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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
The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning  ...  They confirm a nearly-linear running time, but for reasonable inputs the constant factors make the solver much slower than methods with higher asymptotic complexity.  ...  Hence, our initial question in the paper title can be answered with "yes" and "no" at the same time: The running time is nearly linear, but the constant factors prevent usefulness in practice.  ... 
arXiv:1502.07888v1 fatcat:njs75iwlujapffk4srwgorpp74

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  
Another interesting nearly-linear time SDD solver is the recursive sparsification approach by Peng and Spielman [21] .  ...  The algorithm exploits that a Laplacian matrix corresponds to a graph; solving Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles induced by a spanning  ...  Hence, our initial question in the paper title can be answered with "yes" and "no" at the same time: The running time is nearly linear, but the constant factors prevent usefulness in practice.  ... 
doi:10.1007/978-3-319-20086-6_16 fatcat:7xoduhmupzavtbi24qopsikhz4

Efficient and Practical Tree Preconditioning for Solving Laplacian Systems [chapter]

Luca Castelli Aleardi, Alexandre Nolin, Maks Ovsjanikov
2015 Lecture Notes in Computer Science  
In this paper, we focus on a particular type of linear systems, associated with Laplacian matrices of undirected graphs, and study a class of iterative methods for which it is possible to speed up the  ...  of the iterative linear solvers in practice.  ...  help eventually bridge theory and practice in this field.  ... 
doi:10.1007/978-3-319-20086-6_17 fatcat:qowzuiltjbdnpisdgzguqrzj4y

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  
A central claim of this paper is that SDD-based approaches can now be considered practical and reliable.  ...  However, existing solvers are not always efficient, and in many cases they operate only on restricted topologies.  ...  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  
A central claim of this paper is that SDD-based approaches can now be considered practical and reliable.  ...  However, existing solvers are not always efficient, and in many cases they operate only on restricted topologies.  ...  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

Engineering a Combinatorial Laplacian Solver: Lessons Learned

Daniel Hoske, Dimitar Lukarski, Henning Meyerhenke, Michael Wegner
2016 Algorithms  
The algorithm exploits that a Laplacian matrix (which is SDD) corresponds to a graph; solving symmetric Laplacian linear systems amounts to finding an electrical flow in this graph with the help of cycles  ...  Linear system solving is a main workhorse in applied mathematics.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/a9040072 fatcat:65gvrgtmivflvbewd2f3pua4gm

A fast solver for a class of linear systems

Ioannis Koutis, Gary L. Miller, Richard Peng
2012 Communications of the ACM  
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.  ...  We give an overview of this solver and survey the underlying notions and tools from algebra, probability and graph algorithms. We also discuss some of the many and diverse applications of SDD solvers.  ...  This work showed that such systems can be preconditioned with graph Laplacians and so they can be solved in nearly-linear time.  ... 
doi:10.1145/2347736.2347759 fatcat:suhllo3cvnauja5pyzcz3jwhcu

Quadratic Decomposable Submodular Function Minimization: Theory and Practice (Computation and Analysis of PageRank over Hypergraphs) [article]

Pan Li, Niao He, Olgica Milenkovic
2020 arXiv   pre-print
The proposed hypergraph-based PageRank algorithm can be used for local hypergraph partitioning, and comes with provable performance guarantees.  ...  We approach the problem via a new dual strategy and formulate an objective that can be optimized through a number of double-loop algorithms.  ...  The authors also gratefully acknowledge funding from the NSF CIF 1527636 and the NSF Science and Technology Center (STC) at Purdue University, Emerging Frontiers of Science of Information, 0939370.  ... 
arXiv:1902.10132v4 fatcat:bll5yhjtwnbmzkvjuy3tegjidu

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.  ...  We hope that the simplicity of the algorithm and the insights yielded by its analysis will be useful in both theory and practice.  ...  This work was partially supported by NSF awards 0843915 and 1111109, a Sloan Research Fellowship, and a NSF Graduate Research Fellowship (grant no. 1122374).  ... 
doi:10.1145/2488608.2488724 dblp:conf/stoc/KelnerOSZ13 fatcat:thkzyh7ifzac3mzln7fmjtbu5a

Maximizing the Smallest Eigenvalue of Grounded Laplacian Matrix [article]

