 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2019.3
URN: urn:nbn:de:0030-drops-100966
URL: https://drops.dagstuhl.de/opus/volltexte/2018/10096/
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### On Solving Linear Systems in Sublinear Time

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### Abstract

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.

### BibTeX - Entry

```@InProceedings{andoni_et_al:LIPIcs:2018:10096,
author =	{Alexandr Andoni and Robert Krauthgamer and Yosef Pogrow},
title =	{{On Solving Linear Systems in Sublinear Time}},
booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
pages =	{3:1--3:19},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-095-8},
ISSN =	{1868-8969},
year =	{2018},
volume =	{124},
editor =	{Avrim Blum},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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