CS553 Colorado State University
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Loop Transformations
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10/29/09
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Loop Interchange/Permutation (slide 2)

    - we have seen this transformation before
    - The end result: Will reuse A(2,j) for each iteration of the I loop

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Legality Loop Interchange/Permutation (slides 3-4)

    - recall that one direction vector can summarize an infinite number 
    of distance vectors

    - when is it legal given the direction vectors for a loop?
    
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Loop Interchange Example (slide 5)

    --> have students set up dependence problem
    
    iteration (i,j) writes C(i,j)
    iteration (i',j') reads C(i'+1, j'-1)
    
    i=i'+1
    j=j'-1
    1 <= i,j,i',j' <= n
    (i,j) != (i',j')
    
    assume anti dependence
        iteration (i,j) writes C(i,j), target
        iteration (i',j') reads C(i+1, j-1), source
    
    v = (i,j) - (i',j') = (1,-1) 
    can be represented with direction vector (<,>)
    
        
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Framework for Loop transformation (slide 6)

    - frameworks provide a unifying way to specify various loop transformations
        - can determine the space of possible transformations
            - for unimodular, all unimodular matrices that keep dependences 
            lexicographically positive are legal
            
        - can automatically generate transformed code based on 
        transformation specifications
        
        - can compose transformations

    --> What is the transformation matrix for permutation?
    
    
 
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Legality of loop permutation using unimodular framework (slide 7)

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Loop Reversal (slides 8-10)

    - what does -1 multiplied by < mean? by >?  by =?
        keep in mind that these represent a set of numbers, just take the
        negation of that set and use the appropriate representation for 
        the result

    - first do reversal of j loop
        show on ISG how it intuitively is legal
    
    - then do reversal of i loop and show that it isn't legal



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mstrout@cs.colostate.edu, 10/29/09