A lower bound for the distributed Lovász local lemma

Brandt, Sebastian; Fischer, Orr; Hirvonen, Juho; Keller, Barbara; Lempiäinen, Tuomo; Rybicki, Joel; Suomela, Jukka; Uitto, Jara

doi:10.1145/2897518.2897570

Cited by 101 publications

(151 citation statements)

References 46 publications

(40 reference statements)

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“…We extend Brandt et al's [11] randomized lower bound as follows: on degree-∆ graphs, ∆-coloring takes Ω(log ∆ log n) time in RandLOCAL and Ω(log ∆ n) time in DetLOCAL. The hard graphs in this lower bound have girth Ω(log ∆ n), so by the indistinguishability principle, these lower bounds also apply to ∆-coloring trees.…”

Section: New Resultsmentioning

confidence: 93%

“…On the upper bound side, Barenboim and Elkin [3] showed that for ∆ ≥ 3, ∆-coloring trees takes O(log ∆ n + log * n) time in DetLOCAL. We give an elementary proof that for ∆ ≥ 55, ∆-coloring trees can be done in O(log ∆ log n + log * n) time in RandLOCAL, matching Brandt et al's [11] lower bound up to the log * n. A more complicated algorithm for ∆-coloring trees could be derived from [29], for ∆ > ∆ 0 and some very large constant 4 Linial [24] actually only used the existence of ∆-regular graphs with high girth and chromatic number Ω( √ ∆). See [8] for constructions with chromatic number Ω(∆/ log ∆).…”

Section: New Resultsmentioning

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)

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: New Resultsmentioning

confidence: 93%

Section: New Resultsmentioning

confidence: 99%

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)

184

View full text Add to dashboard Cite

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“…This domain of distributed computing is extremely active and productive this last decade, analyzing a large variety of network problems in the so-called LOCAL model [37], capturing the ability to solve task locally in networks 1 . We refer to [4,5,8,13,18,20,21,24,42] for a non exhaustive list of achievements in context. However, all these achievements were based on an operational approach, using sophisticated algorithmic techniques and tools solely from graph theory.…”

Section: Related Workmentioning

confidence: 99%

“…However, all these achievements were based on an operational approach, using sophisticated algorithmic techniques and tools solely from graph theory. Similarly, the existing lower bounds on the round-complexity of tasks in the LOCAL model [32,35,8,23,3] were obtained using graph theoretical arguments only. The question of whether adopting a higher dimensional approach by using topology would help in the context of local computing, be it for a better conceptual understanding of the algorithms, or providing stronger technical tools for lower bounds, is, to our knowledge, entirely open.…”

Section: Related Workmentioning

confidence: 99%

A Topological Perspective on Distributed Network Algorithms

Castañeda¹,

Fraigniaud

Paz

et al. 2019

Structural Information and Communication Complexity

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More than two decades ago, combinatorial topology was shown to be useful for analyzing distributed fault-tolerant algorithms in shared memory systems and in message passing systems. In this work, we show that combinatorial topology can also be useful for analyzing distributed algorithms in networks of arbitrary structure. To illustrate this, we analyze consensus, setagreement, and approximate agreement in networks, and derive lower bounds for these problems under classical computational settings, such as the LOCAL model and dynamic networks. * Supported by ANR projects DESCARTES and FREDA. Additional support from INRIA project GANG.

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“…Unfortunately, this approach is unlikely to lead to an O(log * n)-time algorithm; LLL is known to be a hard problem for a wide range of parameters [6].…”

Section: Prior Work Related To Partial Coloringsmentioning

confidence: 99%

Locality of Not-so-Weak Coloring

Balliu

Hirvonen

Lenzen

et al. 2019

Structural Information and Communication Complexity

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Many graph problems are locally checkable: a solution is globally feasible if it looks valid in all constant-radius neighborhoods. This idea is formalized in the concept of locally checkable labelings (LCLs), introduced by Naor and Stockmeyer (1995). Recently, Chang et al. (2016) showed that in bounded-degree graphs, every LCL problem belongs to one of the following classes:• "Easy": solvable in O(log * n) rounds with both deterministic and randomized distributed algorithms. • "Hard": requires at least Ω(log n) rounds with deterministic and Ω(log log n) rounds with randomized distributed algorithms.

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A lower bound for the distributed Lovász local lemma

Cited by 101 publications

References 46 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

A Topological Perspective on Distributed Network Algorithms

Locality of Not-so-Weak Coloring

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