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

Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu
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.  ... 
doi:10.1145/2488608.2488724 dblp:conf/stoc/KelnerOSZ13 fatcat:thkzyh7ifzac3mzln7fmjtbu5a

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

Ioannis Koutis, Gary L. Miller, David Tolliver
2011 Computer Vision and Image Understanding  
Several 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  ... 
doi:10.1016/j.cviu.2011.05.013 fatcat:6qzlebpyarbvvfwvwl2n3xuec4

Combinatorial Preconditioners and Multilevel Solvers for Problems in Computer Vision and Image Processing [chapter]

Ioannis Koutis, Gary L. Miller, David Tolliver
2009 Lecture Notes in Computer Science  
Several 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  ... 
doi:10.1007/978-3-642-10331-5_99 fatcat:so3ikivg45dp5oontwl5rjk53y

A fast solver for a class of linear systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2012 Communications of the ACM  
Recent research led to 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.  ... 
doi:10.1145/2347736.2347759 fatcat:suhllo3cvnauja5pyzcz3jwhcu

Approaching optimality for solving SDD systems [article]

Ioannis Koutis and Gary L. Miller and Richard Peng
2010 arXiv   pre-print
The 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] .  ... 
arXiv:1003.2958v3 fatcat:kvyhohikuvggvolltvlld4xz2a

Simple parallel and distributed algorithms for spectral graph sparsification [article]

Ioannis Koutis
2014 arXiv   pre-print
Combining this 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 .  ... 
arXiv:1402.3851v2 fatcat:ha6oaa5kmjhuxdlcebqg4xy7ty

Combinatorial Preconditioners for Scalar Elliptic Finite-Element Problems

Haim Avron, Doron Chen, Gil Shklarski, Sivan Toledo
2009 SIAM Journal on Matrix Analysis and Applications  
We present 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".  ... 
doi:10.1137/060675940 fatcat:g7fuh2un7rgvtlhdko4pxinta4

Engineering a Combinatorial Laplacian Solver: Lessons Learned

Daniel Hoske, Dimitar Lukarski, Henning Meyerhenke, Michael Wegner
2016 Algorithms  
Recently, theoretical computer scientists contributed sophisticated 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.  ... 
doi:10.3390/a9040072 fatcat:65gvrgtmivflvbewd2f3pua4gm

Is Nearly-linear the Same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver [chapter]

Daniel Hoske, Dimitar Lukarski, Henning Meyerhenke, Michael Wegner
2015 Lecture Notes in Computer Science  
the problem of 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.  ... 
doi:10.1007/978-3-319-20086-6_16 fatcat:7xoduhmupzavtbi24qopsikhz4

Approaching Optimality for Solving SDD Linear Systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2014 SIAM journal on computing (Print)  
The 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.  ... 
doi:10.1137/110845914 fatcat:onrjx7drjzcjnbfmvtahmjvziy

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 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.  ... 
doi:10.1109/focs.2010.29 dblp:conf/focs/KoutisMP10 fatcat:666ymvove5addhbkbxramj6hfq

Is Nearly-linear the same in Theory and Practice? A Case Study with a Combinatorial Laplacian Solver [article]

Daniel Hoske, Dimitar Lukarski, Henning Meyerhenke, Michael Wegner
2015 arXiv   pre-print
Recently, theoretical computer scientists have contributed sophisticated 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.  ... 
arXiv:1502.07888v1 fatcat:njs75iwlujapffk4srwgorpp74

Faster Spectral Sparsification and Numerical Algorithms for SDD Matrices

Ioannis Koutis, Alex Levin, Richard Peng
2015 ACM Transactions on Algorithms  
The improved sparsification 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.  ... 
doi:10.1145/2743021 fatcat:j5oqwix7yvhqbposok57cxkqby

Perron-Frobenius Theory in Nearly Linear Time: Positive Eigenvectors, M-matrices, Graph Kernels, and Other Applications [article]

AmirMahdi Ahmadinejad, Arun Jambulapati, Amin Saberi, Aaron Sidford
2018 arXiv   pre-print
eigenvectors of non-negative matrices, and 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.  ... 
arXiv:1810.02348v1 fatcat:grtdmfmt5raztehu3khg2cg7yy

Faster spectral sparsification and numerical algorithms for SDD matrices [article]

Ioannis Koutis, Alex Levin, Richard Peng
2013 arXiv   pre-print
The improved sparsification 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.  ... 
arXiv:1209.5821v3 fatcat:hg54ntgzq5fipfqrxhpi7igz4u
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