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Spectral sparsification of graphs

Joshua Batson, Daniel A. Spielman, Nikhil Srivastava, Shang-Hua Teng
2013 Communications of the ACM  
We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification.  ...  In addition to being an interesting concept, spectral sparsification has been an important tool in the design of nearly linear-time algorithms for solving systems of linear equations in symmetric, diagonally  ...  spectral similarity Motivated by problems in numerical linear algebra and spectral graph theory, Spielman and Teng 34 introduced a notion of spectral similarity for two graphs.  ... 
doi:10.1145/2492007.2492029 fatcat:lzpbmvh6rjh35fncz2stbidjdy

Spectral Sparsification of Graphs

Daniel A. Spielman, Shang-Hua Teng
2011 SIAM journal on computing (Print)  
We explain what it means for one graph to be a spectral approximation of another and review the development of algorithms for spectral sparsification.  ...  In addition to being an interesting concept, spectral sparsification has been an important tool in the design of nearly linear-time algorithms for solving systems of linear equations in symmetric, diagonally  ...  spectral similarity Motivated by problems in numerical linear algebra and spectral graph theory, Spielman and Teng 34 introduced a notion of spectral similarity for two graphs.  ... 
doi:10.1137/08074489x fatcat:dciri57merecpcksh2jkcldhyq

Spectral Sparsification of Graphs [article]

Daniel A. Spielman, Shang-Hua Teng
2010 arXiv   pre-print
We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that  ...  Moreover, we present an algorithm that produces spectral sparsifiers in time m, where m is the number of edges in the original graph.  ...  Motivated by problems in numerical linear algebra and spectral graph theory, we introduce a new notion of sparsification that we call spectral sparsification.  ... 
arXiv:0808.4134v3 fatcat:qy5w5mlfrnhvnfxln75xmhxwia

A Unified Approach to Scalable Spectral Sparsification of Directed Graphs [article]

Ying Zhang, Zhiqiang Zhao, Zhuo Feng
2020 arXiv   pre-print
eigenvalues and eigenvectors of the graph Laplacian, leading to the development of a variety of nearly-linear time numerical and graph algorithms.  ...  In this work, we prove the existence of linear-sized spectral sparsifiers for general directed graphs and introduce a practically-efficient and unified spectral graph sparsification approach that allows  ...  OF DIRECTED GRAPH SPARSIFICATION Spectral graph sparsification algorithms can be potentially applied to accelerate many graph and numerical algorithms [8] .  ... 
arXiv:1812.04165v2 fatcat:pms65uku4rg3zatu4ew6snvi5a

A Distributed Algorithm for Spectral Sparsification of Graphs with Applications to Data Clustering [article]

Fabricio Mendoza-Granada, Marcos Villagra
2020 arXiv   pre-print
In particular, the main application of spectral sparsification is to construct sparse graphs whose spectra are close to a given dense graph.  ...  We study spectral sparsification under the assumption that the edges of a graph are allocated among sites which can communicate among each other.  ...  We give our thanks to the reviewers of CTW 2020 for their comments that helped improve this paper.  ... 
arXiv:2003.10612v2 fatcat:6fbkysvy6be2zed4ioas3biq7y

Spectral Sparsification of Hypergraphs [article]

Tasuku Soma, Yuichi Yoshida
2018 arXiv   pre-print
The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of L_G,  ...  In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph G on n vertices, constructs an ϵ-spectral sparsifier of G with O(n^3 n/ϵ^2) hyperedges/hyperarcs.  ...  Cut sparsification and spectral sparsification of graphs have been studied intensively [1, 2, 17, 21, 22, 29] , and it is known that any graph with n vertices can be spectrally sparsified with O(n/ )  ... 
arXiv:1807.04974v1 fatcat:tssz445dxrbgrnyaje3wp7dm4m

Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation [article]

Braxton Osting, Sourabh Palande, Bei Wang
2019 arXiv   pre-print
We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to simplicial complexes via the up Laplacian.  ...  Finally, we introduce higher-order generalizations of spectral clustering and label propagation for simplicial complexes and demonstrate via experiments the utility of the proposed spectral sparsification  ...  Acknowledgements This work was partially supported by NSF DMS-1461138, NSF IIS-1513616, and the University of Utah Seed Grant 10041533. We would like to thank Todd H.  ... 
arXiv:1708.08436v3 fatcat:zspxcutlorhjbkkysska7yh43u

Spectral Sparsification of Random-Walk Matrix Polynomials [article]

