A Time Hierarchy Theorem for the LOCAL Model

Chang, Yi‐Jun; Pettie, Seth

doi:10.1109/focs.2017.23

Cited by 44 publications

(156 citation statements)

References 27 publications

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“…We present a general technique for constructing distributed graph problems with a wide range of different time complexities. In particular, our work answers many of the open questions of Chang and Pettie [5], and disproves one of their conjectures.…”

Section: Introductionsupporting

confidence: 61%

“…Some classical problems are now also known to have intermediate complexities, even though tight bounds are still missing: ∆-colouring and (2∆ − 2)-edge colouring require Ω(log n) rounds [3,7], and can be solved in time O(polylog n) [24]. Some gaps have been conjectured; for example, Chang and Pettie [5] conjecture that there are no problems with complexity between ω(n 1/(k+1) ) and o(n 1/k ). See Table 1 for an overview of the state of the art.…”

Section: State Of the Artmentioning

confidence: 99%

“…Θ(log * n) exists [8,21] ω(log * n), o(log n) does not exist [6] Θ(log n) exists [3,6,15] ω(log n), n o (1) ? Θ(n 1/k ) exists [5] Θ(n) exists trivial class in the deterministic world. In the higher end of this region, it is known that all LCLs solvable in time o(log n) can be solved in time T LLL (n), the time it takes to solve a relaxed variant of the Lovász local lemma [5].…”

Section: State Of the Artmentioning

confidence: 99%

“…Lemma 2. Let G be a graph that satisfies the constraints (1)- (5) and has at least one node that does not have an edge labelled with D. Then G is an extended link machine encoding graph.…”

Section: Local Checkabilitymentioning

confidence: 99%

See 3 more Smart Citations

New classes of distributed time complexity

Balliu

Hirvonen

Korhonen

et al. 2018

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

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Section: Introductionsupporting

confidence: 61%

Section: State Of the Artmentioning

confidence: 99%

Section: State Of the Artmentioning

confidence: 99%

“…Lemma 2. Let G be a graph that satisfies the constraints (1)- (5) and has at least one node that does not have an edge labelled with D. Then G is an extended link machine encoding graph.…”

Section: Local Checkabilitymentioning

confidence: 99%

See 2 more Smart Citations

New classes of distributed time complexity

Balliu

Hirvonen

Korhonen

et al. 2018

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

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“…There is a natural distinction between locally checkable problems: those who are solvable in O(f (∆)+log * n), that is, easy problems, and those who require Ω(log ∆ n) for deterministic algorithms an Ω(log ∆ log n) for randomized ones, that is, hard problems. We also know from prior work that, for bounded-degree graphs, there cannot be problems in between [20,22]. We proved lower bounds for many easy natural problems, by exploiting the hardness of hard problems, namely, c-coloring for c ≤ ∆.…”

Section: Open Problemsmentioning

confidence: 98%

Distributed $Δ$-Coloring Plays Hide-and-Seek

Balliu¹,

Brandt²,

Kühn³

et al. 2021

Preprint

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We prove several new tight or near-tight distributed lower bounds for classic symmetry breaking graph problems. As a basic tool, we first provide a new insightful proof that any deterministic distributed algorithm that computes a ∆-coloring on ∆-regular trees requires Ω(log ∆ n) rounds and any randomized algorithm requires Ω(log ∆ log n) rounds. We prove this result by showing that a natural relaxation of the ∆-coloring problem is a fixed point in the round elimination framework.As a first application, we show that our ∆-coloring lower bound proof directly extends to arbdefective colorings. An arbdefective c-coloring of a graph G = (V, E) is given by a c-coloring of V and an orientation of E, where the arbdefect of a color i is the maximum number of monochromatic outgoing edges of any node of color i. We exactly characterize which variants of the arbdefective coloring problem can be solved in O(f (∆) + log * n) rounds, for some function f , and which of them instead require Ω(log ∆ n) rounds for deterministic algorithms and Ω(log ∆ log n) rounds for randomized ones.As a second application, which is our main contribution, we use the structure of the fixed point as a building block to prove lower bounds for problems that, in some sense, are much easier than ∆-coloring, as they can be solved in O(log * n) deterministic rounds in bounded-degree graphs. More specifically, we prove lower bounds as a function of ∆ for a large class of distributed symmetry breaking problems, which can all be solved by a simple sequential greedy algorithm. For example, we obtain a tight linear-in-∆ lower bound for computing a maximal independent set in ∆-regular trees. For the case where an initial O(∆)-coloring is given, we obtain a tight Ω(β∆ 1/β )-round lower bound for computing a (2, β)-ruling set (for β = o(log ∆)). Our lower bounds even apply to a much more general family of problems, such as variants of ruling sets where nodes in the set do not need to satisfy the independence requirement, but they only need to satisfy the requirements of some arbdefective coloring.Our lower bounds as a function of ∆ also imply lower bounds as a function of n. We obtain, for example, that the maximal independent set problem, on trees, requires Ω(log n/ log log n) rounds for deterministic algorithms, which is tight.

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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 Time Hierarchy Theorem for the LOCAL Model

Cited by 44 publications

References 27 publications

New classes of distributed time complexity

New classes of distributed time complexity

Distributed $Δ$-Coloring Plays Hide-and-Seek

Locality of Not-so-Weak Coloring

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