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Approaching optimality for solving SDD linear systems (2010)
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BibTeX
@MISC{Koutis10approachingoptimality,
author = {Ioannis Koutis and Gary L. Miller and Richard Peng},
title = {Approaching optimality for solving SDD linear systems },
year = {2010}
}
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Abstract
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 non-zero entries and a vector b, computes a vector ¯x satisfying ||x − A + b||A <ɛ||A + b||A, 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.
