## On Optimizing A Class Of Multi-Dimensional Loops With Reductions For Parallel Execution (1997)

Venue: | Parallel Processing Letters |

Citations: | 29 - 23 self |

### BibTeX

@ARTICLE{Lam97onoptimizing,

author = {Chi-Chung Lam and P. Sadayappan and Rephael Wenger},

title = {On Optimizing A Class Of Multi-Dimensional Loops With Reductions For Parallel Execution},

journal = {Parallel Processing Letters},

year = {1997},

volume = {7},

pages = {157--168}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper addresses the compile-time optimization of a form of nested-loop computation that is motivated by a computational physics application. The computations involve multi-dimensional surface and volume integrals where the integrand is a product of a number of array terms. Besides the issue of optimal distribution of the arrays among the processors, there is also scope for reordering of the operations using the commutativity and associativity properties of addition and multiplication, and the application of the distributive law to significantly reduce the number of operations executed. A formalization of the operation minimization problem and proof of its NPcompleteness is provided. A pruning search strategy for determination of an optimal form is developed. An analysis of the communication requirements and a polynomial-time algorithm for determination of optimal distribution of the arrays are also provided. Keywords: loop parallelization, operation minimization, communication op...

### Citations

11864 |
Computers and Intractability, a Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1991
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716 |
High Performance Compilers for Parallel Computing
- Wolfe
- 1996
(Show Context)
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100 | M c Kinley. Optimizing for parallelism and data locality
- Kennedy, S
- 1992
(Show Context)
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92 |
Automatic array alignment in data-parallel programs
- Chatterjee, Gilbert, et al.
- 1993
(Show Context)
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90 | Complete Register Allocation Problems
- Sethi
- 1975
(Show Context)
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54 |
Arithmetic Complexity of Computations
- Winograd
- 1980
(Show Context)
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53 |
Code generation for expressions with common subexpressions
- Aho, Johnson, et al.
- 1977
(Show Context)
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50 | Optimal Evaluation of Array Expressions on Massively Parallel Machines
- Chatterjee, Gilbert, et al.
- 1995
(Show Context)
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32 |
Code generation for a one-register machine
- Bruno, Sethi
- 1976
(Show Context)
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12 |
Parallel Implementation of Quasiparticle Calculations of Semiconductors and Insulators
- Aulbur
- 1996
(Show Context)
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11 |
A Data Locality Algorithm
- Wolf, Lam
- 1991
(Show Context)
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7 |
Solving bigger problems ] By decreasing the operation count and increasing the computation bandwidth
- Miller
(Show Context)
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3 |
Space-Time Method for Ab-Initio
- Rojas, Godby, et al.
- 1995
(Show Context)
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2 |
Leblanc Jr. Crafting a Compiler. Menlo Park
- Fischer, J
- 1991
(Show Context)
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