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Spectral sparsification of graphs

2013
*
Communications of the ACM
*

We explain what it means for one

doi:10.1145/2492007.2492029
fatcat:lzpbmvh6rjh35fncz2stbidjdy
*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*. ...##
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Spectral Sparsification of Graphs

2011
*
SIAM journal on computing (Print)
*

We explain what it means for one

doi:10.1137/08074489x
fatcat:dciri57merecpcksh2jkcldhyq
*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*. ...##
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Spectral Sparsification of Graphs
[article]

2010
*
arXiv
*
pre-print

We introduce a new notion

arXiv:0808.4134v3
fatcat:qy5w5mlfrnhvnfxln75xmhxwia
*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*. ...##
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A Unified Approach to Scalable Spectral Sparsification of Directed Graphs
[article]

2020
*
arXiv
*
pre-print

eigenvalues

arXiv:1812.04165v2
fatcat:pms65uku4rg3zatu4ew6snvi5a
*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] . ...##
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A Distributed Algorithm for Spectral Sparsification of Graphs with Applications to Data Clustering
[article]

2020
*
arXiv
*
pre-print

In particular, the main application

arXiv:2003.10612v2
fatcat:6fbkysvy6be2zed4ioas3biq7y
*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. ...##
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Spectral Sparsification of Hypergraphs
[article]

2018
*
arXiv
*
pre-print

The proposed

arXiv:1807.04974v1
fatcat:tssz445dxrbgrnyaje3wp7dm4m
*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/ ) ...##
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Spectral Sparsification of Simplicial Complexes for Clustering and Label Propagation
[article]

2019
*
arXiv
*
pre-print

We show that the

arXiv:1708.08436v3
fatcat:zspxcutlorhjbkkysska7yh43u
*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. ...##
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Spectral Sparsification of Random-Walk Matrix Polynomials
[article]

2015
*
arXiv
*
pre-print

We consider a fundamental

arXiv:1502.03496v1
fatcat:k7gm3aaeezghfetowqod72tsqi
*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 ...##
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An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

2 edges)

doi:10.1137/1.9781611975031.85
dblp:conf/soda/SrivastavaT18
fatcat:ddqcdx76dbcmlnsfithadfrrti
*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. ...##
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Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms
[article]

2019
*
arXiv
*
pre-print

We study the potential utility

arXiv:1910.02861v1
fatcat:5gme3m7auvdinc7rcbk3pwh52a
*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 ...##
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Spectral Sparsification in the Semi-streaming Setting

2012
*
Theory of Computing Systems
*

The strongest commonly-used notion

doi:10.1007/s00224-012-9396-1
fatcat:g2pmivg6gfhzjgiej7yf62k4da
*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*. ...##
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Bounds on the Spectral Sparsification of Symmetric and Off-Diagonal Nonnegative Real Matrices
[article]

2020
*
arXiv
*
pre-print

We say that a square real matrix M is off-diagonal nonnegative if

arXiv:2009.11133v1
fatcat:ld3xnhpq7jcghke2acvn42zw5q
*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. ...##
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Nearly Tight Spectral Sparsification of Directed Hypergraphs by a Simple Iterative Sampling Algorithm
[article]

2022
*
arXiv
*
pre-print

Our

arXiv:2204.02537v1
fatcat:ui5darnmubevxit4m46fpkomoq
*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. ...##
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Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

These

doi:10.1142/9789814324359_0164
fatcat:mqplntp7hbf33no4djeyaduqzy
*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*. ...##
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Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

2015
*
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
*

We present the first almost-linear time

doi:10.1109/focs.2015.24
dblp:conf/focs/LeeS15a
fatcat:6zsquzow35eqrfk6ymwi2fnbuq
*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 ...
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