Tricks of the trade: loop transformations

Transcription

Tricks of the trade: From loop
transformations to automatic
optimization
By: Maurice Peemen
5KK73 Embedded Computer Architecture
6-12-2013
Application Excecution Time
1
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High performance computing
Embedded computing
Small portions of the code dominate execution time
for(i=0; i<N; i++){
Hot spots
for(j=0; j<N; j++){
Loops
for(k=0; k<N; k++){
c[i][j] = a[i][k] + b[k][j];
We can do something
}}}
Transformations
Loop Transformations
2
• Can help you in many compute domains
Convert Loops to
GPU Threads
Embedded Devices
Fine-tune for architecture
CPU Code
Parallelism
Memory Hierarchy
HLS languages for HW
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Loop Transformation Classes
3
• Data-Flow-Based Transformations
• Iteration Reordering Transformations
• Loop Restructuring Transformations
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Data-Flow-Based Transformations
4
• Very useful:
• Simple compiler
• Simple architecture not Out-Of-Order Superscalar,
Speculative Branch Prediction
• If you are in a hurry and missing a few percent
• These are the easy tricks
• So we use them to warm-up for the more difficult
* Examples are from:
D.F. Bacon, S.L. Graham, and O.J. Sharp, Compiler transformations for
high-performance computing. ACM Comput. Surv. 26, (1994)
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Loop-Based Strength Reduction
5
•
•
•
•
Replace an expensive operation
Can be multiplication, especially for embedded cores
No HW multiplier, e.g. MicroBlaze
An optimizing compiler can help
c=3;
for(i=0; i<N; i++){
a[i] = a[i] + i*c;
}
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c=3;
T=0;
for(i=0; i<N; i++){
a[i] = a[i] + T;
T = T + c;
}
Induction Variable Elimination
6
• Induction variable
• Compute loop exit conditions
• Compute memory addresses
• Given a known final value and relation to addresses
• Possible to eliminate index calculations
• Again an optimizing compiler can help
for(i=0; i<N; i++){
a[i] = a[i] + c;
}
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A = &a[0];
T = &a[0] + N;
while(A<T){
*A = *A + c;
A++;
}
Loop-Invariant Code Motion
7
•
•
•
•
•
Index of c[] does not change
Move one loop up and store in register
Reuse it over iterations of inner loop
Most compilers can perform this
Example is similar to
Scalar Replacement Transformation
for(i=0; i<N; i++){
for(j=0; j<N; j++){
a[i][j]=b[j][i]+c[i];
}
}
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for(i=0; i<N; i++){
T=c[i];
for(j=0; j<N; j++){
a[i][j]=b[j][i]+T;
}
}
Loop Unswitching
8
• Conditional execution in body
• Speculative branch prediction helps but
• This branch is loop invariant code
• Move it outside the loop
• Each execution path independent loops
• Increases code size
for(i=0; i<N; i++){
a[i] = b[i];
if(x<7){a[i]+=5;}
}
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if(x<7){
for(i=0; i<N; i++){
a[i]= b[i];
a[i]+=5;
}
}
else{
for(i=0; i<N; i++){
a[i]=b[i];
}
}
9
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Iteration Reordering Transformations
10
• Change the relative ordering
• Expose parallelism by altering dependencies
• Improve locality by matching with a memory hierarchy
•
•
•
•
We need more theory to understand these
Or combine the tricks to form a methodology
Normal compilers don’t help you with this
Only state-of-the-art compilers can help
• Often research compilers ;)
*Based on:
M. Wolf, M. Lam, A Data Locality Optimizing Algorithm. ACM SIGPLAN,
PLDI (1991)
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Loop representation as Iteration Space
11
for(i=0; i<N; i++){
for(j=0; j<N; j++){
A[i][j] = B[j][i];
i
• Each position represents an iteration
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j
Visitation order in Iteration Space
12
for(i=0; i<N; i++){
for(j=0; j<N; j++){
A[i][j] = B[j][i];
i
• Iteration space is not data space
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j
But we don’t have to be regular
13
A[j][j] = B[j][i];
•
•
•
•
i
But order can have huge impact
E.g. Reuse of data
Parallelism
Loop or control overhead
j
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Types of data reuse
14
for i := 0 to 2
for j := 0 to 100
A[i][j] = B[j+1][0] + B[j][0];
i
A[i][j]
i
j
Spatial
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B[j+1][0]
j
Temporal
i
hit
miss
B[j][0]
j
Group
(temporal)
When do cache misses occur?
15
for(i=0; i<N; i++){
for(j=0; j<N; j++){
A[i][j] = B[j][i];
i
A
i
j
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hit
miss
B
j
Optimize array accesses for memory hierarchy
16
• Solve the following questions?
• When do misses occur?
− Use “locality analysis”
• Is it possible to change iteration order to produce better
behavior?
− Evaluate the cost of various alternatives
• Does the new ordering produce correct results?
− Use “dependence analysis”
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Different Reordering Transformations
17
• Loop Interchange
• Tiling
Can improve locality
• Skewing
• Loop Reversal
