Distance labeling in graphs

Gavoille, Cyril; Peleg, David; Pérennès, Stéphane; Raz, Ran

doi:10.1016/j.jalgor.2004.05.002

Cited by 234 publications

(258 citation statements)

References 11 publications

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“…Distance labelling. A distance labelling of a graph G is an assignment of unique labels to vertices of G so that the distance between any two vertices can be inferred from their labels alone [16,20].…”

Section: Other Related Workmentioning

confidence: 99%

Ephemeral networks with random availability of links: The case of fast networks

Akrida

Gąsieniec

Mertzios

et al. 2016

Journal of Parallel and Distributed Computing

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Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights• We define the Temporal Diameter (TD) and the expected TD of temporal networks with random availabilities of links• We define fast and slow random temporal networks, according to their expected TD• Random temporal networks on instances of dense (directed) random graphs G n,p are fast• We introduce the critical availability as a measure of periodic availability of links• We give a lower and an upper bound on the critical availability AbstractWe consider here a model of temporal networks, the links of which are available only at certain moments in time, chosen randomly from a subset of the positive integers. We define the notion of the Temporal Diameter of such networks. We also define fast and slow such temporal networks with respect to the expected value of their temporal diameter. We then provide a partial characterisation of fast random temporal networks. We also define the critical availability as a measure of periodic random availability of the links of a network, required to make the network fast. We finally give a lower bound as well as an upper bound on the (critical) availability.

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Section: Other Related Workmentioning

confidence: 99%

Ephemeral networks with random availability of links: The case of fast networks

Akrida

Gąsieniec

Mertzios

et al. 2016

Journal of Parallel and Distributed Computing

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“…In fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances. It is known [23,34] that the vertices of an n-vertex tree T can be labeled in O(n log n) total time with labels of up to O(log 2 n) bits such that given the labels of two vertices v, u of T , it is possible to compute in constant time the distance d T (v, u), by merely inspecting the labels of u and v. Hence, one may assume that each vertex v of G knows, additionally to its neighborhood in G, only its O(log 2 n) bit distance label. This distance label can be viewed as a virtual coordinate of v.…”

Section: To a Neighbor Of Z In G That Is Closest To Y In T (Break Tiementioning

confidence: 99%

Navigating in a Graph by Aid of Its Spanning Tree

Dragan

Matamala

2008

Algorithms and Computation

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Abstract. Let G = (V, E) be a graph and T be a spanning tree of G. We consider the following strategy in advancing in G from a vertex x towards a target vertex y: from a current vertex z (initially, z = x), unless z = y, go to a neighbor of z in G that is closest to y in T (breaking ties arbitrarily). In this strategy, each vertex has full knowledge of its neighborhood in G and can use the distances in T to navigate in G. Thus, additionally to standard local information (the neighborhood NG(v)), the only global information that is available to each vertex v is the topology of the spanning tree T (in fact, v can know only a very small piece of information about T and still be able to infer from it the necessary tree-distances). For each source vertex x and target vertex y, this way, a path, called a greedy routing path, is produced. Denote by gG,T (x, y) the length of a longest greedy routing path that can be produced for x and y using this strategy and T . We say that a spanning tree T of a graph G is an additive r-carcass for G if gG,T (x, y) ≤ dG(x, y) + r for each ordered pair x, y ∈ V . In this paper, we investigate the problem, given a graph family F, whether a small integer r exists such that any graph G ∈ F admits an additive r-carcass. We show that rectilinear p × q grids, hypercubes, distance-hereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0-carcasses. Furthermore, every chordal graph G admits an additive (ω(G) + 1)-carcass (where ω(G) is the size of a maximum clique of G), each 3-sun-free chordal graph admits an additive 2-carcass, each chordal bipartite graph admits an additive 4-carcass. In particular, any k-tree admits an additive (k +2)-carcass. All those carcasses are easy to construct.

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“…In particular, we show that there exist simple labeling schemes with one query supporting the distance function on n-node trees as well as the flow function on n-node general graphs with label size O(log n + log W ), where W is the maximum (integral) capacity of an edge. (We note that the lower bound for labeling schemes without queries for each of these problems is Ω(log 2 n + log n log W ) [32,39].) We also show that there exists a labeling scheme with one query supporting the id-NCA function on n-node trees with label size O(log n).…”

Section: Our Contributionmentioning

confidence: 72%

“…In particular, [50] showed that the family of n vertex weighted trees with integer edge capacity of at most W enjoys a scheme using O(log 2 n + log n log W )-bit labels. This bound was proven in [32] to be asymptotically optimal.…”

Section: Introductionmentioning

confidence: 93%

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Labeling Schemes with Queries

Korman

Kutten

Structural Information and Communication Complexity

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We study the question of "how robust are the known lower bounds of labeling schemes when one increases the number of consulted labels". Let f be a function on pairs of vertices. An f -labeling scheme for a family of graphs F labels the vertices of all graphs in F such that for every graph G ∈ F and every two vertices u, v ∈ G, the value f (u, v) can be inferred by merely inspecting the labels of u and v.This paper introduces a natural generalization: the notion of f -labeling schemes with queries, in which the value f (u, v) can be inferred by inspecting not only the labels of u and v but possibly the labels of some additional vertices. We show that inspecting the label of a single additional vertex (one query) enables us to reduce the label size of many labeling schemes significantly. In particular, we show that to support the distance function on n-node trees as well as the flow function on n-node general graphs, O(log n + log W )-bit labels are sufficient and necessary, where W is the maximum (integral) capacity of an edge. We note that it was shown that any labeling scheme (without queries) supporting either the flow function on general graphs or the distance function on trees, must have label size Ω(log 2 n + log n log W ). Using a single query, we also show a routing labeling scheme in the fixed-port model using O(log n)-bit labels, while the lower bound on the label size of any such routing labeling scheme (without queries) is Ω(log 2 n/log log n). We note that all our labeling schemes with queries have asymptotically optimal label size.We then show several extensions. In particular, we show that our labeling schemes with queries on trees can be extended to the dynamic scenario by some adaptations of known model translation methods. Second, we show that the study of the queries model can help with the traditional model too. That is, using ideas from our routing labeling scheme with one query, we show how to construct a 3-approximation routing scheme without queries in the fixed-port model with Θ(log n)-bit labels. Finally, we turn to a non-distributed environment and first translate known results on finding NCA to supporting distance queries. We then use the above to show that one can preprocess a general weighted graph using almost linear space so that flow queries can be answered in almost constant time.

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Distance labeling in graphs

Abstract: We describe recent work on a variant of a distance labeling problem in graphs, called the forbidden-set labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given

Cited by 234 publications

References 11 publications

Ephemeral networks with random availability of links: The case of fast networks

Ephemeral networks with random availability of links: The case of fast networks

Navigating in a Graph by Aid of Its Spanning Tree

Labeling Schemes with Queries

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