Results 1  10
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88
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 45 (7 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.
A unified framework for approximating and clustering data
, 2011
"... Given a set F of n positive functions over a ground set X, we consider the problem of computing x ∗ that minimizes the expression ∑ f∈F f(x), over x ∈ X. A typical application is shape fitting, where we wish to approximate a set P of n elements (say, points) by a shape x from a (possibly infinite) f ..."
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Cited by 34 (8 self)
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Given a set F of n positive functions over a ground set X, we consider the problem of computing x ∗ that minimizes the expression ∑ f∈F f(x), over x ∈ X. A typical application is shape fitting, where we wish to approximate a set P of n elements (say, points) by a shape x from a (possibly infinite) family X of shapes. Here, each point p ∈ P corresponds to a function f such that f(x) is the distance from p to x, and we seek a shape x that minimizes the sum of distances from each point in P. In the kclustering variant, each x ∈ X is a tuple ofk shapes, andf(x) is the distance frompto its closest shape inx. Our main result is a unified framework for constructing coresets and approximate clustering for such general sets of functions. To achieve our results, we forge a link between the classic and well defined notion of εapproximations from the theory of PAC Learning and VC dimension, to the relatively new (and not so consistent) paradigm of coresets, which are some kind of “compressed representation " of the input set F. Using traditional techniques, a coreset usually implies an LTAS (linear time approximation scheme) for the corresponding optimization problem, which can be computed in parallel, via one pass over the data, and using only polylogarithmic space (i.e, in the streaming model). For several function families F for which coresets are known not to exist, or the corresponding (approximate) optimization problems are hard, our framework yields bicriteria approximations, or coresets that are large, but contained in a lowdimensional space. We demonstrate our unified framework by applying it on projective clustering problems. We obtain new coreset constructions and significantly smaller coresets, over the ones that
Nearoptimal Columnbased Matrix Reconstruction
, 2011
"... We consider lowrank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVDlike decompositions for ..."
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Cited by 34 (4 self)
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We consider lowrank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVDlike decompositions for columnbased matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].
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 33 (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.
Rumour spreading and graph conductance
 IN PROCEEDINGS OF THE 21ST ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA
, 2010
"... We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws ..."
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Cited by 33 (2 self)
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We show that if a connected graph with n nodes has conductance φ then rumour spreading, also known as randomized broadcast, successfully broadcasts a message within O(log 4 n/φ 6) many steps, with high probability, using the PUSHPULL strategy. An interesting feature of our approach is that it draws a connection between rumour spreading and the spectral sparsification procedure of Spielman and Teng [23].
Interlacing families II: mixed characteristic polynomials and the Kadison–Singer problem. arXiv:1306.3969,
, 2015
"... Abstract We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the KadisonSinger problem. The first is Weaver's conjecture KS2, which is known to imply KadisonSinger via a projection paving conjecture of Ak ..."
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Cited by 22 (4 self)
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Abstract We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the KadisonSinger problem. The first is Weaver's conjecture KS2, which is known to imply KadisonSinger via a projection paving conjecture of Akemann and Anderson. The second is a formulation due to Casazza et al. of Anderson's original paving conjecture(s), for which we are able to compute explicit paving bounds. The proof involves an analysis of the largest roots of a family of polynomials that we call the "mixed characteristic polynomials" of a collection of matrices.
A General Framework for Graph Sparsification
, 2011
"... We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results ..."
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Cited by 21 (1 self)
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We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczúr and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczúr and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + Õ(n/ǫ²) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczúr and Karger) of O(m log² n) (for the unweighted case) and O(m log³ n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ǫ²) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n 3 m/ǫ 2) time. A key ingredient of our proofs is a natural generalization of Karger’s bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger’s bound, we suspect that our generalization will also be of independent interest.
Faster generation of random spanning trees
"... Abstract — In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1 + δ) of uniform in expected time e O(m √ n log 1/δ). This improves the sparse graph c ..."
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Cited by 19 (1 self)
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Abstract — In this paper, we set forth a new algorithm for generating approximately uniformly random spanning trees in undirected graphs. We show how to sample from a distribution that is within a multiplicative (1 + δ) of uniform in expected time e O(m √ n log 1/δ). This improves the sparse graph case of the best previously known worstcase bound of O(min{mn, n 2.376}), which has stood for twenty years. To achieve this goal, we exploit the connection between random walks on graphs and electrical networks, and we use this to introduce a new approach to the problem that integrates discrete random walkbased techniques with continuous linear algebraic methods. We believe that our use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is a useful paradigm that will find further applications in algorithmic graph theory. Keywordsspanning trees; random walks on graphs; electrical flows; 1.