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54
Approximating minimum maxstretch spanning trees on unweighted graphs
 In Proc. ACMSIAM Symposium on Discrete Algorithms
, 2004
"... Given a graph G and a spanning tree T of G, we say that T is a tree tspanner of G if the distance between every pair of vertices in T is at most t times their distance in G. The problem of finding a tree tspanner minimizing t is referred to as the Minimum MaxStretch spanning Tree (MMST) problem. ..."
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Cited by 22 (0 self)
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Given a graph G and a spanning tree T of G, we say that T is a tree tspanner of G if the distance between every pair of vertices in T is at most t times their distance in G. The problem of finding a tree tspanner minimizing t is referred to as the Minimum MaxStretch spanning Tree (MMST) problem. This paper concerns the MMST problem on unweighted graphs. The problem is known to be NPhard, and the paper presents an O(log n)approximation algorithm for it. Furthermore, it is established that unless P = NP, the problem cannot be approximated additively by any o(n) factor.
Solving elliptic finite element systems in nearlinear time with support preconditioners
 Manuscript, Sandia National
"... Abstract. We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagon ..."
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Cited by 21 (0 self)
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Abstract. We show in this note how support preconditioners can be applied to a class of linear systems arising from use of the finite element method to solve linear elliptic problems. Our technique reduces the problem, which is symmetric and positive definite, to a symmetric positive definite diagonally dominant problem. Significant theory has already been developed for preconditioners in the diagonally dominant case. We show that the degradation in the quality of the preconditioner using our technique is only a small constant factor. 1. Introduction. Finite
Nearly tight low stretch spanning trees
 In Proceedings of the 49th Annual Symposium on Foundations of Computer Science
, 2008
"... We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT(u, v)/dG(u, v)] is bounded by Õ(log n). Our result is obtained via a new approach of building “highways ” between portals and a new strong diameter probabilis ..."
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Cited by 17 (0 self)
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We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch ET ∼T [dT(u, v)/dG(u, v)] is bounded by Õ(log n). Our result is obtained via a new approach of building “highways ” between portals and a new strong diameter probabilistic decomposition theorem. 1
Fast Elimination of Redundant Linear Equations and Reconstruction of RecombinationFree Mendelian Inheritance on a Pedigree
 Proc. of 18th Annual ACMSIAM Symoposium on Discrete Algorithms (SODA’07
, 2007
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Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices
"... Abstract. 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 this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Lapla ..."
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Cited by 11 (0 self)
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Abstract. 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 this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including lowstretch spanning trees, graph sparsifiers, ultrasparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective.
Finding effective supporttree preconditioners
 in Proceedings of the 17th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA
, 2005
"... In 1995, Gremban, Miller, and Zagha introduced supporttree preconditioners and a parallel algorithm called supporttree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the offdiagonal entries ..."
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Cited by 11 (1 self)
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In 1995, Gremban, Miller, and Zagha introduced supporttree preconditioners and a parallel algorithm called supporttree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the offdiagonal entries are nonpositive, and the row and column sums are zero. A Laplacian A with 2m nonzeros can be interpreted as an undirected positivelyweighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c((i, j)) = −Ai,j = −Aj,i if Ai,j = Aj,i < 0. Gremban et al. showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V, E) with n nodes, m edges, and a weight function c> 0 on the edges, where w.l.o.g., mine∈E c(e) = 1. Equipped with this preconditioner, STCG requires O(log 4 n · � ∆/α) iterations, where α = min U⊂V,U≤V /2 c(U, V \U)/U  is the minimum edge expansion of the graph, and ∆ = maxv∈V c(v) is the maximum incident weight on any vertex. Each iteration requires O(m) work and can be implemented in O(log n) steps in parallel, using only O(m) space. Our results generalize to matrices that are symmetric and diagonallydominant (SDD). 1
Graph partitioning into isolated, high conductance clusters: theory, computation and . . .
, 2008
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Online and Stochastic Survivable Network Design
"... Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is kno ..."
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Cited by 11 (3 self)
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Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edgeconnectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict costshares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rentorbuy version and (b) the (twostage) stochastic version of the edgeconnected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edgecosts are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)strict cost shares, which gives constantfactor rentorbuy and stochastic algorithms for these instances.
Reconstructing Approximate Tree Metrics
 Proceedings of the twentysixth ACM symposium on Principles of distributed computing
, 2007
"... We introduce a novel measure called εfourpoints condition (ε4PC), which assigns a value ε ∈ [0, 1] to every metric space quantifying how close the metric is to a tree metric. Datasets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based ..."
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Cited by 10 (2 self)
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We introduce a novel measure called εfourpoints condition (ε4PC), which assigns a value ε ∈ [0, 1] to every metric space quantifying how close the metric is to a tree metric. Datasets taken from real Internet measurements indicate remarkable closeness of Internet latencies to tree metrics based on this condition. We study embeddings of ε4PC metric spaces into trees and prove tight upper and lower bounds. Specifically, we show that there are constants c1 and c2 such that, (1) every metric (X, d) which satisfies the ε4PC can be embedded into a tree with distortion (1 + ε) c1 log X, and (2) for every ε ∈ [0, 1] and any number of nodes, there is a metric space (X, d) satisfying the ε4PC that does not embed into a tree with distortion less than (1 + ε) c2 log X. In addition, we prove a lower bound on approximate distance labelings of ε4PC metrics, and give tight bounds for tree embeddings with additive error guarantees.
Approaching optimality for solving SDD linear systems ∗
, 2010
"... We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifier ˆ G with n − 1+m/k edges, such that the condition number of G with ˆ G is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m log n ..."
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Cited by 10 (2 self)
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We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifier ˆ G with n − 1+m/k edges, such that the condition number of G with ˆ G is bounded above by Õ(k log2 n), with probability 1 − p. The algorithm runs in time Õ((m log n + n log 2 n) log(1/p)). 1 As a result, we obtain an algorithm that on input an n × n symmetric diagonally dominant matrix A with m + n − 1 nonzero entries and a vector b, computes a vector ¯x satisfying x − A + bA <ɛA + bA, in time Õ(m log 2 n log(1/ɛ)). The solver is based on a recursive application of the incremental sparsifier that produces a hierarchy of graphs which is then used to construct a recursive preconditioned Chebyshev iteration. 1