A Nearly-m log n Time Solver for SDD Linear Systems

Abstract

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n × n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax = b for some (unknown) vectorx, our algorithm computes a vector x such that ||x −x||A < ||x||A 1 in timẽThe 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 timeÕ(m log n), a factor of O(log n) faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.

Efficient Accelerated Coordinate Descent Methods and Faster Algorithms for Solving Linear Systems

“…[15] isÕ(m log 2 n log 1 ) where m is the number of nonzero entries in the matrix, the matrix is n × n, and we useÕ to hide O(poly(log log n)) terms. We show how direct application of ACDM to a simple algorithm in [15] yields a faster SDD solver with an asymptotic runtime ofÕ(m log 1.5 n log 1 ) in the unit-cost RAM model, getting closer to the fastest known running time ofÕ(m log n log 1 ) by Koutis, Miller, and Peng [16] in a less restrictive computational model.…”

Support-theoretic subgraph preconditioners for large-scale SLAM

“…The first is to improve the efficiency of the proposed algorithm by adapting the recent results in finding low-stretch spanning trees [9], ultra-sparsifiers [10] and simple combinatorial solvers [11] for Laplacian matrices. The second is to generalize the proposed metric to handle n-ary factors (n > 2).…”

“…To the best of our knowledge, this is the first attempt to derive theoretically good subgraph preconditioners for SLAM problems. Although the MCMC-based algorithm we develop to find high-quality preconditioners is not practical in its current form, recent developments in finding low-stretch spanning trees [9], ultrasparsifiers [10], and simple combinatorial solvers [11] make us believe that this obstacle will be removed in future work.…”

A fast solver for a class of linear systems

“…Theorem [10]. Symmetric diagonally dominant systems can be solved inÕ(m log n log(1/ )) time, where is a standard measure of the approximation error.…”

“…It is the culmination of a line of work initiated by Vaidya [23], which was brought to near-completion with the breakthrough achievement of Spielman and Teng [19]: the first solver that runs in time O(m log c n) for any graph Laplacian, where c is a large constant. The work discussed here, summarized in the following claim from [10], provides a conceptually simpler, faster and more practical algorithm.…”