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Twice-Ramanujan Sparsifiers
[article]

2009
*
arXiv
*
pre-print

Thus, H approximates G spectrally at least as well as a

arXiv:0808.0163v3
fatcat:szswkz4lcrccjmgeoc2lr7vbbi
*Ramanujan*expander with dn/2 edges approximates the complete graph. ... As linear-sized spectral*sparsifiers*of complete graphs are expanders, our*sparsifiers*of arbitrary graphs can be viewed as generalizations of expander graphs. ... In the case where G is the complete graph, excellent spectral*sparsifiers*are supplied by*Ramanujan*Graphs [12, 13] . ...##
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Twice-Ramanujan Sparsifiers

2012
*
SIAM journal on computing (Print)
*

A

doi:10.1137/090772873
fatcat:rqk3cn4k7jaxbmbw3ga6lvb3je
*sparsifier*of a graph is a sparse graph that approximates it. A spectral*sparsifier*is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. ... We prove that every graph has a spectral*sparsifier*with a number of edges linear in its number of vertices. ... The title of this paper reflects the fact that our*sparsifiers*are at most*twice*as dense as the*Ramanujan*graphs achieving the same approximation quality. ...##
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Twice-Ramanujan Sparsifiers

2014
*
SIAM Review
*

A

doi:10.1137/130949117
fatcat:hsbtnt25cbcvtj5x6ldw7wc4zq
*sparsifier*of a graph is a sparse graph that approximates it. A spectral*sparsifier*is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. ... We prove that every graph has a spectral*sparsifier*with a number of edges linear in its number of vertices. ... The title of this paper reflects the fact that our*sparsifiers*are at most*twice*as dense as the*Ramanujan*graphs achieving the same approximation quality. ...##
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Twice-ramanujan sparsifiers

2009
*
Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09
*

A

doi:10.1145/1536414.1536451
dblp:conf/stoc/BatsonSS09
fatcat:i323wwme7nbwzntusa4woxpv4q
*sparsifier*of a graph is a sparse graph that approximates it. A spectral*sparsifier*is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. ... We prove that every graph has a spectral*sparsifier*with a number of edges linear in its number of vertices. ... The title of this paper reflects the fact that our*sparsifiers*are at most*twice*as dense as the*Ramanujan*graphs achieving the same approximation quality. ...##
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An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Thus, [BSS12] called their construction a "

doi:10.1137/1.9781611975031.85
dblp:conf/soda/SrivastavaT18
fatcat:ddqcdx76dbcmlnsfithadfrrti
*twice**Ramanujan**sparsifier*" because, when applied to a clique, it has*twice*the number of edges (dn instead of dn/2) of a*Ramanujan*graph for the same (1 + ... possible using a true d−regular*Ramanujan*graph. ...##
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Spectral Sparsification of Graphs

2011
*
SIAM journal on computing (Print)
*

Formally, the (weighted) hypercube is a good spectral

doi:10.1137/08074489x
fatcat:dciri57merecpcksh2jkcldhyq
*sparsifier*for the complete graph defined on its nodes. ... To write this symbolically, we first observe that a division of the vertices into two parts can be specified by identifying A previous version of the paper, "*Twice*-*Ramanujan**Sparsifiers*," was published ... sampling and decomposition: every graph has a good*sparsifier**Ramanujan*graphs are members of the family of expander graphs. ...##
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Spectral sparsification of graphs

2013
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Communications of the ACM
*

Formally, the (weighted) hypercube is a good spectral

doi:10.1145/2492007.2492029
fatcat:lzpbmvh6rjh35fncz2stbidjdy
*sparsifier*for the complete graph defined on its nodes. ... To write this symbolically, we first observe that a division of the vertices into two parts can be specified by identifying A previous version of the paper, "*Twice*-*Ramanujan**Sparsifiers*," was published ... sampling and decomposition: every graph has a good*sparsifier**Ramanujan*graphs are members of the family of expander graphs. ...##
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GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis
[article]

2020
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arXiv
*
pre-print

However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the

arXiv:1911.04382v3
fatcat:rpwnqsymhfbq7m4ykfnqqowmne
*sparsifier*. ... Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the*sparsifier*, and then recover small portions of spectrally-critical ... with*twice*as many as the edges in*Ramanujan*graphs [2] , [20] . ...##
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Sparsified Cholesky Solvers for SDD linear systems
[article]

