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12
A SketchBased Distance Oracle for WebScale Graphs
"... We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves t ..."
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We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the timeconsuming shortestpath computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small “sketch ” for each node in the graph, and at querytime we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance. Categories and Subject Descriptors G.2.2 [Graph Theory]: Graph algorithms, path and circuit problems
Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners
 In International Colloquium on Automata, Languages and Programming
, 2007
"... Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm outperforms significantly the stateoftheart algorithm for this task [20]. Specifically, the processing timeperedge of our algorithm is drastically smaller than that of the algorithm of [20], ..."
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Abstract. We present a streaming algorithm for constructing sparse spanners and show that our algorithm outperforms significantly the stateoftheart algorithm for this task [20]. Specifically, the processing timeperedge of our algorithm is drastically smaller than that of the algorithm of [20], and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters for the stateoftheart algorithm. We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has a very small incremental update time, and a nontrivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides nontrivial bounds on both incremental and decremental update time for a wide range of stretch parameter t. 1
Spectral sparsification via random spanners
"... In this paper we introduce a new notion of distance between nodes in a graph that we refer to as robust connectivity. Robust connectivity between a pair of nodes u and v is parameterized by a threshold κ and intuitively captures the number of paths between u and v of length at most κ. Using this new ..."
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Cited by 11 (4 self)
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In this paper we introduce a new notion of distance between nodes in a graph that we refer to as robust connectivity. Robust connectivity between a pair of nodes u and v is parameterized by a threshold κ and intuitively captures the number of paths between u and v of length at most κ. Using this new notion of distances, we show that any black box algorithm for constructing a spanner can be used to construct a spectral sparsifier. We show that given an undirected weighted graph G, simply taking the union of spanners of a few (polylogarithmically many) random subgraphs of G obtained by sampling edges at different probabilities, after appropriate weighting, yields a spectral sparsifier of G. We show how this be done in Õ(m) time, producing a sparsifier with Õ(n/ɛ2) edges. While the cut sparsifiers of Benczur and Karger are based on weighting edges according to (inverse) strong connectivity, and the spectral sparsifiers are based on resistance, our method weights edges using the robust connectivity measure. The main property that we use is that this new measure is always greater than the resistance when scaled by a factor of O(κ) (κ is chosen to be O(log n)), but, just like resistance and connectivity, has a bounded sum, i.e. Õ(n), over all the edges of the graph. 1.
Distributed Algorithms for Ultrasparse Spanners and Linear Size Skeletons
"... We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages ..."
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We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some nontrivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to loglogarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion. 1
Graph stream algorithms: A survey
, 2013
"... Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine ..."
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Over the last decade, there has been considerable interest in designing algorithms for processing massive graphs in the data stream model. The original motivation was twofold: a) in many applications, the dynamic graphs that arise are too large to be stored in the main memory of a single machine and b) considering graph problems yields new insights into the complexity of stream computation. However, the techniques developed in this area are now finding applications in other areas including data structures for dynamic graphs, approximation algorithms, and distributed and parallel computation. We survey the stateoftheart results; identify general techniques; and highlight some simple algorithms that illustrate basic ideas. 1.
Fully Dynamic Randomized Algorithms for Graph Spanners
, 2008
"... Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. ..."
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Spanner of an undirected graph G = (V, E) is a subgraph which is sparse and yet preserves allpairs distances approximately. More formally, a spanner with stretch t ∈Nis a subgraph (V, ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a tspanner of itself, the research as well as applications of spanners invariably deal with a tspanner which has as small number of edges as possible. We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner. Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a tspanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a tspanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.
Spanners and Sparsifiers in Dynamic Streams
"... Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing spaceefficient algorithms for classica ..."
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Cited by 3 (2 self)
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Linear sketching is a popular technique for computing in dynamic streams, where one needs to handle both insertions and deletions of elements. The underlying idea of taking randomized linear measurements of input data has been extremely successful in providing spaceefficient algorithms for classical problems such as frequency moment estimation and computing heavy hitters, and was very recently shown to be a powerful technique for solving graph problems in dynamic streams [AGM’12]. Ideally, one would like to obtain algorithms that use one or a small constant number of passes over the data and a small amount of space (i.e. sketching dimension) to preserve some useful properties of the input graph presented as a sequence of edge insertions and edge deletions. In this paper, we concentrate on the problem of constructing linear sketches of graphs that (approximately) preserve the spectral information of the graph in a few passes over the stream. We do so by giving the first sketchbased algorithm for constructing multiplicative graph spanners in only two passes over the stream. Our spanners use Õ(n1+1/k) bits of space and have stretch 2 k. While this stretch is larger than the conjectured optimal 2k − 1 for this amount of space, we show for an appropriate k that it implies the first 2pass spectral sparsifier with n 1+o(1) bits of space. Previous constructions of spectral sparsifiers in this model with a constant number of passes would require n 1+c bits of space for a constant c> 0. We also give an algorithm for constructing spanners that provides an additive approximation to the shortest path metric using a single pass over the data stream, also achieving an essentially best possible space/approximation tradeoff. 1.
