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264
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 145 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
Nearlylinear time algorithms for graph partitioning, graph sparsification, and solving linear systems (Extended Abstract)
 STOC'04
, 2004
"... We present algorithms for solving symmetric, diagonallydominant linear systems to accuracy ɛ in time linear in their number of nonzeros and log(κf (A)/ɛ), where κf (A) isthe condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with ..."
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Cited by 134 (7 self)
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We present algorithms for solving symmetric, diagonallydominant linear systems to accuracy ɛ in time linear in their number of nonzeros and log(κf (A)/ɛ), where κf (A) isthe condition number of the matrix defining the linear system. Our algorithm applies the preconditioned Chebyshev iteration with preconditioners designed using nearlylinear time algorithms for graph sparsification and graph partitioning.
Efficient routing in alloptical networks
 in Proc. 26 th ACM Symp. Theory of Computing
, 1994
"... Communication in alloptical networks requires novel routing paradigms. The high bandwidth of the optic fiber is utilized through wavelengthdivision multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. We study t ..."
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Cited by 120 (0 self)
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Communication in alloptical networks requires novel routing paradigms. The high bandwidth of the optic fiber is utilized through wavelengthdivision multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. We study the problem of routing a set of requests (each of which is a pair of nodes to be connected by a path) on sparse networks using a limited number of wavelengths, ensuring that different paths using the same wavelength never use the same physical link. The constraints on the selection of paths and wavelengths depend on the type of photonic switches used in the network. We present eflicient routing techniques for the two types of photonic switches that dominate current research in alloptical networks. Our results es
Semidefinite Programming and Combinatorial Optimization
 DOC. MATH. J. DMV
, 1998
"... We describe a few applications of semide nite programming in combinatorial optimization. ..."
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Cited by 96 (1 self)
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We describe a few applications of semide nite programming in combinatorial optimization.
A proof of Alon’s second eigenvalue conjecture
, 2003
"... A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 ..."
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Cited by 92 (1 self)
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A dregular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random dregular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 +ɛwith probability 1 − O(n−τ), where τ = ⌈ � √ d − 1+1 � /2⌉−1. We also show that this probability is at most 1 − c/nτ ′, for a constant c and a τ ′ that is either τ or τ +1 (“more often ” τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1
Optimal Scaling for Various MetropolisHastings Algorithms
, 2001
"... We review and extend results related to optimal scaling of MetropolisHastings algorithms. We present various theoretical results for the highdimensional limit. We also present simulation studies which confirm the theoretical results in finite dimensional contexts. ..."
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Cited by 91 (22 self)
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We review and extend results related to optimal scaling of MetropolisHastings algorithms. We present various theoretical results for the highdimensional limit. We also present simulation studies which confirm the theoretical results in finite dimensional contexts.
Some Applications of Laplace Eigenvalues of Graphs
 GRAPH SYMMETRY: ALGEBRAIC METHODS AND APPLICATIONS, VOLUME 497 OF NATO ASI SERIES C
, 1997
"... In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of ..."
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Cited by 90 (0 self)
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In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the maxcut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.
A random walk construction of uniform spanning trees and uniform labelled trees
 SIAM Journal on Discrete Mathematics
, 1990
"... Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter. ..."
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Cited by 82 (3 self)
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Abstract A random walk on a finite graph can be used to construct a uniformrandom spanning tree. We show how random walk techniques can be applied to the study of several properties of the uniform randomspanning tree: the proportion of leaves, the distribution of degrees, and the diameter.