2015
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arXiv
*
pre-print

In doing so, we give the first nearly-linear work routine for constructing spectral vertex

arXiv:1506.08204v2
fatcat:dd67ioi6fncazbu2ff3tqtucsq
*sparsifiers*---that is, spectral approximations of Schur complements of Laplacian matrices. ... Our algorithm should construct the corresponding*Ramanujan*graph, as described in Theorem A.1 and Proposition A.2. ... Discard every column of M that has more than*twice*the average number of nonzeros per column. Then remove the corresponding rows. The remaining matrix has dimension at least n/2. ...##
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An Efficient Algorithm for Unweighted Spectral Graph Sparsification
[article]

2014
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arXiv
*
pre-print

In this paper, we present an efficient algorithm for the construction of a new type of spectral

arXiv:1410.4273v2
fatcat:zmoz5ivgsncc7nbsggqy3fa2y4
*sparsifier*, the unweighted spectral*sparsifier*. ... Additionally, our algorithm can efficiently compute unweighted graph*sparsifiers*for weighted graphs, leading to*sparsified*graphs that retain the weights of the original graphs. ... Comparison with*Twice*-*Ramanujan**Sparsifiers*. ...##
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Spectral Sparsification of Graphs
[article]

2010
*
arXiv
*
pre-print

We prove that every graph has a spectral

arXiv:0808.4134v3
fatcat:qy5w5mlfrnhvnfxln75xmhxwia
*sparsifier*of nearly linear size. ... This is equivalent to saying that the Laplacian of the*sparsifier*is a good preconditioner for the Laplacian of the original. ... We claim that a good*sparsifier*for G may be obtained by setting G to be the edge e with weight 1, plus (n/d) times a*Ramanujan*graph on each vertex set. ...##
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Sparsified Cholesky and Multigrid Solvers for Connection Laplacians
[article]

2015
*
arXiv
*
pre-print

We introduce the

arXiv:1512.01892v1
fatcat:uaarmuaqqfbezdo4a75n4jmjqa
*sparsified*Cholesky and*sparsified*multigrid algorithms for solving systems of linear equations. ... These algorithms accelerate Gaussian elimination by*sparsifying*the nonzero matrix entries created by the elimination process. ... However, this problem is easily remedied by forbidding algorithm bSDDSubset from choosing any vertex of more than*twice*average degree. ...##
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Approaching optimality for solving SDD systems
[article]

2010
*
arXiv
*
pre-print

The solver is based on repeated applications of the incremental

arXiv:1003.2958v3
fatcat:kvyhohikuvggvolltvlld4xz2a
*sparsifier*that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration. ... We present an algorithm that on input of an n-vertex m-edge weighted graph G and a value k, produces an incremental*sparsifier*Ĝ with n-1 + m/k edges, such that the condition number of G with Ĝ is bounded ... Leaving the envelope of nearly-linear time algorithms Batson, Spielman and Srivastava [BSS09] presented a polynomial time algorithm for the construction of a "*twice*-*Ramanujan*" spectral*sparsifier*with ...##
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Approaching Optimality for Solving SDD Linear Systems

2010
*
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
*

The solver is based on repeated applications of the incremental

doi:10.1109/focs.2010.29
dblp:conf/focs/KoutisMP10
fatcat:666ymvove5addhbkbxramj6hfq
*sparsifier*that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration. ... Leaving the envelope of nearlylinear time algorithms Batson, Spielman and Srivastava [31] presented a polynomial time algorithm for the construction of a "*twice*-*Ramanujan*" spectral*sparsifier*with a ... A good spectral*sparsifier*is a also a good cutpreserving*sparsifier*, but the opposite is not necessarily true. The ST-solver [1] consists of a number of major algorithmic components. ...##
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Approaching Optimality for Solving SDD Linear Systems

2014
*
SIAM journal on computing (Print)
*

The solver is based on repeated applications of the incremental

doi:10.1137/110845914
fatcat:onrjx7drjzcjnbfmvtahmjvziy
*sparsifier*that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration. ... Leaving the envelope of nearlylinear time algorithms Batson, Spielman and Srivastava [31] presented a polynomial time algorithm for the construction of a "*twice*-*Ramanujan*" spectral*sparsifier*with a ... A good spectral*sparsifier*is a also a good cutpreserving*sparsifier*, but the opposite is not necessarily true. The ST-solver [1] consists of a number of major algorithmic components. ...
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