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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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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.
Short Labels by Traversal and Jumping
, 2007
"... In this paper, we propose an efficient implicit representation of caterpillar and bounded degree trees of n vertices. Our scheme, called Traversal & Jumping, assigns to the n vertices of any bounded degree tree distinct binary labels of log 2 n + O(1) bits in O(n) time such that we can compute a ..."
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Cited by 6 (3 self)
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In this paper, we propose an efficient implicit representation of caterpillar and bounded degree trees of n vertices. Our scheme, called Traversal & Jumping, assigns to the n vertices of any bounded degree tree distinct binary labels of log 2 n + O(1) bits in O(n) time such that we can compute adjacency between two vertices only from their labels. We use our result to improve previous known upper bound for size of labels of implicit representation of outerplanar graphs (respectively planar graphs) to 2 log 2 n (respectively (3 log 2 n).
Sparse spanners vs. compact routing.
 In Proc. 23th ACM Symp. on Parallel Algorithms and Architectures (SPAA),
, 2011
"... ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ( ..."
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ABSTRACT Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables ofΘ(
Shorter Implicit Representation for Planar Graphs and Bounded Treewidth Graphs
 IN "15TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2007
"... Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition ..."
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Implicit representation of graphs is a coding of the structure of graphs using distinct labels so that adjacency between any two vertices can be decided by inspecting their labels alone. All previous implicit representations of planar graphs were based on the classical three forests decomposition technique (a.k.a. Schnyder’s trees), yielding asymptotically toa3lognbit 1 label representation where n is the number of vertices of the graph. We propose a new implicit representation of planar graphs using asymptotically 2 log nbit labels. As a byproduct we have an explicit construction of a graph with n 2+o(1) vertices containing all nvertex planar graphs as induced subgraph, the best previous size of such induceduniversal graph was O(n 3). More generally, for graphs excluding a fixed minor, we construct a 2logn + O(log log n) implicit representation. For treewidthk graphs we give a log n + O(k log log(n/k)) implicit representation, improving the O(k log n) representation of Kannan, Naor and Rudich [18] (STOC ’88). Our representations for planar and treewidthk graphs are easy to implement, all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.
Distributed Relationship Schemes for Trees
 In Proc. 18th Int. Symp. on Algorithms and Computation
"... Abstract. We consider a distributed representation scheme for trees, supporting some special relationships between nodes at small distance. For instance, we show that for a tree T and an integer k we can assign local information on nodes such that we can decide for any two nodes u and v if the dist ..."
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Abstract. We consider a distributed representation scheme for trees, supporting some special relationships between nodes at small distance. For instance, we show that for a tree T and an integer k we can assign local information on nodes such that we can decide for any two nodes u and v if the distance between u and v is at most k and if so, compute it only using the local information assigned. For trees with n nodes, the local information assigned by our scheme are binary labels of log n + O(k log(k log(n/k))) bits, improving the results of Alstrup, Bille, and Rauhe (SODA '03).
Exact distance labelings yield additivestretch compact routing schemes.
 In 20th International Symposium on Distributed Computing (DISC),
, 2006
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OPTIMAL DISTANCE LABELING FOR INTERVAL GRAPHS AND RELATED GRAPH FAMILIES
, 2008
"... A distance labeling scheme is a distributed graph representation that assigns labels to the vertices and enables answering distance queries between any pair (x, y) of vertices by using only the labels of x and y. This paper presents an optimal distance labeling scheme with labels of O(log n) bits f ..."
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A distance labeling scheme is a distributed graph representation that assigns labels to the vertices and enables answering distance queries between any pair (x, y) of vertices by using only the labels of x and y. This paper presents an optimal distance labeling scheme with labels of O(log n) bits for the nvertex interval graphs family. It improves by log n factor the best known upper bound of [M. Katz, N. A. Katz, and D. Peleg, Distance labeling schemes for wellseparated
Distance Labeling for Permutation Graphs
, 2005
"... We show that every permutation graph with n elements can be preprocessed in O(n) time, if two linear extensions of the corresponding poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure is localized and given as a distance labeling, ..."
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We show that every permutation graph with n elements can be preprocessed in O(n) time, if two linear extensions of the corresponding poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure is localized and given as a distance labeling, that is each vertex receives a label of O(log n) bits so that distance queries between any two vertices are answered by inspecting on their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [20] in the STACS ’00 by a log n factor.
Poset, competition numbers, and interval graph
"... Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ � � V (D) ..."
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Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ � � V (D)