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217
Approximate distance oracles
, 2004
"... 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 ..."
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Cited by 273 (9 self)
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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.
Compact routing schemes
 in SPAA ’01: Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
"... We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extrem ..."
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Cited by 229 (4 self)
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We describe several compact routing schemes for general weighted undirected networks. Our schemes are simple and easy to implement. The routing tables stored at the nodes of the network are all very small. The headers attached to the routed messages, including the name of the destination, are extremely short. The routing decision at each node takes constant time. Yet, the stretch of these routing schemes, i.e., the worst ratio between the cost of the path on which a packet is routed and the cost of the cheapest path from source to destination, is a small constant. Our schemes achieve a nearoptimal tradeoff between the size of the routing tables used and the resulting stretch. More specifically, we obtain: 1. A routing scheme that uses only ~ O(n 1=2) bits of memory at each node of an nnode network that has stretch 3. The space is optimal, up to logarithmic factors, in the sense that
LowDistortion Embeddings of Finite Metric Spaces
 in Handbook of Discrete and Computational Geometry
, 2004
"... INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their diss ..."
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Cited by 66 (1 self)
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INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their dissimilarity (computed, say, by comparing their DNA). It is dicult to see any structure in a large table of numbers, and so we would like to represent a given metric space in a more comprehensible way. For example, it would be very nice if we could assign to each x 2 X a point f(x) in the plane in such a way that D(x; y) equals the Euclidean distance of f(x) and f(y). Such a representation would allow us to see the structure of the metric space: tight clusters, isolated points, and so on. Another advantage would be that the metric would now be represented by only 2n real numbers, the coordinates of the n points in the plane, instead of numbers as before. Moreover, many quantities concern
Normgraphs: variations and applications
 J. Combin. Theory Ser. B
, 1999
"... We describe several variants of the normgraphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n5/3 edges, containing no copy of K3,3, thus slightly impr ..."
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Cited by 53 (9 self)
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We describe several variants of the normgraphs introduced by Kollár, Rónyai, and Szabó and study some of their extremal properties. Using these variants we construct, for infinitely many values of n, a graph on n vertices with more than 1 2 n5/3 edges, containing no copy of K3,3, thus slightly improving an old construction of Brown. We also prove that the maximum number of vertices in a complete graph whose edges can be colored by k colors with no monochromatic copy of K3,3 is (1 + o(1))k 3. This answers a question of Chung and Graham. In addition we prove that for every fixed t, there is a family of subsets of an n element set whose socalled dual shatter function is O(m t) and whose discrepancy is Ω(n 1/2−1/2t √ log n). This settles a problem of Matouˇsek.
Symmetry analysis of reversible markov chains
 INTERNET MATHEMATICS
, 2005
"... We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to ..."
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Cited by 51 (14 self)
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We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a maxdegree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.
Approximate Distance Labeling Schemes
, 2000
"... We consider the problem of labeling the nodes of an nnode graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such con ..."
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Cited by 47 (19 self)
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We consider the problem of labeling the nodes of an nnode graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, kchordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and ATfree graphs). We also show lower bounds, and prove that most of our schemes are optimal in length of labels generated and in the quality of the approximation, leaving some open problems.
Some extremal problems in graph theory
, 1969
"... We consider only graphs without loops and multiple edges. G n denotes a graph of n vertices, v(G) , e(G) and X(G) denote the number of vertices, edges, and the chromatic number of the graph G respectively. The star of a vertex x will be denoted by st x (that is the set of vertices joined to x), the ..."
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Cited by 40 (0 self)
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We consider only graphs without loops and multiple edges. G n denotes a graph of n vertices, v(G) , e(G) and X(G) denote the number of vertices, edges, and the chromatic number of the graph G respectively. The star of a vertex x will be denoted by st x (that is the set of vertices joined to x), the valency of x will be denoted by 6(x), K(m,n) denotes the complete bichromatic graph with m and n vertices in its classes. {KCm,n)r} is the graph obtained from K(m, n) omitting r ( r< _ min (m, n)) independent edges. Thus { K(4,4) 4} = C is the graph formed by the vertices and edges of a cube. a graph G n Let us denote by f ( n; L1 _. L %, ) the maximum number of edges can have if it does not contain any L ~ as a subgraph. abbreviated by f ( n). If it does not cause any confusion, f ( n; L i,..., L), ) will be According to [1] ( 1) f(n; K(Z,m)) = Om(n2 C) (£<_m) exists, and perhaps
A simple linear time algorithm for computing sparse spanners in weighted graphs
 IN PROCEEDINGS OF THE 30TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2003
"... ... edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running ..."
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Cited by 40 (5 self)
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... edges are required in the worst case for any (2k \Gamma 1)spanner, which has been proved for k = 1; 2; 3; 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn 1=k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that computes a (2k \Gamma 1)spanner of size matching the conjectured lower bound. An important feature of our algorithm is its local approach. Unlike all the previous algorithms which require computation of shortest paths, the new algorithm merely explores the edges in the neighborhood of a vertex or a group of vertices. This feature leads to designing simple externalmemory and parallel algorithms for computing sparse spanners, whose running times are optimal up to logarithmic factors.
A simple and linear time randomized algorithm for computing sparse . . .
"... Let G = (V, E) be an undirected weighted graph on V  = n vertices, and E  = m edges. A tspanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Comput ..."
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Cited by 34 (5 self)
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Let G = (V, E) be an undirected weighted graph on V  = n vertices, and E  = m edges. A tspanner of the graph G, for any t ≥ 1, is a subgraph (V, ES), ES ⊆ E, such that the distance between any pair of vertices in the subgraph is at most t times the distance between them in the graph G. Computing a tspanner of minimum size (number of edges) has been a widely studied and well motivated problem in computer science. In this paper we present the first linear time randomized algorithm that computes a tspanner of a given weighted graph. Moreover, the size of the tspanner computed essentially matches the worst case lower bound implied by a 43 years old girth conjecture made independently by Erdős [26], Bollobás [19], and Bondy & Simonovits [21]. Our algorithm uses a novel clustering approach that avoids any distance computation altogether. This feature is somewhat surprising since all the previously existing algorithms employ computation of some sort of local or global distance information which involves growing either breadth first search trees up to θ(t)levels or full shortest path trees on a large fraction of vertices. The truly local approach of our algorithm also leads to equally simple and efficient algorithms for computing spanners in other important computational environments like distributed, parallel, and external memory.