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Twice-Ramanujan Sparsifiers [article]

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava
2009 arXiv   pre-print
Thus, H approximates G spectrally at least as well as a 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] .  ... 
arXiv:0808.0163v3 fatcat:szswkz4lcrccjmgeoc2lr7vbbi

Twice-Ramanujan Sparsifiers

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava
2012 SIAM journal on computing (Print)  
A 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.  ... 
doi:10.1137/090772873 fatcat:rqk3cn4k7jaxbmbw3ga6lvb3je

Twice-Ramanujan Sparsifiers

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava
2014 SIAM Review  
A 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.  ... 
doi:10.1137/130949117 fatcat:hsbtnt25cbcvtj5x6ldw7wc4zq

Twice-ramanujan sparsifiers

Joshua D. Batson, Daniel A. Spielman, Nikhil Srivastava
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
A 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.  ... 
doi:10.1145/1536414.1536451 dblp:conf/stoc/BatsonSS09 fatcat:i323wwme7nbwzntusa4woxpv4q

An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification [chapter]

Nikhil Srivastava, Luca Trevisan
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
Thus, [BSS12] called their construction a "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.  ... 
doi:10.1137/1.9781611975031.85 dblp:conf/soda/SrivastavaT18 fatcat:ddqcdx76dbcmlnsfithadfrrti

Spectral Sparsification of Graphs

Daniel A. Spielman, Shang-Hua Teng
2011 SIAM journal on computing (Print)  
Formally, the (weighted) hypercube is a good spectral 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.  ... 
doi:10.1137/08074489x fatcat:dciri57merecpcksh2jkcldhyq

Spectral sparsification of graphs

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava, Shang-Hua Teng
2013 Communications of the ACM  
Formally, the (weighted) hypercube is a good spectral 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.  ... 
doi:10.1145/2492007.2492029 fatcat:lzpbmvh6rjh35fncz2stbidjdy

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis [article]

Zhuo Feng
2020 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 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] .  ... 
arXiv:1911.04382v3 fatcat:rpwnqsymhfbq7m4ykfnqqowmne

Sparsified Cholesky Solvers for SDD linear systems [article]

Yin Tat Lee, Richard Peng, Daniel A. Spielman
2015 arXiv   pre-print
In doing so, we give the first nearly-linear work routine for constructing spectral vertex 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.  ... 
arXiv:1506.08204v2 fatcat:dd67ioi6fncazbu2ff3tqtucsq

An Efficient Algorithm for Unweighted Spectral Graph Sparsification [article]

David G. Anderson, Ming Gu, Christopher Melgaard
2014 arXiv   pre-print
In this paper, we present an efficient algorithm for the construction of a new type of spectral 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.  ... 
arXiv:1410.4273v2 fatcat:zmoz5ivgsncc7nbsggqy3fa2y4

Spectral Sparsification of Graphs [article]

Daniel A. Spielman, Shang-Hua Teng
2010 arXiv   pre-print
We prove that every graph has a spectral 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.  ... 
arXiv:0808.4134v3 fatcat:qy5w5mlfrnhvnfxln75xmhxwia

Sparsified Cholesky and Multigrid Solvers for Connection Laplacians [article]

Rasmus Kyng, Yin Tat Lee, Richard Peng, Sushant Sachdeva, Daniel A. Spielman
2015 arXiv   pre-print
We introduce the 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.  ... 
arXiv:1512.01892v1 fatcat:uaarmuaqqfbezdo4a75n4jmjqa

Approaching optimality for solving SDD systems [article]

Ioannis Koutis and Gary L. Miller and Richard Peng
2010 arXiv   pre-print
The solver is based on repeated applications of the incremental 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  ... 
arXiv:1003.2958v3 fatcat:kvyhohikuvggvolltvlld4xz2a

Approaching Optimality for Solving SDD Linear Systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science  
The solver is based on repeated applications of the incremental 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.  ... 
doi:10.1109/focs.2010.29 dblp:conf/focs/KoutisMP10 fatcat:666ymvove5addhbkbxramj6hfq

Approaching Optimality for Solving SDD Linear Systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2014 SIAM journal on computing (Print)  
The solver is based on repeated applications of the incremental 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.  ... 
doi:10.1137/110845914 fatcat:onrjx7drjzcjnbfmvtahmjvziy
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