCAL model and every optimal RandLOCAL algorithm for instances of size n must, in some way, encode an optimal DetLOCAL algorithm on poly(log n)-size instances. It is therefore impossible to improve the 2 O( √ log log n) terms in the RandLOCAL MIS and coloring algorithms of [5,16,18,14] without also improving the 2 O( √ log n) -time DetLOCAL algorithms of Panconesi and Srinivasan [28], and it is impossible to improve the O(log 4 log n) term in the RandLOCAL maximal matching algorithm of [5] without also improving the O(log 4 n) DetLOCAL maximal matching algorithm of [17].…”

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confidence: 99%

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An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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184

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Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message:Fast ∆-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al.[11], we prove that the randomized complexity of ∆-coloring a tree with maximum degree ∆ is Θ(log ∆ log n), for any ∆ ≥ 55, whereas its deterministic complexity is Θ(log ∆ n) for any ∆ ≥ 3. 1 This also establishes a large separation between the deterministic complexity of ∆-coloring and (∆ + 1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(log ∆ n) rounds can be transformed to run in O(log * n − log * ∆ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(log ∆ n) time deterministically. (This gives an alternate proof that deterministically ∆-coloring a tree with small ∆ takes Ω(log ∆ n) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √ log n. This shows that a deterministic Ω(log ∆ n) lower bound for any problem (∆-coloring a tree, for example) implies a randomized Ω(log ∆ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2 O( √ log log n) terms in the complexities of the best MIS and (∆ + 1)-coloring algorithms without also improving the 2 O( √ log n) -round Panconesi-Srinivasan algorithms.

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Section: The Necessity Of Graph Shatteringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Chang

Kopelowitz

Pettie

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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184

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“…This round complexity was improved to O(log ∆) + e O( √ log log n) by Barenboim et al [BEPS16]. This was further improved to just 2 O( √ log log n) by Elkin, Pettie, and Su [EPS15]. In the randomized world, there are also algorithms for finding colorings with smaller number of colors.…”

Section: Degree Splitting and Edge Orientationsmentioning

confidence: 99%

Distributed Degree Splitting, Edge Coloring, and Orientations

Ghaffari¹,

Su²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes:• We present a poly log n round deterministic algorithm for (2∆−1)·(1+o (1) • We show that sinkless orientation-i.e., orienting edges such that each node has at least one outgoing edge-on ∆-regular graphs can be solved in O(log ∆ log n) rounds randomized and in O(log ∆ n) rounds deterministically. These prove the corresponding lower bounds by Brandt et al. [STOC'16] and Chang, Kopelowitz, and Pettie [FOCS'16] to be tight. Moreover, these show that sinkless orientation exhibits an exponential separation between its randomized and deterministic complexities, akin to the results of Chang et al. for ∆-coloring ∆-regular trees.• We present a randomized O(log 4 n) round algorithm for orienting a-arboricity graphs with maximum out-degree a(1 + ε). This can be also turned into a decomposition into a(1 + ε) forests when a = Ω(log n) and into a(1 + ε) pseduo-forests when a = o(log n). Obtaining an efficient distributed decomposition into less than 2a forests was stated as the 10th open problem in the book by Barenboim and Elkin.

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“…Since the Barenboim-Elkin algorithms do not solve the general listcoloring problem, we have to start Phase II with a "fresh" palette of unused colors. This fact leads to (∆ + Ω(λ))-coloring algorithms whose running time is sublinear in λ, and (∆ + 1)-coloring algorithms whose running time is at least linear in λ. Elkin, Pettie, and Su [14] recently considered randomized distributed algorithms for coloring locally sparse graphs. One consequence of their results is that (∆ + 1)-coloring can be computed in O(log λ) + 2 O( √ log log n) time for all λ, ∆, n, and in O(log * n) time for certain ranges of the parameters.…”

Section: Mis and Ruling Setsmentioning

confidence: 99%

The Locality of Distributed Symmetry Breaking

Barenboim

Elkin

Pettie

et al. 2016

J. ACM

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123

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Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes

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(2Δ — l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting

Cited by 51 publications

References 0 publications

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

Distributed Degree Splitting, Edge Coloring, and Orientations

The Locality of Distributed Symmetry Breaking

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