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 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Boston
Springer US
01.10.2014
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 1432-4350, 1433-0490 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
|---|---|
| Bibliographie: | 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 |