Results 1  10
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43
TwiceRamanujan sparsifiers
 IN PROC. 41ST STOC
, 2009
"... We prove that for every d> 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d V  ⌉ edges such that for every x ∈ IR V, 1 ≤ xT LHx x T LGx ≤ d + 1 + 2 √ d d + 1 − 2 √ d, where LG and LH are the Laplacian matrices of G and H, respectively. ..."
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Cited by 88 (12 self)
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We prove that for every d> 1 and every undirected, weighted graph G = (V, E), there exists a weighted graph H with at most ⌈d V  ⌉ edges such that for every x ∈ IR V, 1 ≤ xT LHx x T LGx ≤ d + 1 + 2 √ d d + 1 − 2 √ d, where LG and LH are the Laplacian matrices of G and H, respectively.
Using petaldecompositions to build a low stretch spanning tree
 in Proceedings of ACM STOC
, 2012
"... We prove that any graph G = (V,E) with n points and m edges has a spanning tree T such that∑ (u,v)∈E(G) dT (u, v) = O(m log n log log n). Moreover such a tree can be found in timeO(m log n log log n). Our result is obtained using a new petaldecomposition approach which guarantees that the radius o ..."
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Cited by 18 (1 self)
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We prove that any graph G = (V,E) with n points and m edges has a spanning tree T such that∑ (u,v)∈E(G) dT (u, v) = O(m log n log log n). Moreover such a tree can be found in timeO(m log n log log n). Our result is obtained using a new petaldecomposition approach which guarantees that the radius of each cluster in the tree is at most 4 times the radius of the induced subgraph of the cluster in the original graph.
An almostlineartime algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations
"... In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general ..."
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Cited by 14 (7 self)
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In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general framework that also allows us to achieve a running time of O(m 1+o(1) ε −2 k 2) for the maximum concurrent kcommodity flow problem, the first such algorithm with an almost linear dependence on m. We also note that independently Jonah Sherman has produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.
A Local Spectral Method for Graphs: With Applications to Improving Graph Partitions and Exploring Data Graphs Locally
"... The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local r ..."
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Cited by 13 (5 self)
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The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many applications, however, graphs that arise have several local regions of interest, and the second eigenvector will typically fail to provide information finetuned to each local region. In this paper, we introduce a locallybiased analogue of the second eigenvector, and we demonstrate its usefulness at highlighting local properties of data graphs in a semisupervised manner. To do so, we first view the second eigenvector as the solution to a constrained optimization problem, and we incorporate the local information as an additional constraint; we then characterize the optimal solution to this new problem and show that it can be interpreted as a generalization of a Personalized PageRank vector; and finally, as a consequence, we show that the solution can be computed in nearlylinear time. In addition, we show that this locallybiased vector can be used to compute an approximation to the best partition near an input seed set in a manner analogous to the way in which the second eigenvector of the Laplacian can be used to obtain an approximation to the best partition in the entire input graph. Such a primitive is useful
Lean algebraic multigrid (LAMG): Fast graph Laplacian linear solver
 ARXIV EPRINTS
"... Laplacian matrices of graphs arise in largescale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on un ..."
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Cited by 8 (0 self)
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Laplacian matrices of graphs arise in largescale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the symmetric linear system Ax = b is presented, where A is a graph Laplacian. LAMG’s run time and storage are empirically demonstrated to scale linearly with the number of edges. LAMG consists of a setup phase during which a sequence of increasinglycoarser Laplacian systems is constructed, and an iterative solve phase using multigrid cycles. General graphs pose algorithmic challenges not encountered in traditional multigrid applications. LAMG combines a lean piecewiseconstant interpolation, judicious node aggregation based on a new node proximity measure (the affinity), and an energy correction of coarselevel systems. This results in fast convergence and substantial setup and memory savings. A serial LAMG implementation scaled linearly for a diverse set of 3774 realworld graphs with up to 47 million edges, with no parameter tuning. LAMG was more robust than the UMFPACK direct solver and Combinatorial Multigrid (CMG), although CMG was faster than LAMG on average. Our methodology is extensible to eigenproblems and other graph
How user behavior is related to social affinity
 In Proceedings of the 5th ACM International Conference on Web Search and Data Mining (WSDM). ACM
, 2012
"... Previous research has suggested that people who are in the same social circle exhibit similar behaviors and tastes. The rise of social networks gives us insights into the social circles of web users, and recommendation services (including search engines, advertisement engines, and collaborative fil ..."