Can enable above
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Loop Interchange or Loop Permutation
18
for(i=0; i<N; i++){
for(j=0; j<N; j++){
A[j][i] = i*j;
i
for(j=0; j<N; j++){
for(i=0; i<N; i++){
A[j][i] = i*j;
j
j
• N is large with respect to cache size
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hit
miss
i
Tiling the Iteration Order
19
• Result in Iteration Space
for(i=0; i<N; i++){
for(j=0; j<N; j++){
f(A[i],A[j]);
i
i
j
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for(jj=0; jj<N; jj+=Tj){
for(i=0; i<N; i++){
for(j=jj; j<jj+Tj; j++){
f(A[i],A[j]);
j
Cache Effects of Tiling
20
for(jj=0; jj<N; jj+=Tj){
for(i=0; i<N; i++){
for(j=jj; j<jj+Tj; j++){
f(A[i],A[j]);
for(i=0; i<N; i++){
for(j=0; j<N; j++){
f(A[i],A[j]);
i
A[i]
i
j
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A[j]
i
j
A[i]
i
j
hit
miss
A[j]
j
Is this always possible?
21
• No, dependencies can prevent transformations
• Interchange and Tiling is not always valid!
• Example Interchange
• Spatial locality
• Dependency (1,-1)
• Should be lexicographically
correct
for(i=1; i<N; i++){
for(j=0; j<N; j++){
a[j][i] = a[j+1][i-1] + 1;
}
}
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i
j
Lexicographical ordering
22
• Each iteration is a node identified by its index vector:
p = (p1, p2, …, pn)
• Iterations are executed in lexicographic order
• For p = (p1, p2, …, pn) and q = (q1, q2, …, qn)
• p is lexicographically greater than q written as p q
• Therefore iteration p must be executed after q
• Example:
(1,1,1), (1,1,2), (1,1,3), …,
(1,2,1), (1,2,2), (1,2,3), …,
…,
(2,1,1), (2,1,2), (2,1,3), …,
• (1,2,1) (1,1,2), (2,1,1) (1,4,2)
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Dependence vectors
23
• Dependence vector in an n-nested loop is denoted as
a vector: d = (d1, d2, …, dn)
• A single dependence vector represents a set of
distance vectors
• A distance vector defines a distance in the iteration
space
• Distance vector is difference between target and
source iterations:
• d = I T – IS
• The distance of the dependence is:
• IS + d = I T
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Example of dependence vector
24
For(i=0; i<N; i++){
for(j=0; j<N; j++)
A[i,j] = …;
= A[i,j];
B[i,j+1] = …;
= B[i,j];
C[i+1,j] = …;
= C[i,j+1];
0 
A yields:  
0 
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A2,0= =A2,0 A2,1= =A2,1 A2,2= =A2,2
B2,1= =B2,0 B2,2= =B2,1 B2,3= =B2,2
C3,0= =C2,1 C3,1= =C2,2 C3,2= =C2,3
i A1,0= =A1,0 A1,1= =A1,1 A1,2= =A1,2
B1,1= =B1,0 B1,2= =B1,1 B1,3= =B1,2
C2,0= =C1,1 C2,1= =C1,2 C2,2= =C1,3
A0,0= =A0,0 A0,1= =A0,1 A0,2= =A0,2
B0,1= =B0,0 B0,2= =B0,1 B0,3= =B0,2
C1,0= =C0,1 C1,1= =C0,2 C1,2= =C0,3
0 
B yields:  
1 
j
1
C yields: 
 1
Plausible dependence vectors
25
• A dependence vector is plausible iff it is
lexicographically non-negative
• Plausible: (1,-1)
• Implausible: (-1,0)
i
(0,-1)
(1,-1)
(-1,0)
(0,1)
j
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Loop Reversal
26
• Dependency (1,-1)
Dependency (1,1)
for(i=0; i<N; i++){
for(i=0; i<N; i++){
for(j=N-1; j>=0; j--){
for(j=0; j<N; j++){
a[j][i] = a[j+1][i-1] + 1;
a[j][i] = a[j+1][i-1] + 1;
}
}
}
}
i
• Interchange is valid now
j
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More difficult example
27
For(i=0; i<6; i++){
D={(0,1),(1,0),(1,-1)}
for(j=0; j<7; j++){
A[j+1] += 1/3 * (A[j] + A[j+1] + A[j+2]);
i
• What are other legal
visit orders?
j
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Loop Skewing
28
for(i=0 i<6; i++){
D = {(0,1),(1,0),(1,-1)}
for(j=0; j<7; j++){
A[j+1] = 1/3 * (A[j] + A[j+1] + A[j+2]);
i
i
j
j
for(i=0; i<6; i++){
D = {(0,1),(1,1),(1,0)}
for(j=i; j<7+i; j++){
A[j-i+1] = 1/3 * (A[j-i] + A[j-i+1] + A[j-i+2]);
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Enables Tiling
29
• Apply Skew Transformation
• Apply Tiling
i
j
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Is It Really That Easy ?
30
• No, we forget a few things
• Cache capacity and reuse distance
• If caches were infinitely large, we would be finished
• For deep loopnests analisis can be very difficult
• It would be much better to automate these steps
• We need more theory and tools
Cost
models
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Reuse distance
31
• The number of unique accesses between a use and a
reuse is defined as the reuse distance
[a[0] a[6] a[20] b[1] a[6] a[0]] RD = 3
* C. Ding, Y. Zhong, Predicting Whole-Program Locality through Reuse
Distance Analysis, PLDI (2003)
•
•
•
•
If fully associative cache with n=4 entries
Optimal least recently used replacement policy
Cache hit if RD < n
Cache miss if RD ≥ n
• We can measure reuse distance with profiling tools:
• Suggestions for Locality Optimization (SLO)
Beyls, K., D'Hollander, E. Refactoring for Data Locality, IEEE Computer
Vol. 42, no. 2 2009
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Study a matrix multiplication kernel
32
• Loop interchange
for(j=0; j<Bj; j++){
for(i=0; i<Bi; i++){
for(k=0; k<Bk; k++){
C[i][j] += A[i][k] * B[k][j];
C[i][j] += A[i][k] * B[k][j];
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Loop tiling