Fast constructions of lightweight spanners for general graphs
 In Proc. of 24th SODA
, 2013
"... Since the pioneering works of Peleg and Schäffer [32], Althöfer et al. [4], and Chandra et al. [13], it is known that for every weighted undirected nvertex medge graph G = (V, E), and every integer k ≥ 1, there exists a ((2k −1) ·(1+ǫ))spanner with O(n 1+1/k) edges and weight O(k · n 1/k) · ω(MS ..."
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Since the pioneering works of Peleg and Schäffer [32], Althöfer et al. [4], and Chandra et al. [13], it is known that for every weighted undirected nvertex medge graph G = (V, E), and every integer k ≥ 1, there exists a ((2k −1) ·(1+ǫ))spanner with O(n 1+1/k) edges and weight O(k · n 1/k) · ω(MST(G)), for an arbitrarily small constant ǫ> 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) Nearly linear time algorithms for constructing (2k − 1)spanners with nearly O(n 1+1/k) edges were devised in [11, 38, 37]. However, these algorithms fail to guarantee any meaningful upper bound on the weight of the constructed spanners. To our knowledge, there are only two known algorithms for constructing sparse and light spanners for general graphs.
Multiplicative approximations of random walk transition probabilities
, 2011
"... We study the space and time complexity of approximating distributions of lstep random walks in simple (possibly directed) graphs G. While very efficient algorithms for obtaining additive ɛapproximations have been developed in the literature, non nontrivial results with multiplicative guarantees a ..."
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We study the space and time complexity of approximating distributions of lstep random walks in simple (possibly directed) graphs G. While very efficient algorithms for obtaining additive ɛapproximations have been developed in the literature, non nontrivial results with multiplicative guarantees are known, and obtaining such approximations is the main focus of this paper. Specifically, we ask the following question: given a bound S on the space used, what is the minimum threshold t> 0 such that lstep transition probabilities for all pairs u, v ∈ V such that P l uv ≥ t can be approximated within a 1 ± ɛ factor? How fast can an approximation be obtained? We show that the following surprising behavior occurs. When the bound on the space is S = o(n 2 /d), where d is the minimum outdegree of G, no approximation can be achieved for nontrivial values of the threshold t. However, if an extra factor of s space is allowed, i.e. S = ˜ Ω(sn 2 /d) space, then the threshold t is exponentially small in the length of the walk l and even very small transition probabilities can be approximated up to a 1±ɛ factor. One instantiation of these guarantees is as follows: any almost regular directed graph can be represented in Õ(ln3/2+ɛ) space such that all probabilities larger than n −10 can be approximated within a (1 ± ɛ) factor as long as l ≥ 40/ɛ 2. Moreover, we show how
Algorithms for bipartite matching problems . . .
, 2012
"... The problem of finding maximum matchings in bipartite graphs is a classical problem in combinatorial optimization with a long algorithmic history. Graph sparsification is a more recent paradigm of replacing a graph with a smaller subgraph that preserves some useful properties of the original graph, ..."
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The problem of finding maximum matchings in bipartite graphs is a classical problem in combinatorial optimization with a long algorithmic history. Graph sparsification is a more recent paradigm of replacing a graph with a smaller subgraph that preserves some useful properties of the original graph, perhaps approximately. Traditionally, sparsification has been used for obtaining faster algorithms for cutbased optimization problems. The contributions of this thesis are centered around new algorithms for bipartite matching problems, in which, surprisingly, graph sparsification plays a major role, and efficient algorithms for constructing sparsifiers in modern data models. In the first part of the thesis we develop sublinear time algorithms for finding perfect matchings in regular bipartite graphs. These graphs have been studied extensively in the context of expander constructions, and have several applications in combinatorial optimization. The problem of finding perfect matchings in regular bipartite graphs has seen almost 100 years of algorithmic history, with the first algorithm dating back to König in 1916 and an algorithm with runtime linear in the number of edges in the graph discovered in 2000. In this thesis we show that, even though traditionally the