Run Wang, Xiaotian Zhou, Wei Li, Zhongzhi Zhang
2022 arXiv   pre-print
For a connected graph 𝒢=(V,E) with n nodes, m edges, and Laplacian matrix 𝐿, a grounded Laplacian matrix 𝐿(S) of 𝒢 is a (n-k) × (n-k) principal submatrix of 𝐿, obtained from 𝐿 by deleting k rows  ...  Our naïve heuristic algorithm takes Õ(knm) time, while the fast greedy heuristic has a nearly linear time complexity of Õ(km).  ...  Our algorithm is a greedy heuristic one, which is established based on the tools of the derivative matrix, matrix perturbations, and Laplacian solvers, and has a nearly linear time complexity of Õ(km),  ... 
arXiv:2110.12576v2 fatcat:bxnc6p5qn5f67c3yb2lmzjggfa

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  ...  The major new notion introduced by Spielman and Teng [ST04] in their nearly-linear time algorithm was that of a spectral sparsifier, i.e. a graph with a nearly-linear number of edges that α-approximates  ... 
arXiv:1003.2958v3 fatcat:kvyhohikuvggvolltvlld4xz2a

Approximation of the Diagonal of a Laplacian's Pseudoinverse for Complex Network Analysis

Eugenio Angriman, Maria Predari, Alexander van der Grinten, Henning Meyerhenke, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms  
For small-world networks, our algorithm obtains a ± ε-approximation with high probability, in a time that is nearly-linear in |E| and quadratic in 1 / ε.  ...  In this paper, we present a novel approximation algorithm that requires the solution of only one Laplacian linear system.  ...  Acknowledgements We thank our colleague Fabian Brandt-Tumescheit for his technical support for the experiments.  ... 
doi:10.4230/lipics.esa.2020.6 dblp:conf/esa/AngrimanPGM20 fatcat:ucyir4c7bbaala4zhfsxp6ocmq

Approximation of the Diagonal of a Laplacian's Pseudoinverse for Complex Network Analysis [article]

Eugenio Angriman, Maria Predari, Alexander van der Grinten, Henning Meyerhenke
2021 arXiv   pre-print
For small-world networks, our algorithm obtains a ±ϵ-approximation with high probability, in a time that is nearly-linear in |E| and quadratic in 1 / ϵ.  ...  In this paper, we present a novel approximation algorithm that requires the solution of only one Laplacian linear system.  ...  Acknowledgements This work is partially supported by German Research Foundation (DFG) grant ME 3619/3-2 within Priority Programme 1736 Algorithms for Big Data and by DFG grant ME 3619/4-1 (Accelerating  ... 
arXiv:2006.13679v2 fatcat:e6c57yt5nnepbitwrqljuozeqe

Hardness Results for Laplacians of Simplicial Complexes via Sparse-Linear Equation Complete Gadgets [article]

Ming Ding, Rasmus Kyng, Maximilian Probst Gutenberg, Peng Zhang
2022 arXiv   pre-print
It is known that nearly-linear time solvers exist for graph Laplacians. However, nearly-linear time solvers for combinatorial Laplacians are only known for restricted classes of complexes.  ...  We study linear equations in combinatorial Laplacians of k-dimensional simplicial complexes (k-complexes), a natural generalization of graph Laplacians.  ...  The dimension of H k is the kth Betti number of K, which plays an important role in understanding the homology spaces. Hodge Theory and Combinatorial Laplacians.  ... 
arXiv:2202.05011v1 fatcat:uywzjonex5bdxh4noylvuplyl4

An Empirical Study of Cycle Toggling Based Laplacian Solvers [article]

Kevin Deweese, John R. Gilbert, Gary Miller, Richard Peng, Hao Ran Xu, Shen Chen Xu
2016 arXiv   pre-print
We study the performance of linear solvers for graph Laplacians based on the combinatorial cycle adjustment methodology proposed by [Kelner-Orecchia-Sidford-Zhu STOC-13].  ...  The primary difficulty faced by this approach, updating and querying long cycles, motivated us to study an important special case: instances where all cycles are formed by fundamental cycles on a length  ...  Dual cycle-toggling Laplacian solvers have until now been considered mainly in the realm of theory.  ... 
arXiv:1609.02957v1 fatcat:mgyyx5hfqrgebgp7i7wow4bxxa
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