33
for(jj=0; jj<Bj; jj+=Tj){
C[i][j] += A[i][k] * B[k][j];
for(j=jj; j<jj+Tj; j++)
C[i][j] += A[i][k] * B[k][j];
• Can be done in multiple dimensions
• Manual exploration is intractable
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A Practical Intermezzo
34
• Accelerators for energy efficiency
• Special Purpose HW
• ~100x better performance, 100x less energy than CPU
• A simple processor for control
• Multiple accelerators for heavy compute workload
• Easy to develop with current HLS languages
External RAM
Host Processor
Accelerator
1
Accelerator
2
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Accelerator
3
Bandwidht Hunger
35
• Simple RISC core
• Accelerator Operations
• Interconnect Resources
External RAM
Aethereal Interconnect 6 cores
Host Processor
Accelerator
1
Accelerator
3
Accelerator
2
ld A
ld B
A+B
ld C
Processor
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A<<4
A * C A<<4
ld A
ld B
A+B
Accelerator
ld C
A*C
A<<4
What can we do?
36
• Exploit the data reuse in accelerator workload
• Use local buffers
• Reduce communication with external memory
External RAM
Host Processor
Accelerator
2
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+
Local
Buffer
acc
+
Accelerator
3
*
Accelerator
1
*
Data path
acc
A Different Approach: Inter-Tile Reuse
37
CPU
DDR
• But how about
DATA OVERLAP?
• Accelerator workload is
often static
• We know the contents of
the next tile
• Optimize Iteration Order
• Formulate the ordering as
optimization problem
Optimal
Tile IN
Local
Memory
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Optimal
Tile Out
Accelerator
Compute
Demosaic Benchmark
38
• Convert an 8 Mpix image: 0.6 GMAC
•
•
•
•
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Intel-i7
ARM-A9
Microblaze
Accelerator
0.56 s
5.75 s
22.10 s
1.35 s @ 50 MB/s
Demosaic Benchmark
39
•
•
•
•
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Intel-i7
ARM-A9
Microblaze
Accelerator
0.56 s
5.75 s
22.10 s
1.35 s @ 50 MB/s
Demosaic Benchmark
40
•
•
•
•
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Intel-i7
ARM-A9
Microblaze
Accelerator
0.56 s
5.75 s
22.10 s
1.35 s @ 50 MB/s
Motion Estimation
41
• H264 video encoder ME ~ 30 Gops
for(frame){
for(macroblocks){
for(reference frame){
for(searchrange){
for(pixel){
SAD_operation();
for(find max){
Store vector();
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Block Matching
42
•
•
•
•
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Intel-i7
ARM A9
Microblaze
Accelerator
8.1 sec
72.3 sec
283.9 sec
3.4 sec @ 50MB/s
Block Matching
43
•
•
•
•
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Intel-i7
ARM A9
Microblaze
Accelerator
8.1 sec
72.3 sec
283.9 sec
3.4 sec @ 50MB/s
What do you we achieve
44
Host Processor
Cost
models
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Local
Buffer
*
Accelerator
2
Data path
+
Accelerator
3
acc
+
Accelerator
1
*
External RAM
acc
45
• Alter the form of the loop
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Loop Unrolling
46
• Duplicate loop body and adjus loop header
• Increases ILP
• Reduces loop overhead
• Possibilities for common subexpression elimination
• If partial unroll, loop should stay in original bound!
• Downsides
• Code size increases, can increase I-cache miss-rate
• Global determination is difficult
for(i=1; i<N; i++){
a[i]=a[i-1]+a[i]+a[i+1];
}
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for(i=2; i<N-1; i+=2){
a[i]=a[i-1]+a[i]+a[i+1];
a[i+1]=a[i]+a[i+1]+a[i+2];
}
if(mod(N-1,2)==1){
a[N-1]=a[N-2]+a[N-1]+a[N];
Loop Merge or Fusion
47
• Why?
• Improve locality
• Reduce loop overhead
for(i=0; i<N; i++){
b[i]=a[i]+1;
}
for(j=0; j<N; j++){
c[j]=b[j]+a[j];
}
Location
Production/Consumption
Consumption(s)
Time
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for(i=0; i<N; i++){
b[i]=a[i]+1;
c[j]=b[j]+a[j];
}
Location
Production/Consumption
Consumption(s)
Time
Merging is not always allowed
48
• Data dependencies from first to second loop
• Allowed if
•  I: cons(I) in loop 2  prod(I) in loop 1
• Enablers: Bump, Reverse, Skew
for(i=0; i<N; i++){
b[i]=a[i]+1;
}
N-1>=i
for(i=0; i<N; i++){
c[i]=b[N-1]+a[i];
}
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for(i=0; i<N; i++){
b[i]=a[i]+1;
}
i-2 < i
for(i=0; i<N; i++){
c[i]=b[i-2]+a[i];
}
Loop Bump: Example as enabler
49
for(i=2; i<N; i++){
b[i]=a[i]+1;
}
for(i=0; i<N-2; i++){
c[i]=b[i+2]*0.5;
}
i+2 > i => direct merging not
possible
for(i=2; i<N; i++){
b[i]=a[i]+1;
}
for(i=2; i<N; i++){
c[i-2]=b[i+2-2]*0.5;
}
for(i=2; i<N; i++){
b[i]=a[i]+1;
c[i-2]=b[i]*0.5;
}
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i+2-2 = i =>
possible to merge
Automatic frameworks
50
• These are the tools to abstract away from code:
• Model reuse (potential for locality)
• Local iteration space
• Transform loops to take advantage of reuse
• Does this still give valid results?
• Loop transformation theory
• Iteration space
• Dependence vectors
• Unimodular transformations
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Uniformly generated references
51
• f() and g() are indexing functions:
n

