5,122 research outputs found
A nearly-mlogn time solver for SDD linear systems
We present an improved algorithm for solving symmetrically diagonally
dominant linear systems. On input of an symmetric diagonally
dominant matrix with non-zero entries and a vector such that
for some (unknown) vector , our algorithm computes a
vector such that
{ denotes the A-norm} in time
The solver utilizes in a standard way a `preconditioning' chain of
progressively sparser graphs. To claim the faster running time we make a
two-fold improvement in the algorithm for constructing the chain. The new chain
exploits previously unknown properties of the graph sparsification algorithm
given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning
properties. We also present an algorithm of independent interest that
constructs nearly-tight low-stretch spanning trees in time
, a factor of faster than the algorithm in
[Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the
construction time of the preconditioning chain.Comment: to appear in FOCS1
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Iterative Row Sampling
There has been significant interest and progress recently in algorithms that
solve regression problems involving tall and thin matrices in input sparsity
time. These algorithms find shorter equivalent of a n*d matrix where n >> d,
which allows one to solve a poly(d) sized problem instead. In practice, the
best performances are often obtained by invoking these routines in an iterative
fashion. We show these iterative methods can be adapted to give theoretical
guarantees comparable and better than the current state of the art.
Our approaches are based on computing the importances of the rows, known as
leverage scores, in an iterative manner. We show that alternating between
computing a short matrix estimate and finding more accurate approximate
leverage scores leads to a series of geometrically smaller instances. This
gives an algorithm that runs in
time for any , where the term is comparable
to the cost of solving a regression problem on the small approximation. Our
results are built upon the close connection between randomized matrix
algorithms, iterative methods, and graph sparsification.Comment: 26 pages, 2 figure
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