Routing in Trees

Fraigniaud, Pierre; Gavoille, Cyril

doi:10.1007/3-540-48224-5_62

Cited by 136 publications

(192 citation statements)

References 9 publications

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“…In [6,15], a shortest path routing labeling scheme for trees is described that assigns each vertex of an n-vertex tree a O(log 2 n/ log log n)-bit label. Given the label of a source vertex and the label of a destination, it is possible to compute in constant time, based solely on these two labels, the neighbor of the source that heads in the direction of the destination.…”

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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“…These are schemes that label the vertices of a graph with short labels (describing some global topology information) in such a way that given the label of a source vertex and the label of a destination, it is possible to compute efficiently the edge from the source that heads in the direction of the destination. In [18,40], a shortest path routing scheme for trees with O(log 2 n/ log log n)-bit labels is described. For general graphs, the most general result to date is a multiplicative (4k − 5)-stretched routing labeling scheme that uses labels of sizeÕ(kn 1/k ) bits 1 is obtained in [40] for every k ≥ 2.…”

Section: Some Known Strategiesmentioning

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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“…Our focus is thus on informative labeling schemes using relatively short labels (say, of length polylogarithmic in n). Labeling schemes of this type were recently developed for different graph families and for a variety information types, including vertex adjacency [Alstrup and Rauhe 2002;Breuer 1966;Breuer and Folkman 1967;Kannan et al 1992;Korman et al 2006], distance [Alstrup et al 2005;Katz et al 2000;Kaplan and Milo 2001;Korman et al 2006;Peleg 1999;Thorup 2004], tree routing [Fraigniaud and Gavoille 2001;Fraigniaud and Gavoille 2002;Thorup and Zwick 2001], vertex-connectivity [Alstrup and Rauhe 2002;Katz et al 2004], flow Katz et al 2004], tree ancestry [Abiteboul at al. 2001;Abiteboul at al.…”

Section: Problem and Motivationmentioning

confidence: 99%

Labeling Schemes for Vertex Connectivity

Korman

2007

Automata, Languages and Programming

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This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of assigning short labels to the nodes of any n-node graph is such a way that given the labels of any two nodes u and v, one can decide whether u and v are k-vertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such a labeling scheme is 2 k log n.

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Routing in Trees

Cited by 136 publications

References 9 publications

Collective Tree 1-Spanners for Interval Graphs

Collective Tree 1-Spanners for Interval Graphs

Navigating in a Graph by Aid of Its Spanning Tree

Labeling Schemes for Vertex Connectivity

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