On designing optimal parallel triangular solvers

This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent o...

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Vydáno v:Information and computation Ročník 161; číslo 2; s. 172 - 210
Hlavní autor: Santos, Eunice E.
Médium: Journal Article
Jazyk:angličtina
Vydáno: San Diego, CA Elsevier Inc 2000
Elsevier
Témata:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
distributed-memory
Exact sciences and technology
LogP model
Mathematics
matrix computation
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
numerical methods
parallel algorithmsand complexity
Sciences and techniques of general use
Theoretical computing
triangular solvers
matrix computation
distributed-memory
LogP model
numerical methods
triangular solvers
parallel algorithmsand complexity
Triangular matrix
Parallel algorithm
Triangular linear system
Numerical method
Forward substitution scheme
Complexity
Latency
Equation system
Distributed memory
Distributed memory multiprocessor system
Linear system
Computer program
Matrix calculus
Backward substitution scheme
Time complexity
Message transmission
ISSN:0890-5401, 1090-2651
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Abstract This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent of the data layout, lower bounds for data layouts in which the number of data items per processor is bounded, and lower bounds for specific data layouts commonly used in designing parallel algorithms for this problem are presented in this paper. Furthermore, algorithms are provided which have running times within a constant factor of the lower bounds described. One interesting result is that the popular two-dimensional block matrix layout necessarily results in significantly longer running times than simpler one-dimensional schemes. Finally, a generalization of the lower bounds to banded triangular linear systems is presented.
AbstractList This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic bounds for solving these systems using forward/backward substitution are presented. Specifically, lower bounds on execution time independent of the data layout, lower bounds for data layouts in which the number of data items per processor is bounded, and lower bounds for specific data layouts commonly used in designing parallel algorithms for this problem are presented in this paper. Furthermore, algorithms are provided which have running times within a constant factor of the lower bounds described. One interesting result is that the popular two-dimensional block matrix layout necessarily results in significantly longer running times than simpler one-dimensional schemes. Finally, a generalization of the lower bounds to banded triangular linear systems is presented.
Author Santos, Eunice E.
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  givenname: Eunice E.
  surname: Santos
  fullname: Santos, Eunice E.
  email: santos@cs.vt.edu
  organization: Department of Computer Science, Virginia Polytechnic Institute & State University, Blacksburg, Virginia, USA
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Cites_doi 10.1006/jpdc.1999.1529
10.1137/0909037
10.1137/0714076
10.1137/0910025
10.1016/0167-8191(88)90009-9
10.1145/240455.240477
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Issue 2
Keywords matrix computation
distributed-memory
LogP model
numerical methods
triangular solvers
parallel algorithmsand complexity
Triangular matrix
Parallel algorithm
Triangular linear system
Numerical method
Forward substitution scheme
Complexity
Latency
Equation system
Distributed memory
Distributed memory multiprocessor system
Linear system
Computer program
Matrix calculus
Backward substitution scheme
Time complexity
Message transmission
Language English
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Karp, Sahay, Santos, Schauser (bib11) 1993
Culler, Karp, Patterson, Sahay, Santos, Schauser, Subramonian, von Eicken (bib2) November 1996
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Borodin (10.1006/inco.2000.2866_bib1) 1975
Higham (10.1006/inco.2000.2866_bib10) 1993
Santos (10.1006/inco.2000.2866_bib16) 1995
Heller (10.1006/inco.2000.2866_bib8) 1987; 29
Karp (10.1006/inco.2000.2866_bib11) 1993
Li (10.1006/inco.2000.2866_bib13) 1989
Santos (10.1006/inco.2000.2866_bib17) 1999; 57
Gallivan (10.1006/inco.2000.2866_bib6) 1990
Sameh (10.1006/inco.2000.2866_bib15) 1977
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Culler (10.1006/inco.2000.2866_bib2) 1996
Eisenstat (10.1006/inco.2000.2866_bib5) 1998
Karp (10.1006/inco.2000.2866_bib12) 1990
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Snippet This paper explores the problem of solving triangular linear systems on parallel distributed-memory machines. Working within the LogP model, tight asymptotic...
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SubjectTerms Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science; control theory; systems
distributed-memory
Exact sciences and technology
LogP model
Mathematics
matrix computation
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
numerical methods
parallel algorithmsand complexity
Sciences and techniques of general use
Theoretical computing
triangular solvers
Title On designing optimal parallel triangular solvers
URI https://dx.doi.org/10.1006/inco.2000.2866
Volume 161
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