Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs

We present the design and analysis of a nearly-linear work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD n -by- n matrix A with m nonzero entries and a vector b , our algorithm computes a vector such that in work and depth for any θ >0, where A...

Full description

Saved in:
Bibliographic Details
Published in:Theory of computing systems Vol. 55; no. 3; pp. 521 - 554
Main Authors: Blelloch, Guy E., Gupta, Anupam, Koutis, Ioannis, Miller, Gary L., Peng, Richard, Tangwongsan, Kanat
Format: Journal Article
Language:English
Published: Boston Springer US 01.10.2014
Springer Nature B.V
Subjects:
Algorithms
Approximation
Computer architecture
Computer Science
Decomposition
Graph theory
Linear programming
Linear systems
Mathematical analysis
Solvers
Studies
Theory of Computation
Vectors (mathematics)
United States > US
Linear systems
Low-stretch spanning trees
Low-diameter decomposition
Low-stretch subgraphs
Parallel algorithms
ISSN:1432-4350, 1433-0490
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We present the design and analysis of a nearly-linear work parallel algorithm for solving symmetric diagonally dominant (SDD) linear systems. On input an SDD n -by- n matrix A with m nonzero entries and a vector b , our algorithm computes a vector such that in work and depth for any θ >0, where A + denotes the Moore-Penrose pseudoinverse of  A . The algorithm relies on a parallel algorithm for generating low-stretch spanning trees or spanning subgraphs. To this end, we first develop a parallel decomposition algorithm that in O ( m log O (1) n ) work and polylogarithmic depth, partitions a graph with n nodes and m edges into components with polylogarithmic diameter such that only a small fraction of the original edges are between the components. This can be used to generate low-stretch spanning trees with average stretch O ( n α ) in O ( m log O (1) n ) work and O ( n α ) depth for any α >0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O ( m log O (1) n ) work and polylogarithmic depth. We apply this subgraph construction to derive a parallel linear solver. By using this solver in known applications, our results imply improved parallel randomized algorithms for several problems, including single-source shortest paths, maximum flow, minimum-cost flow, and approximate maximum flow.
Bibliography:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ObjectType-Article-1
ObjectType-Feature-2
content type line 23
ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-013-9444-5