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Constructing LinearSized Spectral Sparsification in AlmostLinear Time
"... We present the first almostlinear time algorithm for constructing linearsized spectral sparsification for graphs. This improves all previous constructions of linearsized spectral sparsification, which requires Ω(n2) time [1], [2], [3]. A key ingredient in our algorithm is a novel combination of t ..."
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We present the first almostlinear time algorithm for constructing linearsized spectral sparsification for graphs. This improves all previous constructions of linearsized spectral sparsification, which requires Ω(n2) time [1], [2], [3]. A key ingredient in our algorithm is a novel combination
Spectral Sparsification and Restricted Invertibility
, 2010
"... In this thesis we prove the following two basic statements in linear algebra. Let B be an arbitrary n × m matrix where m ≥ n and suppose 0 < ε < 1 is given. 1. Spectral Sparsification. There is a nonnegative diagonal matrix Sm×m with at most ⌈n/ε2 ⌉ nonzero entries for which (1 − ε) 2BBT ≼ BSB ..."
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Cited by 11 (1 self)
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In this thesis we prove the following two basic statements in linear algebra. Let B be an arbitrary n × m matrix where m ≥ n and suppose 0 < ε < 1 is given. 1. Spectral Sparsification. There is a nonnegative diagonal matrix Sm×m with at most ⌈n/ε2 ⌉ nonzero entries for which (1 − ε) 2BBT
Spectral Sparsification and Spectrally Thin Trees
, 2012
"... We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs w ..."
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We provide results of intensive experimental data in order to investigate the existence of spectrally thin trees and unweighted spectral sparsifiers for graphs with small expansion. In addition, we also survey and prove some partial results on the existence of spectrally thin trees on dense graphs
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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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
Spectral Sparsification in Dynamic Graph Streams
"... Abstract. We present a new bound relating edge connectivity in a simple, unweighted graph with effective resistance in the corresponding electrical network. The bound is tight. While we believe the bound is of independent interest, our work is motivated by the problem of constructing combinatorial a ..."
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Cited by 4 (2 self)
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in the combinatorial sparsifier construction, we also preserve the spectral properties of the graph. Combining this with the algorithms of Ahn et al. (SODA 2012, PODS 2012) gives rise to the first data stream algorithm for the construction of spectral sparsifiers in the dynamic setting where edges can be added
On the Limits of Sparsification ⋆
"... Abstract. Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. ..."
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Abstract. Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for kCNFs: every kCNF is a subexponential size disjunction of kCNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem
LinearSize Approximations to the VietorisRips
"... The VietorisRips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the u ..."
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itspersistencediagramisagoodapproximationto that of the VietorisRips filtration. This new filtration can be constructed in O(nlogn) time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes
Improved Sparsification
, 1993
"... In previous work we introduced sparsification, a technique that transforms fully dynamic algorithms for sparse graphs into ones that work on any graph, with a logarithmic increase in complexity. In this work we describe an improvement on this technique that avoids the logarithmic overhead. Using ..."
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Cited by 29 (5 self)
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. Using our improved sparsification technique, we keep track of the following properties: minimum spanning forest, best swap, connectivity, 2edgeconnectivity, and bipartiteness, in time O(n 1/2 ) per edge insertion or deletion; 2vertexconnectivity and 3vertexconnectivity, in time O(n) per
Results 1  10
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637,482