d
• n is depth of loop nest
• d is dimensions of array, A
• Two references A[f(i)] and A[g(i)] are uniformly
generated if
f(i) = Hi + cf and g(i) = Hi + cg
• H is a linear transform
• cf and cg are constant vectors
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Example: detect reuse and factor
52
for i = 0 to 5
for j = 0 to 6
A[j+1] = 1/3 * (A[j] + A[j+1] + A[j+2]);
H
i
c
A[j+1]
i 
0 1    1
 j
A[j]
i 
0 1    0
 j
A[j+2]
i 
0 1     2
 j
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All references belong to
the same uniformly
generated set H = [0 1]
Quantify reuse
53
• Make use of vector spaces to find loops with reuse
• Convert that reuse into locality, how?
• Make the best loop the inner loop
• Metric to find the best loop
• Memory accesses / iteration of innermost loop
• No locality results in memory accesses
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Temporal reuse
54
• For reference: A[Hi+c]
• Between iteration m and n when Hm+c = Hn+c
• H(m-n) = 0
• The direction or reuse is: r = m-n
• The self-temporal reuse vector space is:
• RST = ker H
ker A = nullspace(A) = {x | Ax=0}
• Locality if RST is in the localized vector space L
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Self-temporal reuse (2)
55
• Amount of reuse: sdim(Rst)
• RST intersect L = locality
• # of memory references = 1/ s dim(RST L)
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Example of self-temporal reuse
56
for i = 1 to n
for j = 1 to n
for k = 1 to n
C[i,k] += A[i,j] * B[j,k];
Acces
reuse
L=span{(0,0,1)}
1 0 0 
C[i,k] 
span{(0,1,0)}