Dehua Cheng, Yu Cheng, Yan Liu, Richard Peng, Shang-Hua Teng
2015 arXiv   pre-print
We consider a fundamental algorithmic question in spectral graph theory: Compute a spectral sparsifier of random-walk matrix-polynomial L_α(G)=D-∑_r=1^dα_rD(D^-1A)^r where A is the adjacency matrix of  ...  In this paper, we develop the first nearly linear time algorithm for this sparsification problem: For any G with n vertices and m edges, d coefficients α, and ϵ > 0, our algorithm runs in time O(d^2m^2n  ...  The computation of a (nearly) linear size spectral sparsifier of a dense Laplacian matrix is a fundamental algorithmic problem in spectral graph theory that has been used in solving linear systems [ST14  ... 
arXiv:1502.03496v1 fatcat:k7gm3aaeezghfetowqod72tsqi

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  
2 edges) and girth g > 2d 1/8 + 1, and if λ 1 ≤ λ 2 ≤ · · · λ n are the eigenvalues of the (non-normalized) Laplacian of G, then  ...  We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if G is an n-node weighted undirected graph of average combinatorial degree d (that is, G has dn/  ...  Acknowledgments We would like to thank Alexandra Kolla for helpful conversations, as well as the Simons Institute for the Theory of Computing, where this work was carried out.  ... 
doi:10.1137/1.9781611975031.85 dblp:conf/soda/SrivastavaT18 fatcat:ddqcdx76dbcmlnsfithadfrrti

Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms [article]

Steven Herbert, Sathyawageeswar Subramanian
2019 arXiv   pre-print
We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation.  ...  Our results indicate that spectral sparsification of a graph with n nodes through a sampling method, e.g. as in using effective resistances, gives, with high probability, a locally computable matrix H̃  ...  Introduction In classical graph theory and signal processing, sparsifying dense matrices and performing algorithms thereon to reduce the computational load is a key idea, which has received significant  ... 
arXiv:1910.02861v1 fatcat:5gme3m7auvdinc7rcbk3pwh52a

Spectral Sparsification in the Semi-streaming Setting

Jonathan A. Kelner, Alex Levin
2012 Theory of Computing Systems  
The strongest commonly-used notion of sparsification is spectral sparsification; H is a spectral sparsifier of G if the quadratic forms induced by the Laplacians of G and H approximate one another well  ...  In this paper, we introduce an algorithm for constructing a spectral sparsifier of G with O(n log n/ 2 ) edges (where is a parameter measuring the quality of the sparsifier), taking O(m) time and requiring  ...  Recently, Spielman and Teng [10] defined the notion of spectral sparsification.  ... 
doi:10.1007/s00224-012-9396-1 fatcat:g2pmivg6gfhzjgiej7yf62k4da

Bounds on the Spectral Sparsification of Symmetric and Off-Diagonal Nonnegative Real Matrices [article]

Sergio Mercado, Marcos Villagra
2020 arXiv   pre-print
We say that a square real matrix M is off-diagonal nonnegative if and only if all entries outside its diagonal are nonnegative real numbers.  ...  In this note we show that for any off-diagonal nonnegative symmetric matrix M, there exists a nonnegative symmetric matrix M which is sparse and close in spectrum to M.  ...  Spectral Sparsification of Graphs Let G = (V, E, w) be a simple undirected graph with vertices v 1 , v 2 , ..., v n ∈ V . An edge in E between vertices v i and v j is denoted ij.  ... 
arXiv:2009.11133v1 fatcat:ld3xnhpq7jcghke2acvn42zw5q

Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm [article]

Kazusato Oko, Shinsaku Sakaue, Shin-ichi Tanigawa
2022 arXiv   pre-print
Our algorithm can be regarded as an extension of the spanner-based graph sparsification by Koutis and Xu (2016).  ...  Spectral hypergraph sparsification, which is an attempt to extend well-known spectral graph sparsification to hypergraphs, has been extensively studied over the past few years.  ...  Acknowledgements This work was supported by JST ERATO Grant Number JPMJER1903 and JSPS KAKENHI Grant Number 20H05961.  ... 
arXiv:2204.02537v1 fatcat:ui5darnmubevxit4m46fpkomoq

Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices

Daniel A. Spielman
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
These algorithms motivate and rely upon fascinating primitives in graph theory, including low-stretch spanning trees, graph sparsifiers, ultra-sparsifiers, and local graph clustering.  ...  The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems.  ...  In Spectral Graph Theory, one studies graphs by examining the eigenvalues and eigenvectors of matrices related to these graphs.  ... 
doi:10.1142/9789814324359_0164 fatcat:mqplntp7hbf33no4djeyaduqzy

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Yin Tat Lee, He Sun
2015 2015 IEEE 56th Annual Symposium on Foundations of Computer Science  
We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs.  ...  A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance [SS11], and adaptive construction  ...  We thank Zeyuan Allen-Zhu, Zhenyu Liao, and Lorenzo Orecchia for sending us their manuscript of [AZLO15] and the inspiring talk Zeyuan Allen-Zhu gave at the Simons Institute for the Theory of Computing  ... 
doi:10.1109/focs.2015.24 dblp:conf/focs/LeeS15a fatcat:6zsquzow35eqrfk6ymwi2fnbuq
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