Results 1  10
of
36
Approximate distance oracles
 J. ACM
"... Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in ..."
Abstract

Cited by 279 (10 self)
 Add to MetaCart
Let G = (V, E) be an undirected weighted graph with V  = n and E  = m. Let k ≥ 1 be an integer. We show that G = (V, E) can be preprocessed in O(kmn 1/k) expected time, constructing a data structure of size O(kn 1+1/k), such that any subsequent distance query can be answered, approximately, in O(k) time. The approximate distance returned is of stretch at most 2k − 1, i.e., the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k−1. A 1963 girth conjecture of Erdős, implies that Ω(n 1+1/k) space is needed in the worst case for any real stretch strictly smaller than 2k + 1. The space requirement of our algorithm is, therefore, essentially optimal. The most impressive feature of our data structure is its constant query time, hence the name “oracle”. Previously, data structures that used only O(n 1+1/k) space had a query time of Ω(n 1/k). Our algorithms are extremely simple and easy to implement efficiently. They also provide faster constructions of sparse spanners of weighted graphs, and improved tree covers and distance labelings of weighted or unweighted graphs. 1
Reachability and Distance Queries via 2Hop Labels
, 2002
"... Reachability and distance queries in graphs are fundamental to numerous applications, ranging from geographic navigation systems to Internet routing. Some of these applications involve huge graphs and yet require fast query answering. We propose a new data structure for representing all distances in ..."
Abstract

Cited by 142 (1 self)
 Add to MetaCart
(Show Context)
Reachability and distance queries in graphs are fundamental to numerous applications, ranging from geographic navigation systems to Internet routing. Some of these applications involve huge graphs and yet require fast query answering. We propose a new data structure for representing all distances in a graph. The data structure is distributed in the sense that it may be viewed as assigning labels to the vertices, such that a query involving vertices u and v may be answered using only the labels of u and v.
ForbiddenSet Labeling on Graphs
"... We describe recent work on a variant of a distance labeling problem in graphs, called the forbiddenset labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given {L(x)  x ∈ X} for any X ⊂ V ∪ E and L(u), L(v) for u, v ∈ V, we can decide if a ..."
Abstract

Cited by 130 (28 self)
 Add to MetaCart
We describe recent work on a variant of a distance labeling problem in graphs, called the forbiddenset labeling problem. Given a graph G = (V, E), we wish to assign labels L(x) to vertices and edges of G so that given {L(x)  x ∈ X} for any X ⊂ V ∪ E and L(u), L(v) for u, v ∈ V, we can decide if a property holds in the graph G \ X, or compute a value like the distance between u, v in G \ X. The problem is motivated by routing in networks where some nodes or edges may fail, or where nodes may decide to route on paths avoiding some ‘forbidden’ set of nodes or edges.
Gem: graph embedding for routing and datacentric storage in sensor networks without geographic information
, 2003
"... Information ..."
Proximity search in databases
 In VLDB
, 1998
"... An information retrieval (IR) engine can rank documents based on textual proximityofkeywords within each document. In this paper we apply this notion to search across an entire database for objects that are \near " other relevant objects. Proximity search enables simple \focusing " ..."
Abstract

Cited by 60 (1 self)
 Add to MetaCart
An information retrieval (IR) engine can rank documents based on textual proximityofkeywords within each document. In this paper we apply this notion to search across an entire database for objects that are \near &quot; other relevant objects. Proximity search enables simple \focusing &quot; queries based on general relationships among objects, helpful for interactive query sessions. We view the database as a graph, with data in vertices (objects) and relationships indicated by edges. Proximity is dened based on shortest paths between objects. We have implemented a prototype search engine that uses this model to enable keyword searches over databases, and we have found it very e ective for quickly nding relevant information. Computing the distance between objects in a graph stored on disk can be very expensive. Hence, we show how to build compact indexes that allow us to quickly nd the distance between objects at search time. Experiments show that our algorithms are ecient and scale well. 1
Proof Labeling Schemes
 Proc. the 24th Annual ACM Symposium on Principles of Distributed Computing (PODC 2005), Las Vegas
, 2005
"... This paper addresses the problem of locally verifying global properties. Several natural questions are studied, such as “how expensive is local verification? ” and more specifically “how expensive is local verification compared to computation? ” A suitable model is introduced in which these questio ..."
Abstract

Cited by 34 (20 self)
 Add to MetaCart
This paper addresses the problem of locally verifying global properties. Several natural questions are studied, such as “how expensive is local verification? ” and more specifically “how expensive is local verification compared to computation? ” A suitable model is introduced in which these questions are studied in terms of the number of bits a node needs to communicate. In addition, approaches are presented for the efficient construction of schemes, and upper and lower bounds are established on the cost of schemes for multiple basic problems. The paper also studies the role and cost of unique identities in terms of impossibility and complexity. Previous studies on related questions deal with distributed algorithms that simultaneously compute a configuration and verify that this configuration has a certain desired property. It turns out that this combined approach enables verification to be less costly, since the configuration is typically generated so as to be easily verifiable. In contrast, our approach separates the configuration design from the verification. That is, it first generates the desired configuration without bothering with the need to verify, and then handles the task of constructing a suitable verification scheme. Our approach thus allows for a more modular design of algorithms, and has the potential to aid in verifying properties even when the original design of the structures for maintaining them was done without verification in mind.
Online exact shortest distance query processing
 Proceedings of the International Conference on Extending Database Technology (EDBT), 2009
"... Shortestpath query processing not only serves as a long established routine for numerous applications in the past but also is of increasing popularity to support novel graph applications in very large databases nowadays. For a large graph, there is the new scenario to query intensively against ar ..."
Abstract