0 0 1 
n in j
1
1 0 0 
A[i,j] 
span{(0,0,1)}

0 1 0 
n in k
1/n
0 1 0 
B[j,k] 
span{(1,0,0)}

0 0 1 
n in I
1
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H
ker H
Next step
57
• These are the tools to abstract away from code:
• Model reuse (potential for locality)
• Local iteration space
• Transform loops to take advantage of reuse
• Does this still give valid results?
• Loop transformation theory
• Iteration space
• Dependence vectors
• Unimodular transformations
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Unimodular Transformations
58
• Interchange
• Permute the nesting order
• Reversal
• Reverse the order of iterations
• Skewing
• Scale iterations by an outer loop index
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When is loop interchange legal?
59
• Change order of loops
• For some permutation p of 1 … n
for I1 = …
for I2 = …
…
for In = …
statement;
• Use matrix notation
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for Ip(1) = …
for Ip(2) = …
…
for Ip(n) = …
statement;
Transform and matrix notation
60
• If dependences are vectors in iteration space,
transforms can be represented as matrix transforms
• For example a 2-deep loop interchange is:
0 1 
T

1
0


0 1   p1   p2 
1 0   p    p 

 2  1
• Because, T is a linear transform:
• Td is transformed dependence
0 1   d1   d 2 
1 0   d    d 

 2  1
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Reversal
61
• Reversal of the ith loop reverses its iterations
• Can be represented as:
• Diagonal matrix with ith element = -1
• Example for 2 deep loop, reversal of outer loop
 1 0 
T

0
1


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 1 0   p1    p1 
 0 1  p    p 

 2  2 
Skewing
62
• Skew loop Ij by a factor f w.r.t. loop Ii iterations
• (p1, …, pi, …, pj, …)
(p1, …, pi, …, pj + f pi, …)
• Example for a 2-level nested loop
1 0 
T

1
1


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1 0   p1   p1 
1 1   p    p  p 

 2  1
2
Check transformation legality
63
• Distance vectors give a partial order among points in
the iteration space
• A loop transform changes the order in which points
are visited
• The new visit order must respect the dependence
partial order
6-12-2013
Example
64
for i = 0 to N-1
D={(1,0)}
for j = 0 to N-1
A[i+1][j] += A[i][j]
i
j
0 1 
Loop interchange T  

