eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-01-08
3:1
3:19
10.4230/LIPIcs.ITCS.2019.3
article
On Solving Linear Systems in Sublinear Time
Andoni, Alexandr
1
Krauthgamer, Robert
2
Pogrow, Yosef
2
Columbia University, New York, NY, USA
Weizmann Institute of Science, Rehovot, Israel
We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S in R^{n x n} and a vector b in R^n in the range of S, and the goal is to output x in R^n satisfying Sx=b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting.
Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u in [n] together with S and b as above, the goal is to output an approximation x^_u for x^*_u, where x^* is a fixed solution to Sx=b.
Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x_{u} in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive | x^_u-x^*_u | <=epsilon | x^* |_infty for accuracy parameter epsilon>0. We further prove that the condition-number assumption is necessary and tight.
In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n (for the same additive approximation), even if S has bounded sparsity and condition number.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol124-itcs2019/LIPIcs.ITCS.2019.3/LIPIcs.ITCS.2019.3.pdf
Linear systems
Laplacian solver
Sublinear time
Randomized linear algebra