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Cited by 5 (0 self)
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Previous research has suggested that people who are in the same social circle exhibit similar behaviors and tastes. The rise of social networks gives us insights into the social circles of web users, and recommendation services (including search engines, advertisement engines, and collaborative filtering engines) provide a motivation to adapt recommendations to the interests of the audience. An important primitive for supporting these applications is the ability to quantify how connected two users are in a social network. The shortestpath distance between a pair of users is an obvious candidate measure. This paper introduces a new measure of “affinity ” in social networks that takes into account not only the distance between two users, but also the number of edgedisjoint paths between them, i.e. the “robustness ” of their connection. Our measure is based on a sketchbased approach, and affinity queries can be answered extremely efficiently (at the expense of a onetime offline sketch computation). We compare this affinity measure against the “approximate shortestpath distance”, a sketchbased distance measure with similar efficiency characteristics. Our empirical study is based on a Hotmail email exchange graph combined with demographic information and Bing query history, and a Twitter mentiongraph together with the text of the underlying tweets. We found that users who are close to each other – either in terms of distance or affinity – have a higher similarity in terms of demographics, queries, and tweets.
Near LinearWork Parallel SDD Solvers, LowDiameter Decomposition, and LowStretch Subgraphs
"... This paper presents the design and analysis of a near linearwork parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input of a SDD nbyn matrix A with m nonzero entries and a vector b, our algorithm computes a vector ˜x such that ‖˜x − A + b‖A ≤ ε · ‖A + b‖A in O ..."
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Cited by 5 (3 self)
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This paper presents the design and analysis of a near linearwork parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input of a SDD nbyn matrix A with m nonzero entries and a vector b, our algorithm computes a vector ˜x such that ‖˜x − A + b‖A ≤ ε · ‖A + b‖A in O(m log O(1) n log 1 ε and O(m 1/3+θ log 1) depth for any fixed θ> 0.) work ε The algorithm relies on a parallel algorithm for generating lowstretch spanning trees or spanning subgraphs. To this end, we first develop a parallel decomposition algorithm that in polylogarithmic depth and Õ(E) work1, partitions a graph into components with polylogarithmic diameter such that only a small fraction of the original edges are between the components. This can be used to generate lowstretch spanning trees with average stretch O(n α) in O(n 1+α) work and O(n α) depth. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in Õ(E) work and polylogarithmic depth. We apply this subgraph construction to derive our solver. By using the linear system solver in known applications, our results imply improved parallel randomized algorithms for several problems, including singlesource shortest paths, maximum flow, mincost flow, and approximate maxflow.
Uniform sampling for matrix approximation
 In Proceedings of the 6th Annual Conference on Innovations in Theoretical Computer Science (ITCS
, 2015
"... ar ..."
NEARLY LINEAR TIME ALGORITHMS FOR PRECONDITIONING AND SOLVING SYMMETRIC, DIAGONALLY DOMINANT LINEAR SYSTEMS
, 2014
"... We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse p ..."
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Cited by 3 (0 self)
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We present a randomized algorithm that on input a symmetric, weakly diagonally dominant nbyn matrix A with m nonzero entries and an nvector b produces an x ̃ such that ‖x ̃ − A†b‖A ≤ ‖A†b‖A in expected time O(m logc n log(1/)) for some constant c. By applying this algorithm inside the inverse power method, we compute approximate Fiedler vectors in a similar amount of time. The algorithm applies subgraph preconditioners in a recursive fashion. These preconditioners improve upon the subgraph preconditioners first introduced by Vaidya in 1990. For any symmetric, weakly diagonally dominant matrix A with nonpositive offdiagonal entries and k ≥ 1, we construct in time O(m logc n) a preconditioner B of A with at most 2(n − 1) +O((m/k) log39 n) nonzero offdiagonal entries such that the finite generalized condition number κf (A,B) is at most k, for some other constant c. In the special case when the nonzero structure of the matrix is planar the corresponding linear system solver runs in expected time O(n log2 n+n logn log logn log(1/)). We hope that our introduction of algorithms of low asymptotic complexity will lead to the development of algorithms that are also fast in practice.