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TwiceRamanujan sparsifiers
 IN PROC. 41ST STOC
, 2009
"... We prove that for every d> 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d V  ⌉ edges such that for every x ∈ IR V, 1 ≤ xT LHx x T LGx ≤ d + 1 + 2 √ d d + 1 − 2 √ d, where LG and LH are the Laplacian matrices of G and H, respectively. ..."
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Cited by 88 (12 self)
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We prove that for every d> 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d V  ⌉ edges such that for every x ∈ IR V, 1 ≤ xT LHx x T LGx ≤ d + 1 + 2 √ d d + 1 − 2 √ d, where LG and LH are the Laplacian matrices of G and H, respectively.
Single pass sparsification in the streaming model with edge deletions. arXiv preprint arXiv:1203.4900
, 2012
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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 2 (1 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.
On Sketching Quadratic Forms
, 2015
"... We undertake a systematic study of sketching a quadratic form: given an n × n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1 + ε)approximation to xTAx for any desired query x ∈ Rn. While a general matrix does not admit nontrivial sketch ..."
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We undertake a systematic study of sketching a quadratic form: given an n × n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1 + ε)approximation to xTAx for any desired query x ∈ Rn. While a general matrix does not admit nontrivial sketches, positive semidefinite (PSD) matrices admit sketches of size Θ(ε−2n), via the JohnsonLindenstrauss lemma, achieving the “for each ” guarantee, namely, for each query x, with a constant probability the sketch succeeds. (For the stronger “for all” guarantee, where the sketch succeeds for all x’s simultaneously, again there are no nontrivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting O(ε−2n) edges in a graph, one achieves the “for all ” guarantee. Our main results advance this front. 1. For the “for all ” guarantee, we prove that Batson et al.’s bound is optimal even when we
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
Bypassing Erdős’ Girth Conjecture: Hybrid Stretch and Sourcewise Spanners
, 2014
"... An (α, β)spanner of an nvertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v,H) ≤ α ·dist(u, v,G)+β for every pair (u, v) ∈ V × V, where dist(u, v,G′) denotes the distance between u and v in G ′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially co ..."
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An (α, β)spanner of an nvertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v,H) ≤ α ·dist(u, v,G)+β for every pair (u, v) ∈ V × V, where dist(u, v,G′) denotes the distance between u and v in G ′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1, 0)spanner of size O(n1+1/k). This sizestretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass ” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of khybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative kstretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted nvertex graph G with m edges, there is a (polynomially constructible) khybrid spanner with O(k2 · n1+1/k) edges. This should be compared against the current best (α, β) spanner construction of [5] that obtains (k, k − 1) stretch with O(k · n1+1/k) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S × V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more).
Single pass graph sparsification in distributed stream processing systems∗
, 2011
"... We give a distributed one pass streaming algorithm for graph sparsification. Besides producing a sparsifier, our algorithmmaintains a hierarchy of UNIONFIND data structures in a distributed manner that efficiently support queries of strong connectivities between pairs of vertices. An important comp ..."
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We give a distributed one pass streaming algorithm for graph sparsification. Besides producing a sparsifier, our algorithmmaintains a hierarchy of UNIONFIND data structures in a distributed manner that efficiently support queries of strong connectivities between pairs of vertices. An important component of the algorithm is an implementation of UNIONFIND queries over an Active Distributed Hash Table that guarantees good load balancing properties. This is achieved via a single step of what is known in the literature as the zigzag heuristic. We provide theoretical guarantees for the load balancing achieved by this heuristic, and show how the structure of our sparsification scheme ensures good load balancing across the hierarchy of UNIONFIND data structures maintained by the algorithm. We also present simulation results on synthetic as well as real world data verifying the load balancing properties and the quality of approximation of strong connectivities achieved by the algorithm. 1