Download Cycle Shrinking - Program Optimization for Multi Core Architectures - Lecture Slides and more Slides Computer Science in PDF only on Docsity!
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
The Lecture Contains:
Cycle Shrinking …
Cycle Shrinking in Distance Varying Loops
Loop Peeling
Index Set Splitting
Loop Fusion
Loop Fission
Loop Reversal
Loop Skewing
Iteration Space of The Loop
Example
The Final Code After Skewing is
Loop Blocking or Strip Mining
Loop Tiling
The Tiled Iteration Space
Circular Loop Skewing
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
Cycle Shrinking …
Dependence cycle with distance > 1 Transform a serial loop into two nested loops (outer serial and inner parallel) Consider the loop
for I = 1,n A[i+k] = B[i] – B[i+k] = A[i] + C[i] endfor
for i=1, n, k forall j=1, i+k- A[j+k]=B[j]- B[j+k]=A[j]+C[j] endforall endfor
For I = 3,n A[i]=B[i-2]- B[i]=A[i-3]*k Endfor
For j =3, n, 2 forall I = j, j+ A[i]=B[i-2]- B[i]=A[i-3]*k endforall Endfor
A3 = B1 -
B3 = A0 * k A4 = B2 - B4 = A1 * k A5 = B3 - B5 = A2 * k A6 = B4 - B6 = A3 * k A7 = B5 - B7 = A4 * k A8 = B6 - B8 = A5 * k
Cycle Shrinking in Distance Varying Loops
The distance may not be constant Cycle may be reduced by the minimum distance
For I = 1,n X[i]=Y[i]+Z[i] Y[i+3]=X[i-4]*W[i] Endfor
for j=1, n, 3 forall I = j, j+ X[i]=Y[i]+Z[i] Y[i+3]=X[i-4]*W[i] endforall endfor
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
Loop Fusion
When two adjacent countable loops have the same loop limits they can sometimes be fused Reduces cost of test and branch Fusing loops which refer to the same data enhances temporal locality It has significant impact on cache and virtual memory performance Loop fusion may increase size of the loop which can reduce instruction locality (noticeable with very small cache memories) Fusion is legal if all the dependence relations are preserved Before fusion all relations must flow from body1 to body2 (unless carried by an outer loop)
For I = 1,n A[i]=B[i]+ Endfor For I = 1,n C[i]=A[i]/ Endfor For I = 1,n D[i]=1/C[i+1] Endfor S2 S S5 S
For I = 1,n A[i]=B[i]+ C[i]=A[i]/ D[i]=1/C[i+1] Endfor
after fusion the second dependence is violated
For I = 1,n A[i]=B[i]+ C[i]=A[i]/ Endfor For I = 1,n D[i]=1/C[i+1] Endfor
for I = 1, A[i]=B[i]+ Endfor for I = 1, C[i]=A[i+1]* Endfor
A[1]=B[1]+
for I = 2, A[i]=B[i]+ Endfor for I = 1, C[i]=A[i+1]* 2 Endfor
A[1]=B[1]+
for j = 0, A[j+2]=B[j+2]+ C[j+1]=A[j+2]* Endfor
Loop Fission
A single loop may be broken into smaller loops (inverse of loop fusion) Used on machines which have very small instruction cache Improves memory locality Construct a statement level dependence graph of the body of the loop Dependence relations carried by outer loop need not be preserved Inner loops are treated as single nodes If there are no cycles then loop fission can divide the loop into separate loops around each node The loops are ordered in topological order of the dependence graph
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
For I = 1,n A[i] = A[i] + B[i-1] B[i] = C[i-1]*x + y C[i] = 1/B[i] D[i] = sqrt(C[i]) endfor
For ib = 0,n- B[ib+1] = C[ib]*x + y C[ib+1] = 1/B[ib+1] Endfor For ib = 0,n- A[ib+1] = A[ib+1] + B[ib] Endfor For ib = 0,n- D[ib+1] = sqrt(C[i]) Endfor I = n+
Loop Reversal
Compiler can decide to run a loop backward Always legal for parallel loops Illegal for sequential loop if it has loop carried dependence Allows loop fusion to proceed where it might otherwise fail
for I = 1,n A[i]=B[i]+ C[i]=A[i]/ endfor for i=1,n D[i]=1/C[i+1] endfor
for i=n downto 1 A[i]=B[i]+ C[i]=A[i]/ D[i]=1/C[i+1] endfor
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
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
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
The Final Code After Skewing is
For js = 2, n+m- for is = max(0, js-m+2), min(n-2, js) I = is+ j = js-is+ A[I,j] = 0.5 * (A[i-1, j-1] + A[i-1, j+1]) endfor endfor
Loop Blocking or Strip Mining
Creates doubly nested loops out of single loops Organizes computation into chunks of approximately equal sizes Used to overcome size limitations of caches and local memory
for I = 1,n A[i] = B[i] + C[i] endfor
for j = 1, n, k for I = j, min(j+k, n) A[i]=B[i]+C[i] endfor endfor
Example
for I = 1, 16 A[i+3]=A[i]+B[i] endfor
for It = 1, 16, 5 for i=it, min(16, it+4) A[i+3]=A[i]+B[i] endfor endfor
file:///D|/...ry,%20Dr.%20Sanjeev%20K%20Aggrwal%20&%20Dr.%20Rajat%20Moona/Multi-core_Architecture/lecture%2036/36_10.htm[6/14/2012 12:13:25 PM]
Module 18: Loop Optimizations
Lecture 36: Cycle Shrinking
Interchange jt loop with I loop Compiler finds the new lower limits for jt For It = -15, 45, 20 for jt = max(-15,it), 45, 20 for i = max(I,it), min(50,it+19) for j=max(I,jt), min(60, jt+19) A[I,j] = A[I,j]+ endfor endfor endfor endfor
Circular Loop skewing
A variation of loop skewing Skew the inner loop iterations such that they wrap around a cylinder The shape of the iteration space does not change but the relative positions change Backward dependencies with large distances make tiling unprofitable Circular loop skewing shortens backward dependencies
For I = 0, n- for j = 0, n- A[i] = A[i] + B[i] * C[i] endfor endfor
The circular loop skewing does not change the shape of the iteration space. It changes the iterations computed at each point.
