## A characterization of one-to-one modular mappings (1996)

Venue: | PARALLEL PROCESSING LETTERS |

Citations: | 8 - 7 self |

### BibTeX

@INPROCEEDINGS{Darte96acharacterization,

author = {Alain Darte and Michele Dion and Yves Robert},

title = {A characterization of one-to-one modular mappings},

booktitle = {PARALLEL PROCESSING LETTERS},

year = {1996},

pages = {5--1},

publisher = {Parallel}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we deal with modular mappings as introduced by LeeandFortes [14, 13, 12], and we build upon their results. Our main contribution is a characterization of one-to-one modular mappings that is valid even when the source domain and the target domain of the transformation have the same size but not the same shape. This characterization is constructive, and a procedure to test the injectivity of a given transformation is presented.

### Citations

402 | The High Performance Fortran Handbook - Koelbel, Loveman, et al. - 1994 |

389 |
A Loop Transformation Theory and an Algorithm to Maximize Parallelism
- Wolf, Lam
- 1991
(Show Context)
Citation Context ...dulo a constant vector. A ne mappings are timespace transformations that have been used extensively by avariety of researchers to derive e cient time-space transformations for loop nest programs (see =-=[1, 17, 6,5,7,8,20,11, 15, 19, 21]-=- among others). However, the systematic derivation of programs that can take advantage of wraparound connectivity in networks such as rings and 2D- or 3D-torus remains out of the scope of a ne mapping... |

245 | Global optimizations for parallelism and locality on scalable parallel machines
- Anderson, Lam
- 1993
(Show Context)
Citation Context ...dulo a constant vector. A ne mappings are timespace transformations that have been used extensively by avariety of researchers to derive e cient time-space transformations for loop nest programs (see =-=[1, 17, 6,5,7,8,20,11, 15, 19, 21]-=- among others). However, the systematic derivation of programs that can take advantage of wraparound connectivity in networks such as rings and 2D- or 3D-torus remains out of the scope of a ne mapping... |

171 |
Integral Matrices
- Newman
- 1972
(Show Context)
Citation Context ...diag( d gcd(d�si) ). The matrix Q2 (Q 2p) i = d gcd(d� si) k00 i p = Q 2 ;1 S 0 k 00 ;1 S 0 generates S 0 . Besides, if we letS( ) = diag( 0 i) and S( )= diag( 0 0 0 i), we know that i n;i+1 = d (see =-=[16]-=- 0 p.40) and that i divides si (if A and B are two nonsingular integer n n matrices, then the k-th element sk(AB) of the Smith normal form of AB is divisible by sk(A) andsk(B), see [16] p.33). 0 i div... |

152 |
VLSI Array Processors
- Kung
- 1988
(Show Context)
Citation Context ...dulo a constant vector. A ne mappings are timespace transformations that have been used extensively by avariety of researchers to derive e cient time-space transformations for loop nest programs (see =-=[1, 17, 6,5,7,8,20,11, 15, 19, 21]-=- among others). However, the systematic derivation of programs that can take advantage of wraparound connectivity in networks such as rings and 2D- or 3D-torus remains out of the scope of a ne mapping... |

130 |
The MasPar MP-1 architecture
- BLANK
- 1990
(Show Context)
Citation Context ...sing one-to-one modular mappings. We point out that many other BLAS3-like kernels have been implemented onto 2D processor meshes using wraparound connections (e.g. the scienti c library of the MasPar =-=[2, 4]-=-). We refer to Section 2 for the automatic synthesis of Cannon's algorithm using modular mappings, thereby providing the reader with a complete example to demonstrate the usefulness of modular mapping... |

