Graph spanners

Peleg, David; Schäffer, Alejandro A.

doi:10.1002/jgt.3190130114

Cited by 503 publications

(370 citation statements)

References 11 publications

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“…(A similar definition can be given for multiplicative c-spanners [1,14,13]; however since we are only concerned with additive spanners, we will often omit "additive".) In this paper, we continue the approach taken in [5,4,7] of studying collective tree spanners.…”

Section: Introductionmentioning

confidence: 99%

Collective Tree 1-Spanners for Interval Graphs

Corneil

Dragan

Köhler

et al. 2005

Graph-Theoretic Concepts in Computer Science

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Abstract. In this paper we study the existence of a small set T of spanning trees that collectively "1-span" an interval graph G. In particular, for any pair of vertices u, v we require a tree T ∈ T such that the distance between u and v in T is at most one more than their distance in G. We show that:-there is no constant size set of collective tree 1-spanners for interval graphs (even unit interval graphs), -interval graph G has a set of collective tree 1-spanners of size O(log D), where D is the diameter of G, -interval graphs have a 1-spanner with fewer than 2n − 2 edges. Furthermore, at the end of the paper we state other results on collective tree c-spanners for c > 1 and other more general graph classes.

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Section: Introductionmentioning

confidence: 99%

Collective Tree 1-Spanners for Interval Graphs

Corneil

Dragan

Köhler

et al. 2005

Graph-Theoretic Concepts in Computer Science

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“…The lower bound of the routing cost of a k-edge subgraph is 2a * − k, and such a routing cost is achieved only when k requirements are routed by one edge and all others by two edges. Therefore, every edge in H is also an edge in G and d H (i, j) = 2 for any edge (i, j) in G but not in H. It also implies that H is a 2-spanner of G. Since the ORG problem is obviously in NP and the 2-spanner problem is NP-complete [4,16], by the above reduction, the ORG problem is NP-complete.…”

Section: The Org Problemmentioning

confidence: 93%

Light graphs with small routing cost

Chao²,

Tang

2002

Networks

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Let G = ({1,…, n}, E, w) be an undirected graph with nonnegative edge weights w and let aij be the nonnegative requirement between vertices i and j. For any spanning subgraph H of G, the weight of H is the total weight of its edges and the routing cost of H is Σi<j aijdH(i, j), where dH(i, j) is the distance between i and j in H. In this paper, we investigated two special cases of the problem of finding a spanning subgraph with small weight and small routing cost. For the case where all the distances in G are 1, we show that the problem is NP‐complete, and give a simple approximation algorithm for it. Furthermore, we define some sufficient conditions for the problem to be polynomial‐time solvable. For the case where all the requirements are 1, we develop an algorithm for finding a spanning tree with small weight and small routing cost. The algorithm provides trade‐offs among tree weights, routing costs, and time complexity. We also extend the results to some other related problems. © 2002 Wiley Periodicals, Inc.

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“…We believe that finding such subgraphs can be useful in other applications, e.g., in obtaining scalable-degree spanners [21,22] -fundamental subgraphs that preserve distances between nodes up to some stretch.…”

Section: φC(g)mentioning

confidence: 99%

Fast Information Spreading in Graphs with Large Weak Conductance

Censor-Hillel¹,

Shachnai²

2012

SIAM J. Comput.

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Gathering data from nodes in a network is at the heart of many distributed applications, most notably, while performing a global task. We consider information spreading among n nodes of a network, where each node v has a message m(v) which must be received by all other nodes. The time required for information spreading has been previously upper-bounded with an inverse relationship to the conductance of the underlying communication graph. This implies high running times for graphs with small conductance.The main contribution of this paper is an information spreading algorithm which overcomes communication bottlenecks and thus achieves fast information spreading for a wide class of graphs, despite their small conductance. As a key tool in our study we use the recently defined concept of weak conductance, a generalization of classic graph conductance which measures how well-connected the components of a graph are. Our hybrid algorithm, which alternates between random and deterministic communication phases, exploits the connectivity within components by first applying partial information spreading, after which messages are sent across bottlenecks, thus spreading further throughout the network. This yields substantial improvements over the best known running times of algorithms for information spreading on any graph that has a large weak conductance, from polynomial to polylogarithmic number of rounds.We demonstrate the power of fast information spreading in accomplishing global tasks on the leader election problem, which lies at the core of distributed computing. Our results yield an algorithm for leader election that has a scalable running time on graphs with large weak conductance, improving significantly upon previous results.

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Graph spanners

Cited by 503 publications

References 11 publications

Collective Tree 1-Spanners for Interval Graphs

Collective Tree 1-Spanners for Interval Graphs

Light graphs with small routing cost

Fast Information Spreading in Graphs with Large Weak Conductance

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