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Collective tree spanners of graphs
 SWAT 2004
, 2004
"... In this paper we introduce a new notion of collective tree spanners. We say that a graph G =(V,E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈T(G) exists such that d ..."
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Cited by 14 (10 self)
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In this paper we introduce a new notion of collective tree spanners. We say that a graph G =(V,E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈T(G) exists such that dT (x, y) ≤ dG(x, y) +r. Among other results, we show that any chordal graph, chordal bipartite graph or cocomparability graph admits a system of at most log 2 n collective additive tree 2–spanners and any cchordal graph admits a system of at most log 2 n collective additive tree (2⌊c/2⌋)–spanners. Towards establishing these results, we present a general property for graphs, called (α, r)– decomposition, and show that any (α, r)–decomposable graph G with n vertices admits a system of at most log 1/α n collective additive tree 2r– spanners. We discuss also an application of the collective tree spanners to the problem of designing compact and efficient routing schemes in graphs.
Additive Spanners and Distance and Routing Labeling Schemes for Hyperbolic Graphs
"... δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, g ..."
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Cited by 11 (1 self)
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δHyperbolic metric spaces have been defined by M. Gromov in 1987 via a simple 4point condition: for any four points u, v, w, x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x),d(u, x) + d(v, w) differ by at most 2δ. They play an important role in geometric group theory, geometry of negatively curved spaces, and have recently become of interest in several domains of computer science, including algorithms and networking. For example, (a) it has been shown empirically that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension, (b) every connected finite graph has an embedding in the hyperbolic plane so that the greedy routing based on the virtual coordinates obtained from this embedding is guaranteed to work. A connected graph G = (V, E) equipped with standard graph metric dG is δhyperbolic if the metric space (V, dG) is δhyperbolic. In this paper, using our Layering Partition technique, we provide a simpler construction of distance approximating trees of δhyperbolic graphs on n vertices with an additive error O(δ log n) and show that every nvertex δhyperbolic graph has an additive O(δ log n)spanner with at most O(δn) edges. As a consequence, we show that the family of δhyperbolic graphs with n vertices enjoys an O(δ log n)additive routing labeling scheme with O(δ log 2 n) bit labels and O(log δ) time routing protocol, and an easier constructable O(δ log n)additive distance labeling scheme with O(log 2 n) bit labels and constant time distance decoder.
Collective Tree Spanners in Graphs with Bounded Parameters
 ALGORITHMICA
, 2006
"... In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ colle ..."
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Cited by 4 (4 self)
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In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded treewidth, graphs with bounded cliquewidth, and graphs with bounded chordality. We say that a graph G = (V, E) admits a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ∈ T (G) exists such that dT (x, y) ≤ dG(x, y) + r. We describe a general method for constructing a “small” system of collective additive tree rspanners with small values of r for “well ” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O ( √ n) collective additive tree 0spanners, any weighted graph with treewidth at most k − 1 admits a system of k log 2 n collective additive tree 0spanners, any weighted graph with cliquewidth at most k admits a system of k log 3/2 n collective additive tree (2w)spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log 2 n collective additive tree (2⌊c/2⌋w)spanners and a system of 4 log 2 n collective additive tree (2(⌊c/3⌋+1)w)spanners (here, w is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4 log 2 n collective additive tree (2w)spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs. Results of this paper were partially presented at the ISAAC’05 conference [14].
Collective tree 1spanners for interval graphs
 in GraphTheoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 3787
, 2005
"... Abstract. In this paper we study the existence of a small set T of spanning trees that collectively “1span ” an interval graph G. Inparticular, 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 ..."
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Abstract. In this paper we study the existence of a small set T of spanning trees that collectively “1span ” an interval graph G. Inparticular, 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 1spanners for interval graphs (even unit interval graphs), – interval graph G has a set of collective tree 1spanners of size O(log D), where D is the diameter of G, – interval graphs have a 1spanner with fewer than 2n − 2edges. Furthermore, at the end of the paper we state other results on collective tree cspanners for c>1 and other more general graph classes. 1
Exact distance labelings yield additivestretch compact routing schemes
 In 20 th International Symposium on Distributed Computing (DISC
, 2006
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Navigating in a graph by aid of its spanning tree
, 2008
"... 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, gotoaneighborofz in G that is closest to y in T (breaking ties arbitrarily). In this stra ..."
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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, gotoaneighborofz 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 treedistances). 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 rcarcass 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 ∈Fadmits an additive rcarcass. We show that rectilinear p × q grids, hypercubes, distancehereditary graphs, dually chordal graphs (and, therefore, strongly chordal graphs and interval graphs), all admit additive 0carcasses. 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 3sunfree chordal graph admits an additive 2carcass, each chordal bipartite graph admits an additive 4carcass. In particular, any ktree admits an additive (k+2)carcass. All those carcasses are easy to construct.
Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs
"... Abstract. In this paper, we propose a new compact and low delay routing labeling scheme for Unit Disk Graphs (UDGs) which often model wireless ad hoc networks. We show that one can assign each vertex of an nvertex UDG G acompactO(log 2 n)bit label such that, given the label of a source vertex and ..."
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Abstract. In this paper, we propose a new compact and low delay routing labeling scheme for Unit Disk Graphs (UDGs) which often model wireless ad hoc networks. We show that one can assign each vertex of an nvertex UDG G acompactO(log 2 n)bit label such that, given the label of a source vertex and the label of a destination, it is possible to compute efficiently, based solely on these two labels, a neighbor of the source vertex that heads in the direction of the destination. We prove that this routing labeling scheme has a constant hop routestretch ( = hop delay), i.e., for each two vertices x and y of G, it produces a routing path with h(x, y) hops (edges) such that h(x, y) ≤ 3·dG(x, y)+12, where dG(x, y) is the hop distance between x and y in G. To the best of our knowledge, this is the first compact routing scheme for UDGs which not only guaranties delivery but has a low hop delay and polylog label size. Furthermore, our routing labeling scheme has a constant length routestretch. 1
On Advice Complexity of the kserver Problem under Sparse Metrics
"... Abstract. We consider the kServer problem under the advice model of computation when the underlying metric space is sparse. On one side, we show that an advice of size Ω(n) is required to obtain a 1competitive algorithm for sequences of size n, even for the 2server problem on a path metric of siz ..."
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Abstract. We consider the kServer problem under the advice model of computation when the underlying metric space is sparse. On one side, we show that an advice of size Ω(n) is required to obtain a 1competitive algorithm for sequences of size n, even for the 2server problem on a path metric of size N ≥ 5. Through another lower bound argument, we show that at least n 2 (logα − 1.22) bits of advice is required to obtain an optimal solution3 for metric spaces of treewidth α, where 4 ≤ α < 2k. On the other side, we introduce Θ(1)competitive algorithms for a wide range of sparse graphs, which require advice of (almost) linear size. Namely, we show that for graphs of size N and treewidth α, there is an online algorithm which receives O(n(logα+log logN)) bits of advice and optimally serves a sequence of length n. With a different argument, we show that if a graph admits a system of µ collective tree (q, r) spanners, then there is a (q+ r)competitive algorithm which receives O(n(logµ+ log logN)) bits of advice. Among other results, this gives a 3competitive algorithm for planar graphs, provided with O(n log logN) bits of advice. 1
Additive Spanners for Circle Graphs and Polygonal Graphs
, 2008
"... A graph G =(V, E) is said to admit a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices u, v of G a spanning tree T ∈ T (G) exists such that the distance in T between u and v is at most r plus their distance in G ..."
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A graph G =(V, E) is said to admit a system of μ collective additive tree rspanners if there is a system T (G) of at most μ spanning trees of G such that for any two vertices u, v of G a spanning tree T ∈ T (G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the problem of finding “small” systems of collective additive tree rspanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show n that every nvertex circle graph admits a system of at most 2 log 3 2 collective additive tree 2spanners and every nvertex kpolygonal graph admits a system of at most 2 log 3 k+7 collective additive tree 2spanners. 2 Moreover, we show that every nvertex kpolygonal graph admits an additive (k + 6)spanner with at most 6n − 6 edges and every nvertex 3polygonal graph admits a system of at most 3 collective additive tree 2spanners and an additive tree 6spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time.
Collective Additive Tree Spanners of Homogeneously Orderable Graphs (Extended Abstract)
, 2008
"... In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distancehereditary graphs and to show that the Steiner tree problem can still be solve ..."
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In this paper we investigate the (collective) tree spanners problem in homogeneously orderable graphs. This class of graphs was introduced by A. Brandstädt et al. to generalize the dually chordal graphs and the distancehereditary graphs and to show that the Steiner tree problem can still be solved in polynomial time on this more general class of graphs. In this paper, we demonstrate that every nvertex homogeneously orderable graph G admits – a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤ dG(x, y) + 3 (i.e., an additive tree 3spanner) and – a system T (G) of at most O(log n) spanning trees such that, for any two vertices x, y of G, a spanning tree T ∈T(G) exists with dT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collective additive tree 2spanners). These results generalize known results on tree spanners of dually chordal graphs and of distancehereditary graphs. The results above are also complemented with some lower bounds which say that on some nvertex homogeneously orderable graphs any system of collective additive tree 1spanners must have at least Ω(n) spanning trees and there is no system of collective additive tree 2spanners with constant number of trees.