69 |
Time Optimal Linear Schedules for Algorithms with Uniform Dependencies
- Shang, Fortes
- 1991
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65 | Constructive Methods for Scheduling Uniform Loop Nests
- Darte, Robert
- 1994
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53 | Toward automatic distribution
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39 |
Systolic algorithms and architectures
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- 1990
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37 | Mapping uniform loop nests onto distributed memory architectures
- Darte, Robert
- 1994
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31 |
Software to support massively parallel computing on the MasPar MP-1
- Christy
(Show Context)
Citation Context ...sing one-to-one modular mappings. We point out that many other BLAS3-like kernels have been implemented onto 2D processor meshes using wraparound connections (e.g. the scienti c library of the MasPar =-=[2, 4]-=-). We refer to Section 2 for the automatic synthesis of Cannon's algorithm using modular mappings, thereby providing the reader with a complete example to demonstrate the usefulness of modular mapping... |

30 |
einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem Wurfelgitter
- Hajós, ‘Uber
- 1941
(Show Context)
Citation Context ...m S 1 Sk to G is one-to-one. If G =[g 1]k1 [gr]kr then at least one of the [gi]ki is a subgroup of G. Proof This result has been proved by Hajos in its works on one of the Minkowski's conjecture, see =-=[9]-=-. Lemma 5 If Tb is a one-to-one modular mapping on Jb, then 8x 2 Z n ,there exists (x 1�x 2) 2 S 0 Jb such that x = x 1 + x 2, and this decomposition is unique. Proof We rst prove the existence of suc... |

19 | Data alignment: Transformations to reduce communication on distributed memory architectures
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14 |
Some e cient solutions to the a ne scheduling problem, I, one dimensional time
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8 |
On the injectivity of modular mappings
- Lee, Fortes
- 1994
(Show Context)
Citation Context ...CONTINUE Cannon's algorithm (except data movement) can therefore be described by a modular transformation applied to the original program. We refer the reader to the original papers of Lee and Fortes =-=[14, 13]-=- for several interesting variants of this standard parallelization, as well as for a method to derive data communications. 3 Review of Lee and Fortes results 3.1 De nitions In this section, we use the... |

6 |
Area-Time Optimal VLSI Networks for Multiplying Matrices
- Preparata, Vuillemin
- 1980
(Show Context)
Citation Context ...pical example is Cannon's matrix-matrix product algorithm on a 2D-torus of processors [3]: this well-known algorithm (whose counterpart in the systolic eld is the Preparata-Vuillemin 2Dsystolic array =-=[18]-=-) cannot be synthesized using a ne transformations, whereas Lee and Fortes [14, 13] demonstrate how tosynthesize it, as well as many interesting variants, using one-to-one modular mappings. We point o... |

3 |
Acellular computer to implement the Kalman lter algorithm
- Cannon
- 1969
(Show Context)
Citation Context ...aparound connectivity in networks such as rings and 2D- or 3D-torus remains out of the scope of a ne mappings. Atypical example is Cannon's matrix-matrix product algorithm on a 2D-torus of processors =-=[3]-=-: this well-known algorithm (whose counterpart in the systolic eld is the Preparata-Vuillemin 2Dsystolic array [18]) cannot be synthesized using a ne transformations, whereas Lee and Fortes [14, 13] d... |

2 |
Modular mappings of rectangular algorithms
- Lee, Fortes
- 1994
(Show Context)
Citation Context ...p@lip.ens−lyon.frsA Characterization of One-to-One Modular Mappings Alain Darte Michele Dion Yves Robert April 1995 Abstract In this paper, we deal with modular mappings as introduced by LeeandFortes =-=[14, 13, 12]-=-, and we build upon their results. Our main contribution is a characterization of one-to-one modular mappings that is valid even when the source domain and the target domain of the transformation have... |

1 |
and Jose A.B.Fortes. Data distribution independent parallel programs for matrix multiplication
- Lee
- 1994
(Show Context)
Citation Context ...p@lip.ens−lyon.frsA Characterization of One-to-One Modular Mappings Alain Darte Michele Dion Yves Robert April 1995 Abstract In this paper, we deal with modular mappings as introduced by LeeandFortes =-=[14, 13, 12]-=-, and we build upon their results. Our main contribution is a characterization of one-to-one modular mappings that is valid even when the source domain and the target domain of the transformation have... |

1 | Partitioning and mapping algorithms into xed-size systolic arrays - Moldovan, Fortes - 1986 |