Table Of ContentLOOP PARALLELIZATION
A Book Series On
LOOP TRANSFORMATIONS FOR
RESTRUCTURING COMPILERS
Utpal Banerjee
Series Titles:
Loop Transformations for Restructuring
Compilers: The Foundations
Loop Parallelization
LOOP PARALLELIZATION
Utpal Banerjee
Intel Corporation
Loop Transformations for Restructuring Compilers
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
ISBN 978-1-4419-5141-0 ISBN 978-1-4757-5676-0 (eBook)
DOI 10.1007/978-1-4757-5676-0
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is available
from the Library of Congress.
Copyright © 1994 by Springer Science+Business Media New York
OriginaIly published by Kluwer Academic Publishers in 1994
AII rights reserved. No part of this publication may be reproduced, stored in
a retrieval system or transmitted in any form or by any means, mechanical,
photo-copying, recording, or otherwise, without the prior written permission of
the publisher, Springer Science+Business Media, LLC
Printed on acid-free pap er.
Contents
Preface xiii
Acknowledgments xvii
1 Background 1
1.1 Introduction. 1
1.2 Program Model 2
1.3 Dependence .. 9
1.4 Loop Transformation 19
1.5 Loop Parallelization . 26
2 Loop Permutations 29
2.1 Introduction . . . 29
2.2 Basic Concepts 34
2.3 Preventing Permutations 40
2.4 Parallelization by Permutation . 51
2.5 Computation of Loop Limits 55
2.6 Optimization Problems . . . 62
3 Unimodular Transformations 67
3.1 Introduction . . . . . . . . . 67
3.2 Basic Concepts ....... 73
3.3 Elementary Transformations 78
3.4 Inner Loop Parallelization 86
3.5 Outer Loop Parallelization 97
v
vi
3.6 Computation of Loop Limits. .106
4 Remainder Transformations 113
4.1 Introduction........ · 113
4.2 Single-Loop Transformation · 114
4.3 GCD Transformation . . · 124
4.4 Echelon Transformation · 134
5 Program Partitioning 147
5.1 Introduction.... · 147
5.2 Vertical Partitions · 148
5.3 Horizontal Partitions · 156
5.4 Vertical and Horizontal Parallelism · 164
5.5 Suggested Reading ........ . · 164
Bibliography 167
Index 173
List of Figures
1.1 Index space of Example 1.1. . . . . . . . . . . . . .. 6
1.2 Index space of Example 1.2. . . . . . . . . . . . . .. 8
1.3 Dependence graph of Example 1.4 and the major
weakly connected components. ............ 15
1.4 Dependence graph of Example 1.5. ........ 17
1.5 Iterations of individual loops (Example 1.6). . . . 21
1.6 Execution order for iterations of L (Example 1.6). 21
2.1 Index space of L in Example 2.1. ... 31
2.2 Index space of Lp in Example 2.1.. . . 33
2.3 Index space of (L1' L in Example 2.7. 57
2)
2.4 Index space of (L1' L in Example 2.8. 58
2)
3.1 Dependence graph for Example 3.1. . . 69
3.2 A wave through the index space of Example 3.1. 70
3.3 Index space of Lv. . . . . . . . . . . . . . 71
3.4 Index space with typical distance vectors. . 80
3.5 Index space after outer loop reversal. . . 81
3.6 Index space after inner loop reversal. . . 82
3.7 Index space after an upper loop skewing. 83
3.8 Index space after a lower loop skewing. . 85
4.1 Dependence graphs of the loop nests of Example 4.1. 117
vii
List of Tables
1.1 Iterations of individual loops (Example 1.3). . . .. 11
2.1 Permutations and direction vectors in a triple Loop. 64
2.2 Level change under a permutation in a triple Loop. 64
2.3 Direction vectors, permutations, and change in de-
pendence levels in a triple loop. . . . . . . . . . . .. 65
4.1 Values of I and (K, Y) in Example 4.1. ........ 118
ix
List of Notations
In the following, i = (i1,i2, ... ,im) and j = (jI,i2, ... ,jm) are two
vectors of size m, and 1 Sf S m.
Xf--+y A function that maps an element x of
its domain to an element y of its range . 27
u+ max( u, 0) (positive part of u) . . . . . . . . 76
u max( -u, 0) (negative part of u) .... 76
alb The exact result of division of a by b . 5
sig(i) Sign of an integer i. . . . . . 13
Z Set of all integers. . . . . . . . . . . . . 3
zm
Set of all integer m-vectors ....... 3
R Set of all real numbers . . . . . . . . . . . 3
Rm
Set of all real m-vectors ........ 3
sig(i) Sign of a vector i . . . . . . . . . . . . 13
0 Zero vector (size implied by context) 13
(i;j) Vector formed by concatenating
elements of i with elements of j ... 27
isj ir S jr for each r in 1 S r sm. . . . 4
i -<e j i =jl,i =i2, ... ,ie- =je-l,ie <je· 14
1 2 1
i-<j i -<e j for some f. 9
i >-d j -<e i .................... 14
i>-j j -< i .................... 13
det(A) Determinant of a (square) matrix A . . 67
xi
Xll
A' Transpose of a matrix A . . . . . . 98
Column r of matrix U . . . . . . . 87
Row t of matrix U. . . . . . . . 100
The m x m identity matrix. 5
An identity matrix
(size implied by context) . . 39
p A typical permutation matrix 27
U A typical unimodular matrix. 27
L A typical loop nest (L1' L2, •.. ,Lm). 2
I Index vector (II, 12, ... ,1m) of L .. 2
H(I) Body of L ....... . 2
R Index space of L .. . . . . 3
Po Lower limit vector of L 4
P Lower limit matrix of L . . 4
qo Upper limit vector of L .. 4
Q Upper limit matrix of L .. 4
d A typical distance vectors of L . . 13
N N umber of distance vectors of L . . 13
D Set of distance vectors of L . . . . 13
'D Distance matrix of L . . . . . . . . 13
A typical direction vector of L . . 13
Direction matrix of L .. . . . . . 13
L doall loop corresponding to a do loop L . 22
L' A typical mixed loop nest. . . . . . . . 22
Lp Transformed program of L defined by
a permutation matrix P. . . . . . . . . . . . . .. 35
Lv Transformed program of L defined by
a unimodular matrix U . . . . . . . . . . . . . .. 73
LG Transformed program of L defined by
the gcd transformation . . . . . . . . . . . . . . . 125
Ls Transformed program of L defined by
the echelon transformation . . . . . 136
Ls Mixed loop nest obtained from Ls. . . 139
