68 research outputs found
An Efficient Parallel Solver for SDD Linear Systems
We present the first parallel algorithm for solving systems of linear
equations in symmetric, diagonally dominant (SDD) matrices that runs in
polylogarithmic time and nearly-linear work. The heart of our algorithm is a
construction of a sparse approximate inverse chain for the input matrix: a
sequence of sparse matrices whose product approximates its inverse. Whereas
other fast algorithms for solving systems of equations in SDD matrices exploit
low-stretch spanning trees, our algorithm only requires spectral graph
sparsifiers
Simple parallel and distributed algorithms for spectral graph sparsification
We describe a simple algorithm for spectral graph sparsification, based on
iterative computations of weighted spanners and uniform sampling. Leveraging
the algorithms of Baswana and Sen for computing spanners, we obtain the first
distributed spectral sparsification algorithm. We also obtain a parallel
algorithm with improved work and time guarantees. Combining this algorithm with
the parallel framework of Peng and Spielman for solving symmetric diagonally
dominant linear systems, we get a parallel solver which is much closer to being
practical and significantly more efficient in terms of the total work.Comment: replaces "A simple parallel and distributed algorithm for spectral
sparsification". Minor change
Improving information centrality of a node in complex networks by adding edges
The problem of increasing the centrality of a network node arises in many
practical applications. In this paper, we study the optimization problem of
maximizing the information centrality of a given node in a network
with nodes and edges, by creating new edges incident to . Since
is the reciprocal of the sum of resistance distance
between and all nodes, we alternatively consider the problem of minimizing
by adding new edges linked to . We show that the
objective function is monotone and supermodular. We provide a simple greedy
algorithm with an approximation factor and
running time. To speed up the computation, we also present an
algorithm to compute -approximate
resistance distance after iteratively adding edges, the
running time of which is for any
, where the notation suppresses the factors. We experimentally demonstrate the effectiveness and
efficiency of our proposed algorithms.Comment: 7 pages, 2 figures, ijcai-201
Fast, Accurate Second Order Methods for Network Optimization
Dual descent methods are commonly used to solve network flow optimization
problems, since their implementation can be distributed over the network. These
algorithms, however, often exhibit slow convergence rates. Approximate Newton
methods which compute descent directions locally have been proposed as
alternatives to accelerate the convergence rates of conventional dual descent.
The effectiveness of these methods, is limited by the accuracy of such
approximations. In this paper, we propose an efficient and accurate distributed
second order method for network flow problems. The proposed approach utilizes
the sparsity pattern of the dual Hessian to approximate the the Newton
direction using a novel distributed solver for symmetric diagonally dominant
linear equations. Our solver is based on a distributed implementation of a
recent parallel solver of Spielman and Peng (2014). We analyze the properties
of the proposed algorithm and show that, similar to conventional Newton
methods, superlinear convergence within a neighbor- hood of the optimal value
is attained. We finally demonstrate the effectiveness of the approach in a set
of experiments on randomly generated networks.Comment: arXiv admin note: text overlap with arXiv:1502.0315
An SDP-Based Algorithm for Linear-Sized Spectral Sparsification
For any undirected and weighted graph with vertices and
edges, we call a sparse subgraph of , with proper reweighting of the
edges, a -spectral sparsifier if holds for any , where and
are the respective Laplacian matrices of and . Noticing that
time is needed for any algorithm to construct a spectral sparsifier and a
spectral sparsifier of requires edges, a natural question is to
investigate, for any constant , if a -spectral
sparsifier of with edges can be constructed in time,
where the notation suppresses polylogarithmic factors. All previous
constructions on spectral sparsification require either super-linear number of
edges or time.
In this work we answer this question affirmatively by presenting an algorithm
that, for any undirected graph and , outputs a
-spectral sparsifier of with edges in
time. Our algorithm is based on three novel
techniques: (1) a new potential function which is much easier to compute yet
has similar guarantees as the potential functions used in previous references;
(2) an efficient reduction from a two-sided spectral sparsifier to a one-sided
spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a
semi-definite program.Comment: To appear at STOC'1
Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time
We present the first almost-linear time algorithm for constructing
linear-sized spectral sparsification for graphs. This improves all previous
constructions of linear-sized spectral sparsification, which requires
time.
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, and adaptive constructions based on barrier functions.Comment: 22 pages. A preliminary version of this paper is to appear in
proceedings of the 56th Annual IEEE Symposium on Foundations of Computer
Science (FOCS 2015
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