1
0


Loop reversal
6-12-2013
 0 1  1   0 
1 0  0   1  ok

   
1 0 
T

0

1


1 0  1  1  ok
0 1 0   0 

   
 1 0 
T

0
1


 1 0  1   1 false
 0 1  0    0 

   
Loop nest locality optimization
65
• For a given iteration space:
• A set of dependence vectors
• Uniformly generated reference sets
• Find the unimodular and/or tiling transform, subject
to data dependences, that minimizes the number of
memory accesses per iteration
• Very hard problem exponential in the number of loops
• Simplification:
• Use vector space spanned by only elementary vectors
• Use heuristic algorithm for finding transforms
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SRP transform
66
• Prune the search: no reuse loops to outer loops
Contains no reuse
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SRP transform
67
• Make loops with reuse inner loops
Loops for
tiling
Loops for
tiling
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SRP transform
68
• Transform the reuse into locality
• Using Skew, Reversal and Permutation
FullyFullyFullypermutable
permutable
permutable
loop
loopnest
nest
loop nest
Legal to tile
6-12-2013
Tricks of the Trade
69
• Good compilers can help
•
•
•
•
High-level transformations
Compilers are often not enough
Huge gains in parallelims and locality
Be carefull with dependencies
• Similar to reordering
• Tools can help you: Check out SLO
• Optimizing source to source compilers as well
• If you have luck, these don’t support all constructions
• Check out Polyhedral compilers, e.g. POCC
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Extra Enabler Transformations
70
6-12-2013
Loop Extend
for I = exp1 to exp2
A(I)

@HC 5KK73 Embedded Computer Architecture
exp3  exp1
exp4  exp2
if Iexp1 and Iexp2
A(I)
71
Loop Extend: Example as enabler
for (i=0;
B[i] =
for (i=2;
C[i-2]
i<N; i++)
f(A[i]);
i<N+2; i++)
= g(B[i]);
for (i=0; i<N+2; i++)
if(i<N)
Loop Extend
B[i] = f(A[i]);
for (i=0; i<N+2; i++)
if(i>=2)
C[i-2] = g(B[i]);


Loop Merge
for (i=0; i<N+2; i++)
if(i<N)
B[i] = f(A[i]);
if(i>=2)
C[i-2] = g(B[i]);
72
Loop Reduce
if Iexp3 and Iexp4
A(I)

for I = max(exp1,exp3) to
min(exp2,exp4)
A(I)
73
Loop Body Split
A(I)
B(I)

A(I) must be
single-assignment:
its elements should
be written once
for Ia = exp1 to exp2
A(Ia)
for Ib = exp1 to exp2
B(Ib)
74
Loop Body Split: Example as enabler
for (i=0; i<N; i++)
Loop Body Split
A[i] = f(A[i-1]);
B[i] = g(in[i]);
for (j=0; j<N; j++)
for (i=0; i<N; i++)
C[j] = h(B[j],A[N]);
A[i] = f(A[i-1]);
for (k=0; k<N; k++)
B[k] = g(in[k]);
for (j=0; j<N; j++)
for (i=0; i<N; i++)
C[j] = h(B[j],A[N]);
A[i] = f(A[i-1]);
for (j=0; j<N; j++)
Loop Merge
B[j] = g(in[j]);
C[j] = h(B[j],A[N]);
75
Loop Reverse
A(I)
Location
Production
Consumption
Time

for I = exp2 downto exp1
Location
A(I)
Or written differently:
A(exp2-(I-exp1))
Time
76
Loop Reverse: Satisfy dependencies
for (i=0; i<N; i++)
A[i] = f(A[i-1]);
No loop-carried dependencies
allowed ! Can not reverse
the loop
A[0] = …
Enabler: data-flow
for (i=1; i<=N; i++)
transformations; exploit
A[i]= A[i-1]+ f(…);
associativity of + operator
… = A[N] …
A[N] = …
for (i=N-1; i>=0; i--)
A[i]= A[i+1]+ f(…);
… = A[0] …
77
Loop Reverse: Example as enabler
for (i=0;
B[i] =
for (i=0;
C[i] =
i<=N; i++)
f(A[i]);
i<=N; i++)
g(B[N-i]);
N–i > i  merging not possible
Loop Reverse for (i=0; i<=N; i++)

N-(N-i) = i 
B[i] = f(A[i]);
merging possible
for (i=0; i<=N; i++)
C[N-i] = g(B[N-(N-i)]);

Loop Merge for (i=0; i<=N; i++)
B[i] = f(A[i]);
C[N-i] = g(B[i]);
78
Loop Interchange
for I1 = exp1 to exp2
A(I1, I2)

A(I1, I2)
79
Loop Interchange: index traversal
j
j
Loop
Interchange
i
for(i=0; i<W; i++)
for(j=0; j<H; j++)
A[i][j] = …;
i
for(j=0; j<H; j++)
for(i=0; i<W; i++)
A[i][j] = …;
80

Tricks of the trade: loop transformations

Transcription

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