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...
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| Veröffentlicht in: | Theory of computing systems Jg. 55; H. 3; S. 521 - 554 |
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| Sprache: | Englisch |
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01.10.2014
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| ISSN: | 1432-4350, 1433-0490 |
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| Abstract | 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. |
|---|---|
| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).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 [thetas]>0, where A super(+) 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(mlog super( )O1)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 super( ) alpha in O(mlog super( )O1)n) work and O(n super( ) alpha depth for any alpha >0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O(mlog super( )O1)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. 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. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) 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 [theta]>0, where A ^sup +^ 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(mlog^sup 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 ^sup [alpha]^) in O(mlog^sup O(1)^ n) work and O(n ^sup [alpha]^) depth for any [alpha]>0. Alternatively, it can be used to generate spanning subgraphs with polylogarithmic average stretch in O(mlog^sup 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.[PUBLICATION ABSTRACT] |
| Author | Gupta, Anupam Miller, Gary L. Peng, Richard Tangwongsan, Kanat Blelloch, Guy E. Koutis, Ioannis |
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| Cites_doi | 10.1002/9781118032701 10.1137/S0097539792224474 10.1137/0222073 10.1080/01621459.1963.10500830 10.1137/0217079 10.1109/SFCS.1993.366822 10.1007/978-3-642-13562-0_2 10.1017/CBO9780511804441 10.1145/4221.4227 10.1016/0012-365X(79)90084-0 10.1006/jagm.1997.0888 10.1137/1.9780898718812 10.1137/0220006 10.1145/1060590.1060665 |
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| Keywords | Linear systems Low-stretch spanning trees Low-diameter decomposition Low-stretch subgraphs Parallel algorithms |
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| Snippet | 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... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We present the design and analysis of a nearly-linear work parallel algorithm for... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We present the design and analysis of a nearly-linear work parallel algorithm for... |
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| SubjectTerms | Algorithms Approximation Computer architecture Computer Science Decomposition Graph theory Linear programming Linear systems Mathematical analysis Solvers Studies Theory of Computation Vectors (mathematics) |
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| Title | Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs |
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