Distributed Degree Splitting, Edge Coloring, and Orientations

Ghaffari, Mohsen; Su, Hsin-Hao

doi:10.1137/1.9781611974782.166

Cited by 67 publications

(92 citation statements)

References 29 publications

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“…They also exhibited an O(log log n) round randomized algorithm and an O(log n) round deterministic algorithm for ∆-coloring trees, hence proving that these complexities are tight and they have an exponential separation. Ghaffari and Su [GS17] later showed that the original problem of sinkless orientation (which is a special case of the Lovász Local Lemma) also exhibits the same exponential separation, by providing a Θ(log log n)-round randomized and a Θ(log n)-round deterministic algorithm for it. We emphasize that this exponential separation is between complexities that are in O(log n).…”

Section: An Overview Of the Recent Developments On Det Vs Randmentioning

confidence: 99%

On the Use of Randomness in Local Distributed Graph Algorithms

Ghaffari

Kühn

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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We attempt to better understand randomization in local distributed graph algorithms by exploring how randomness is used and what we can gain from it:• We first ask the question of how much randomness is needed to obtain efficient randomized algorithms. We show that for all locally checkable problems for which poly log n-time randomized algorithms exist, there are such algorithms even if either (I) there is a only a single (private) independent random bit in each poly log n-neighborhood of the graph, (II) the (private) bits of randomness of different nodes are only poly log n-wise independent, or (III) there are only poly log n bits of global shared randomness (and no private randomness). • Second, we study how much we can improve the error probability of randomized algorithms.For all locally checkable problems for which poly log n-time randomized algorithms exist, we show that there are such algorithms that succeed with probability 1 − n −2 ε(log log n) 2 and more generally T -round algorithms, for T ≥ polylog n, that succeed with probability 1 − n −2 ε log 2 T . We also show that poly log n-time randomized algorithms with success probability 1 − 2 −2 log ε n for some ε > 0 can be derandomized to poly log n-time deterministic algorithms. Both of the directions mentioned above, reducing the amount of randomness and improving the success probability, can be seen as partial derandomization of existing randomized algorithms. In all the above cases, we also show that any significant improvement of our results would lead to a major breakthrough, as it would imply significantly more efficient deterministic distributed algorithms for a wide class of problems.

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Section: An Overview Of the Recent Developments On Det Vs Randmentioning

confidence: 99%

On the Use of Randomness in Local Distributed Graph Algorithms

Ghaffari

Kühn

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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“…Edge splitting (also known as degree splitting) can be defined as coloring all edges red or blue such that each node has at most ∆ 2 (1 + ε) edges in each color. Ghaffari and Su [GS17] provided a poly log n-round algorithm for edge-splitting, which led to the first efficient deterministic distributed 2∆(1 + o(1))-edge-coloring algorithm, thus partially resolving Open Problem 11.4 of [BE13]. A significantly more efficient edge splitting algorithm was later provided in [GHK + 17b].…”

Section: The Splitting Problem and Its Significancementioning

confidence: 99%

On the Complexity of Distributed Splitting Problems

Bamberger

Ghaffari

Kühn

et al. 2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

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One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, poly log n-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the weak splitting problem takes a central role in this context: Each node of a graph G = (V, E) has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient poly log n-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems and we in particular obtain the following results:• We obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if δ and ∆ are the minimum and maximum degrees of G and if δ = Ω(log n), weak splitting can be solved deterministically in time O ∆ δ · poly(log n) . Further, if δ = Ω(log log n) and ∆ ≤ 2 εδ , there is a randomized algorithm with time complexity O ∆ δ · poly(log log n) . • We prove that the following two related problems are also complete in the same sense:(I) Color the nodes of a graph with C ≤ poly log n colors such that each node with a sufficiently large polylogarithmic degree has at least 2 log n colors among its neighbors, and (II) Color the nodes with a large constant number of colors so that for each node of a sufficiently large at least logarithmic degree d(v), the number of neighbors of each color is at most (1 − ε)d(v) for some constant ε > 0.

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“…Due to the observation of (27), X u i and X v i are mutually independent. According to the exponential correlation of (29), by choosing a suitably small t = O(log n), the total variation distance between (σ u i , σ v i ) and (…”

Section: Lower Boundsmentioning

confidence: 99%

“…For the σ ′ returned by a t-round protocol where t ≤ 0.49 · diam(G), according to the property (27), the σ ′ Gx and σ ′ Gy are independent of each other, thus the phases of G x and G y on σ ′ are independent of each other: …”

Section: Proof Of the ω(Diam) Lower Boundmentioning

confidence: 99%

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What Can be Sampled Locally?

Feng

Sun

Yin

2017

Proceedings of the ACM Symposium on Principles of Distributed Computing

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The local computation of Linial [FOCS'87] and Naor and Stockmeyer [STOC'93] concerns with the question of whether a locally definable distributed computing problem can be solved locally: more specifically, for a given local CSP whether a CSP solution can be constructed by a distributed algorithm using local information. In this paper, we consider the problem of sampling a uniform CSP solution by distributed algorithms, and ask whether a locally definable joint distribution can be sampled from locally. More broadly, we consider sampling from Gibbs distributions induced by weighted local CSPs, especially the Markov random fields (MRFs), in the LOCAL model.We give two Markov chain based distributed algorithms which we believe to represent two fundamental approaches for sampling from Gibbs distributions via distributed algorithms. The first algorithm generically parallelizes the single-site sequential Markov chain by iteratively updating a random independent set of variables in parallel, and achieves an O(∆ log n) time upper bound in the LOCAL model, where ∆ is the maximum degree, when the Dobrushin's condition for the Gibbs distribution is satisfied. The second algorithm is a novel parallel Markov chain which proposes to update all variables simultaneously yet still guarantees to converge correctly with no bias. It surprisingly parallelizes an intrinsically sequential process: stabilizing to a joint distribution with massive local dependencies, and may achieve an optimal O(log n) time upper bound independent of the maximum degree ∆ under a stronger mixing condition.We also show a strong Ω(diam) lower bound for sampling: in particular for sampling independent set in graphs with maximum degree ∆ ≥ 6. Independent sets are trivial to construct locally and the sampling lower bound holds even when every node is aware of the entire graph. This gives a strong separation between sampling and constructing locally checkable labelings.

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Distributed Degree Splitting, Edge Coloring, and Orientations

Cited by 67 publications

References 29 publications

On the Use of Randomness in Local Distributed Graph Algorithms

On the Use of Randomness in Local Distributed Graph Algorithms

On the Complexity of Distributed Splitting Problems

What Can be Sampled Locally?

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