Cited by 19 (4 self)
 Add to MetaCart
(Show Context)
Shortestpath query processing not only serves as a long established routine for numerous applications in the past but also is of increasing popularity to support novel graph applications in very large databases nowadays. For a large graph, there is the new scenario to query intensively against arbitrary nodes, asking to quickly return node distance or even shortest paths. And traditional main memory algorithms and shortest paths materialization become inadequate. We are interested in graph labelings to encode the underlying graphs and assign labels to nodes to support efficient query processing. Surprisingly, the existing work of this category mainly emphasizes on reachability query processing, while no sufficient effort has been given to distance labelings to support querying exact shortest distances between nodes. Distance labelings must be developed on the graph in whole to correctly retain node distance information. It makes many existing methods to be inapplicable. We focus on fast computing distanceaware 2hop covers, which can encode the allpairs shortest paths of a graph in O(V  · E1/2) space. Our approach exploits strongly connected components collapsing and graph partitioning to gain speed, while it can overcome the challenges in correctly retaining node distance information and appropriately encoding allpairs shortest paths with small overhead. Furthermore, our approach avoids precomputing allpairs shortest paths, which can be prohibitive over large graphs. We conducted extensive performance studies, and confirm the efficiency of our proposed new approaches. 1.
Treedecompositions with bags of small diameter
, 2007
"... This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
This paper deals with the length of a Robertson–Seymour’s treedecomposition. The treelength of a graph is the largest distance between two vertices of a bag of a treedecomposition, minimized over all treedecompositions of the graph. The study of this invariant may be interesting in its own right because the class of bounded treelength graphs includes (but is not reduced to) bounded chordality graphs (like interval graphs, permutation graphs, ATfree graphs, etc.). For instance, we show that the treelength of any outerplanar graph is ⌈k/3⌉, where k is the chordality of the graph, and we compute the treelength of meshes. More fundamentally we show that any algorithm computing a treedecomposition approximating the treewidth (or the treelength) of an nvertex graph by a factor α or less does not give an αapproximation of the treelength (resp. the treewidth) unless if α = Ω(n 1/5). We complete these results presenting several polynomial time constant approximate algorithms for the treelength. The introduction of this parameter is motivated by the design of compact distance labeling, compact routing tables with nearoptimal route length, and by the construction of sparse additive spanners.
Distance labeling in hyperbolic graphs
 In 16 th Annual International Symposium on Algorithms and Computation (ISAAC
, 2005
"... Abstract. A graph G is δhyperbolic if for any four vertices u, v, x,y of G the two larger of the three distance sums dG(u, v) + dG(x,y), dG(u, x) + dG(v, y),dG(u, y) + dG(v, x) differ by at most δ, and the smallest δ � 0 for which G is δhyperbolic is called the hyperbolicity of G. In this paper, w ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
(Show Context)
Abstract. A graph G is δhyperbolic if for any four vertices u, v, x,y of G the two larger of the three distance sums dG(u, v) + dG(x,y), dG(u, x) + dG(v, y),dG(u, y) + dG(v, x) differ by at most δ, and the smallest δ � 0 for which G is δhyperbolic is called the hyperbolicity of G. In this paper, we construct a distance labeling scheme for bounded hyperbolicity graphs, that is a vertex labeling such that the distance between any two vertices of G can be estimated from their labels, without any other source of information. More precisely, our scheme assigns labels of O(log 2 n) bits for bounded hyperbolicity graphs with n vertices such that distances can be approximated within an additive error of O(log n). The label length is optimal for every additive error up to n ε. We also show a lower bound of Ω(log log n) on the approximation factor, namely every smultiplicative approximate distance labeling scheme on bounded hyperbolicity graphs with polylogarithmic labels requires s = Ω(log log n).
Reconstructing Approximate Tree Metrics
 Proceedings of the twentysixth ACM symposium on Principles of distributed computing
, 2007
"... We introduce a novel measure called εfourpoints condition (ε4PC), which assigns a value ε ∈ [0, 1] to every metric space quantifying how close the metric is to a tree metric. Datasets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
We introduce a novel measure called εfourpoints condition (ε4PC), which assigns a value ε ∈ [0, 1] to every metric space quantifying how close the metric is to a tree metric. Datasets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of ε4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c2 such that, (1) every metric (X, d) which satisfies the ε4PC can be embedded into a tree with distortion (1 + ε) c1 log X, and (2) for every ε ∈ [0, 1] and any number of nodes, there is a metric space (X, d) satisfying the ε4PC that does not embed into a tree with distortion less than (1 + ε) c2 log X. In addition, we prove a lower bound on approximate distance labelings of ε4PC metrics, and give tight bounds for tree embeddings with additive error guarantees.