• Using normalized iteration vectors the shape of the iteration space changes as shown below
  

  	The dependence distance are (0,1) and (1,-1)
  	This dependence prevents loop interchange

• If normalization can prevent interchange then un-normalization can enable loop interchange
• This is called loop skewing
• Skewing changes the iteration vector of each iteration by adding the outer loop index value to the inner loop index
  • (I,j) becomes (I, j+i)
  • A dependence relation from (i1, j1) to (i2, j2) will have distance (i1, j1) –(i2-j2) = (d1, d2)
  • After skewing the distance will change to (i1, j1+i1) –(i2, j2+i2) = (d1, d2+d1)
• In general loops can be skewed by a factor changing iteration label from (I,j) to (I, j+fi)
  • This changes distance from (d1, d2) to (d1, d2+fd1)
  • F can also be negative
• Choosing whether to skew and the factor by which to skew depends upon the goal to enable other transformations

Example

• Interchange following loop using skewing
  

  for I = 2, n for j = 2, m A[I,j] = 0.5 * (A[i-1, j-1]+A[i-1, j+1]) endfor Endfor

• The two dependence distances are (1,1) and (1,-1)
• The second one prevents the interchange
• Skewing the loop would change the dependence distance to (1,2) and (1,0) allowing the interchange
• The compiler must generate the correct limits using FM method