Deterministic Distributed Ruling Sets of Line Graphs

Kühn, Fabian; Weidner, Simon

doi:10.1007/978-3-030-01325-7_19

Cited by 24 publications

(28 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this deterministic distributed version, we select vertices by constructing a ruling set for the set W i . We use the algorithm given in [SEW13,KMW18], on the set W i with parameters q = 2δ i , c = ρ −1 . The following theorem summarizes the properties of the returned ruling set RS i .…”

Section: Superclusteringmentioning

confidence: 99%

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Elkin

Matar

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

Given a pair of parameters α ≥ 1, β ≥ 0, a subgraph G = (V, H) of an n-vertex unweighted undirected graph G = (V, E) is called an (α, β)-spanner if for every pair u, v ∈ V of vertices, we have d G (u, v) ≤ αd G (u, v) + β. If β = 0 the spanner is called a multiplicative α-spanner, and if α = 1 + , for an arbitrarily small > 0, the spanner is said to be a near-additive one.Graph spanners [Awe85, PS89] are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Nearadditive spanners, introduced in [EP01], preserve large distances much more faithfully than the more traditional multiplicative spanners. Also, recent lower bounds [AB15] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with α = 1), and therefore showed that essentially near-additive spanners provide the best approximation of distances that one can hope for.Numerous distributed algorithms, for constructing sparse near-additive spanners, were devised in [Elk01, EZ06, DGPV09, Pet10, EN17]. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [EN17], and also there are efficient deterministic algorithms in the LOCAL model [DGPV09]. However, the only known deterministic CONGEST-model algorithm for the problem [Elk01] requires super-linear time in n. In this paper we remedy the situation and devise an efficient deterministic CONGESTmodel algorithm for constructing arbitrarily sparse near-additive spanners.The running time of our algorithm is low polynomial, i.e., roughly O(β ·n ρ ), where ρ > 0 is an arbitrarily small positive constant that affects the additive term β. In general, the parameters of our new algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm for the problem due to [EN17].In this paper, we focus on unweighted, undirected graphs. Given a graph G = (V, E), a subgraph G = (V, H), H ⊆ E, is said to be a (multiplicative) t-spanner of G, if for every pair u, v ∈ V of vertices, we have d G (u, v) ≤ t · d G (u, v), for a parameter t ≥ 1, where d G (respectively, d G ) stands for the distance in the graph G (respectively, in the subgraph G ).A fundamental tradeoff for multiplicative spanners is that for any κ = 1, 2, . . . , and any n-vertex graph G = (V, E), there exists a (2κ−1)-spanner with O(n 1+1/κ ) edges [ADD + 93, PS89]. Assuming Erdös-Simonovits girth conjecture (see e.g., [ES82]), this tradeoff is optimal. Efficient distributed algorithms for constructing multiplicative spanners that (nearly) realize this tradeoff were given in [ABCP93, Coh98, BS07, Elk07, DGPV08, EN17, DMZ10, GP17, GK18].A different variety of spanners, called near-additive spanners, was presented by Elkin and Peleg in [EP01]. They showed that for every > 0 and κ = 1, 2, . . . , there exists β = β(κ, ), such that for every n-vertex graph G = (V, E) there exists a (1 + , β)In [EP01], the additive error β i...

show abstract

Section: Superclusteringmentioning

confidence: 99%

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Elkin

Matar

2019

Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

show abstract

“…the graph G. See Section 1.5.2 for the definition of ruling sets. This is done using the algorithm of [32,42]. Theorem 3.2 summarizes the properties of the returned ruling set S i .…”

Section: Superclusteringmentioning

confidence: 99%

“…Theorem 3.2. [32,42] Given a graph G = (V , E), a set of vertices W i ⊆ V and parameters q ∈ {1, 2, . .…”

Section: Superclusteringmentioning

confidence: 99%

Ultra-Sparse Near-Additive Emulators

Elkin

Matar

2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

View full text Add to dashboard Cite

Near-additive (aka (1 + ϵ, β)-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings.Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch β. Specifically, for any pair of parameters ϵ > 0, κ = 1, 2, . . . , one can have a (1 + ϵ, β)-emulator with O(n 1+ 1 κ ) edges, with β = log κ ϵ log κ

show abstract

“…sums up all the results on MIS. A (3, 2)-ruling set can be computed by computing an MIS over G 2 , and more general ruling sets have been studied in [2,12,13,19,21].…”

Section: Related Workmentioning

confidence: 99%

Distributed reconfiguration of maximal independent sets

Censor-Hillel

Rabie

2020

Journal of Computer and System Sciences

View full text Add to dashboard Cite

In this paper, we investigate a distributed maximal independent set (MIS) reconfiguration problem, in which there are two maximal independent sets for which every node is given its membership status, and the nodes need to communicate with their neighbors in order to find a reconfiguration schedule that switches from the first MIS to the second. Such a schedule is a list of independent sets that is restricted by forbidding two neighbors to change their membership status at the same step. In addition, these independent sets should provide some covering guarantee.We show that obtaining an actual MIS (and even a 3-dominating set) in each intermediate step is impossible. However, we provide efficient solutions when the intermediate sets are only required to be independent and 4-dominating, which is almost always possible, as we fully characterize.Consequently, our goal is to pin down the tradeoff between the possible length of the schedule and the number of communication rounds. We prove that a constant length schedule can be found in O(MIS + R32) rounds, where MIS is the complexity of finding an MIS in a worst-case graph and R32 is the complexity of finding a (3, 2)-ruling set. For bounded degree graphs, this is O(log * n) rounds and we show that it is necessary. On the other extreme, we show that with a constant number of rounds we can find a linear length schedule. IntroductionConsider a distributed setting in which each node of a network receives an input from a higher-level application which tells it whether it is selected or not, such that the set of selected nodes is a maximal independent set (MIS), which we will denote by α. The reason that the application requires an MIS is because it needs the set of selected nodes to dominate all nodes for the sake of, say, monitoring the network, but without having violations of two neighbors being in the set, because they may cause conflicting actions. Now, because of changes in the network traffic, the energy consumption, or any one of various conditions that may change, the application needs to change the selected set of nodes. Once a new input MIS, denoted by β, is given to the nodes by the application, the nodes need to reconfigure their states to that set while never sacrificing the safety condition of independence. In fact, for compatibility reasons, neighboring nodes cannot change their membership in the set at the same time, so a sequence of changes is needed for converging into the new MIS. We call such a sequence a reconfiguration schedule.The length of the schedule is clearly a measure that is required to be minimized. Hence, an extreme solution would be to have all nodes declare themselves as unselected, and then the new set of nodes declare that they are selected. However, this very fast approach suffers from loosing the domination property throughout the reconfiguration schedule. Thus, the structure of the network must be taken into account, but since the topology is unknown, finding a schedule that maintains a good covering at all times necessitates that th...

show abstract

Deterministic Distributed Ruling Sets of Line Graphs

Cited by 24 publications

References 30 publications

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Near-Additive Spanners In Low Polynomial Deterministic CONGEST Time

Ultra-Sparse Near-Additive Emulators

Distributed reconfiguration of maximal independent sets

Contact Info

Product

Resources

About