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A simple, combinatorial algorithm for solving SDD systems in nearly-linear time

2013
*
Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13
*

*In*this paper, we present

*a*

*simple*

*combinatorial*

*algorithm*that

*solves*symmetric diagonally dominant (

*SDD*)

*linear*

*systems*

*in*

*nearly*-

*linear*

*time*. ... After constructing

*a*"nice" spanning tree of

*a*graph associated with the

*linear*

*system*, the entire

*algorithm*consists of the repeated application of

*a*

*simple*(non-recursive) update rule, which it implements ... Acknowledgments We thank Daniel Spielman

*for*many helpful conversations. ...

##
###
Combinatorial preconditioners and multilevel solvers for problems in computer vision and image processing

2011
*
Computer Vision and Image Understanding
*

Several

doi:10.1016/j.cviu.2011.05.013
fatcat:6qzlebpyarbvvfwvwl2n3xuec4
*algorithms**for*problems including image segmentation, gradient inpainting and total variation are based on*solving*symmetric diagonally dominant (*SDD*)*linear**systems*. ... Finally, we outline two new reductions of non-*linear*filtering problems to*SDD**systems*and review the integration of*SDD**systems*into selected*algorithms*. * This work was partially supported by NSF CCF ... Acknowledgment We would like to thank Eduardo Rosa-Molinar and his Biological Imaging Group at the University of Puerto Rico-Rio Piedras that provided us the serial block-face imaging dataset used*in*this ...##
###
Combinatorial Preconditioners and Multilevel Solvers for Problems in Computer Vision and Image Processing
[chapter]

2009
*
Lecture Notes in Computer Science
*

Several

doi:10.1007/978-3-642-10331-5_99
fatcat:so3ikivg45dp5oontwl5rjk53y
*algorithms**for*problems including image segmentation, gradient inpainting and total variation are based on*solving*symmetric diagonally dominant (*SDD*)*linear**systems*. ... Finally, we outline two new reductions of non-*linear*filtering problems to*SDD**systems*and review the integration of*SDD**systems*into selected*algorithms*. * This work was partially supported by NSF CCF ... Acknowledgment We would like to thank Eduardo Rosa-Molinar and his Biological Imaging Group at the University of Puerto Rico-Rio Piedras that provided us the serial block-face imaging dataset used*in*this ...##
###
A fast solver for a class of linear systems

2012
*
Communications of the ACM
*

Recent research led to

doi:10.1145/2347736.2347759
fatcat:suhllo3cvnauja5pyzcz3jwhcu
*a*fast*algorithm**for**solving*symmetric diagonally dominant (*SDD*)*linear**systems*. ... The solution of*linear**systems*is*a*problem of fundamental theoretical importance but also one with*a*myriad of applications*in*numerical mathematics, engineering and science. ...*In*fact, LSSTs are indispensable components of all*nearly*-*linear**time**SDD**system*solvers. ...##
###
Approaching optimality for solving SDD systems
[article]

2010
*
arXiv
*
pre-print

The

arXiv:1003.2958v3
fatcat:kvyhohikuvggvolltvlld4xz2a
*algorithm*runs*in**time*Õ((m n + n^2n)(1/p)). ... As*a*result, we obtain an*algorithm*that on input of an n× n symmetric diagonally dominant matrix*A*with m non-zero entries and*a*vector b, computes*a*vector x satisfying ||x-*A*^+b||_A<ϵ ||*A*^+b||_A ,*in*...*In**a*seminal work, Spielman and Teng showed that*SDD**systems*can be*solved**in**nearly*-*linear**time*[ST04, EEST05, ST06] . ...##
###
Simple parallel and distributed algorithms for spectral graph sparsification
[article]

2014
*
arXiv
*
pre-print

Combining this

arXiv:1402.3851v2
fatcat:ha6oaa5kmjhuxdlcebqg4xy7ty
*algorithm*with the parallel framework of Peng and Spielman*for**solving*symmetric diagonally dominant*linear**systems*, we get*a*parallel solver which is much closer to being practical and ... We describe*a**simple**algorithm**for*spectral graph sparsification, based on iterative computations of weighted spanners and uniform sampling. ... They were introduced by Spielman and Teng [24] as*a*basic component of the first*nearly*-*linear**time*solvers*for**linear**systems*on symmetric diagonally dominant (*SDD*) matrices 1 . ...##
###
Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

2009
*
SIAM Journal on Matrix Analysis and Applications
*

We present

doi:10.1137/060675940
fatcat:g7fuh2un7rgvtlhdko4pxinta4
*a*new preconditioner*for**linear**systems*arising from finite-element discretizations of scalar elliptic partial differential equations (PDE's). ... The splitting idea is*simple*and natural*in*the context of*combinatorial*preconditioners, but hard to exploit*in*other preconditioning paradigms. ... Sivan Toledo's work on this problem started*a*decade ago,*in*1996, when he was working with John Gilbert under DARPA contract DABT63-95-C-0087, "Portable parallel preconditioning". ...##
###
Engineering a Combinatorial Laplacian Solver: Lessons Learned

