Linear-in-
                $$\varDelta $$
                
                    
                                    
                        Δ
                    
                
             lower bounds in the LOCAL model

Göös, Mika; Hirvonen, Juho; Suomela, Jukka

doi:10.1007/s00446-015-0245-8

Cited by 13 publications

(15 citation statements)

References 26 publications

(56 reference statements)

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“…With this result we resolve Open Problem 11.6 in Barenboim and Elkin [5], and present a proof for the conjecture of Göös et al [18].…”

Section: Introductionsupporting

confidence: 61%

“…In particular, they did not tell anything nontrivial about the complexity of MM or MIS in the usual LOCAL model. Now we know that an entirely different kind of approach was needed-even though the present work shares some coauthors with [18,21], the techniques of the present work are entirely unrelated to those.…”

Section: Introductionmentioning

confidence: 99%

“…The same technique was later used to study maximal fractional matchings in the LOCAL model of computing. This is a problem that can be solved in O(∆) rounds (independent of n) in the usual LOCAL model [2], and there was a matching lower bound that shows that the same problem cannot be solved in o(∆) rounds (independent of n) in the same model [18].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lower Bounds for Maximal Matchings and Maximal Independent Sets

Balliu

Brandt

Hirvonen

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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102

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There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(∆ + log * n) communication rounds; here n is the number of nodes and ∆ is the maximum degree. The lower bound by Linial (1987Linial ( , 1992 shows that the dependency on n is optimal: these problems cannot be solved in o(log * n) rounds even if ∆ = 2.However, the dependency on ∆ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds.We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(∆ + log log n/ log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing.As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(∆ + log n/ log log n) rounds; this is an improvement over prior lower bounds also as a function of n.

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“…With this result we resolve Open Problem 11.6 in Barenboim and Elkin [5], and present a proof for the conjecture of Göös et al [18].…”

Section: Introductionsupporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lower Bounds for Maximal Matchings and Maximal Independent Sets

Balliu

Brandt

Hirvonen

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

102

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show abstract

“…2. The lower-bound result aims at establishing that one needs some specific number of rounds, e.g., Ω(∆) rounds [4,10,12]. However, in Theorem 1.2 we aim at proving that even if the round complexity is, say, exponential in ∆, one cannot avoid using fine-grained fractional values.…”

Section: Key New Ideasmentioning

confidence: 99%

Distributed Half-Integral Matching and Beyond

Dahal¹,

Suomela²

2023

Preprint

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By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log * n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from {0, 1 2 , 1}. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree ∆ = 2d, and any distributed graph algorithm with round complexity T (∆) that only depends on ∆ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2 d . We give a new algorithm that shows that this is also sufficient.

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“…• Maximal matching on bipartite graphs can be solved in (Δ) rounds [17], but not in (Δ) rounds [4]. • Maximal fractional matching can be solved in (Δ) rounds [3], but not in (Δ) rounds [14]. • Weak 2-coloring on odd-degree graphs can be solved in (log * Δ) rounds [21], but not in (log * Δ) rounds [5].…”

Section: Related Work 21 Distributed Complexity Of Locally Verifiable Problemsmentioning

confidence: 99%

Efficient Load-Balancing through Distributed Token Dropping

Brandt

Keller

Rybicki

et al. 2021

Proceedings of the 33rd ACM Symposium on Parallelism in Algorithms and Architectures

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We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for stable orientations and more generally for locally optimal semi-matchings. The prior work by Czygrinow et al. (DISC 2012) finds a stable orientation in (Δ 5 ) rounds in graphs of maximum degree Δ, while we improve it to (Δ 4 ) and also prove a lower bound of Ω(Δ). For the more general problem of locally optimal semi-matchings, the prior upper bound is ( 5 ) and our new algorithm runs in ( • 4 ) rounds, which is an improvement for = ( ); here and are the maximum degrees of customers and servers, respectively.

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Linear-in- $$\varDelta $$ Δ lower bounds in the LOCAL model

Cited by 13 publications

References 26 publications

Lower Bounds for Maximal Matchings and Maximal Independent Sets

Lower Bounds for Maximal Matchings and Maximal Independent Sets

Distributed Half-Integral Matching and Beyond

Efficient Load-Balancing through Distributed Token Dropping

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