2016
*
Algorithms
*

Recently, theoretical computer scientists contributed sophisticated

doi:10.3390/a9040072
fatcat:65gvrgtmivflvbewd2f3pua4gm
*algorithms**for**solving**linear**systems*with symmetric diagonally-dominant (*SDD*) matrices*in*provably*nearly*-*linear**time*. ...*Linear**system**solving*is*a*main workhorse*in*applied mathematics. ... Thus, the problem INV-*SDD*of*solving**linear**systems*Ax = b*for*x on*SDD*matrices*A*is of significant importance. ...##
###
Is Nearly-linear the Same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver
[chapter]

2015
*
Lecture Notes in Computer Science
*

the problem of

doi:10.1007/978-3-319-20086-6_16
fatcat:7xoduhmupzavtbi24qopsikhz4
*solving*an*SDD**linear**system*as finding an electrical flow*in**a*graph. ... Elkin et al. [11] provide an*algorithm**for*computing spanning trees with polynomial stretch*in**nearly*-*linear**time*. ... Thus, the problem INV-*SDD*of*solving**linear**systems*Ax = b*for*x on*SDD*matrices*A*is of significant importance. ...##
###
Approaching Optimality for Solving SDD Linear Systems

2014
*
SIAM journal on computing (Print)
*

The

doi:10.1137/110845914
fatcat:onrjx7drjzcjnbfmvtahmjvziy
*algorithm*runs*in**timẽ*O((m log n + n log 2 n) log(1/p)). ...*in*expected*timẽ*O(m log 2 n log(1/ )). ... ACKNOWLEDGEMENTS The authors wish to thank Dan Spielman and Charalampos Tsourakakis*for*their very helpful comments and discussions. ...##
###
Approaching Optimality for Solving SDD Linear Systems

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

The

doi:10.1109/focs.2010.29
dblp:conf/focs/KoutisMP10
fatcat:666ymvove5addhbkbxramj6hfq
*algorithm*runs*in**timẽ*O((m log n + n log 2 n) log(1/p)). ...*in*expected*timẽ*O(m log 2 n log(1/ )). ... ACKNOWLEDGEMENTS The authors wish to thank Dan Spielman and Charalampos Tsourakakis*for*their very helpful comments and discussions. ...##
###
Is Nearly-linear the same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver
[article]

2015
*
arXiv
*
pre-print

Recently, theoretical computer scientists have contributed sophisticated

arXiv:1502.07888v1
fatcat:njs75iwlujapffk4srwgorpp74
*algorithms**for**solving**linear**systems*with symmetric diagonally dominant matrices (*a*class to which Laplacian matrices belong)*in*... provably*nearly*-*linear**time*. ... Thus, the problem INV-*SDD*of*solving**linear**systems*Ax = b*for*x on*SDD*matrices*A*is of significant importance. ...##
###
Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

2015
*
ACM Transactions on Algorithms
*

The improved sparsification

doi:10.1145/2743021
fatcat:j5oqwix7yvhqbposok57cxkqby
*algorithms*are employed to accelerate*linear**system*solvers and*algorithms**for*computing fundamental eigenvectors of dense*SDD*matrices. ... The first*algorithm*is*a**simple*modification of the fastest known*algorithm*and runs inÕ(m log 2 n)*time*, an O(log n) factor faster than before. ... Richard Peng was at Microsoft Research New England*for*part of this work and is supported by*a*Microsoft Research Fellowship. ...##
###
Perron-Frobenius Theory in Nearly Linear Time: Positive Eigenvectors, M-matrices, Graph Kernels, and Other Applications
[article]

2018
*
arXiv
*
pre-print

eigenvectors of non-negative matrices, and

arXiv:1810.02348v1
fatcat:grtdmfmt5raztehu3khg2cg7yy
*solving**linear**systems**in*asymmetric M-matrices,*a*generalization of Laplacian*systems*. ...*In*this paper we provide*nearly**linear**time**algorithms**for*several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative ... Cohen*for*helpful conversations which identified key technical challenges*in**solving*M-matrices that motivated this work. ...##
###
Faster spectral sparsification and numerical algorithms for SDD matrices
[article]

2013
*
arXiv
*
pre-print

The improved sparsification

arXiv:1209.5821v3
fatcat:hg54ntgzq5fipfqrxhpi7igz4u
*algorithms*are employed to accelerate*linear**system*solvers and*algorithms**for*computing fundamental eigenvectors of slightly dense*SDD*matrices. ... We show that the fastest known*algorithm**for*computing*a*sparsifier with O(n n/ϵ^2) edges can actually run*in*Õ(m^2 n)*time*, an O( n) factor faster than before. ... Richard Peng was at Microsoft Research New England*for*part of this work and is supported by*a*Microsoft Research Fellowship. ...
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