Processor Architectures and Program Mapping 5KK70 TU/e Henk Corporaal Jef van Meerbergen Bart Mesman Data Memory Management Part b: Loop transformations & Data Reuse

Thanks to the IMEC DTSE experts: Erik Brockmeyer IMEC, Leuven, Belgium and also Martin Palkovic, Sven Verdoolaege, Tanja van Achteren, Sven Wuytack, Arnout Vandecappelle, Miguel Miranda, Cedric Ghez, Tycho van Meeuwen, Eddy Degreef, Michel Eyckmans, Francky Catthoor, e.a.

@HC 5KK70 Platform-based Design3 DM methodology Dataflow Transformations Analysis/Preprocessing Loop/control-flow transformations Data Reuse Storage Cycle Budget Distribution Memory Allocation and Assignment Memory Layout organisation C-out C-in Address optimization

@HC 5KK70 Platform-based Design4 for (i=0; i < 8; i++) A[i] = …; for (i=0; i < 8; i++) B[7-i] = f(A[i]); Location Time Production Consumption for (i=0; i < 8; i++) A[i] = …; B[7-i] = f(A[i]); Location Time Production Consumption Locality of Reference

@HC 5KK70 Platform-based Design5 Regularity for (i=0; i < 8; i++) A[i] = …; for (i=0; i < 8; i++) B[i] = f(A[7-i]); Location Time for (i=0; i < 8; i++) A[i] = …; for (i=0; i < 8; i++) B[7-i] = f(A[i]); Location Time Production Consumption

@HC 5KK70 Platform-based Design6 for (i=0; i < 8; i++) B[i] = f1(A[i]); for (i=0; i < 8; i++) C[i] = f2(A[i]); Location Time Consumption Location Time Consumption Enabling Reuse for (i=0; i < 8; i++) B[i] = f1(A[i]); C[i] = f2(A[i]);

@HC 5KK70 Platform-based Design7 How to do these loop transformations automatically? Requires cost function Requires technique Let's introduce some terminology - iteration spaces - polytopes - ordering vector / execution order

@HC 5KK70 Platform-based Design8 01 j i Iteration space and polytopes // assume A[][] exists for (i=1; i<6; i++) { for (j=2; j<6; j++) { B[i][j] = g( A[i-1][j-2]); } --- iteration space --- consumption space --- production space --- dependency vector

@HC 5KK70 Platform-based Design9 Example with 3 polytopes A: for (i=1; i<=N; ++i) for (j=1; j<=N-i+1; ++j) a[i][j] = in[i][j] + a[i-1][j]; B: for (p=1; p<=N; ++p) b[p][1] = f( a[N-p+1][p], a[N-p][p] ); C: for (k=1; k<=N; ++k) for (l=1; l<=k; ++k) b[k][l+1] = g (b[k][l]); A B C Algorithm having 3 loops: j i k p l

@HC 5KK70 Platform-based Design10 Common iteration space for (i=1; i<=(2*N+1); ++i) for (j=1; j<=2*N; ++j) if (i>=1 && i =1 && j<=N-i+1) a[i][j] = in[i][j] + a[i-1][j]; if (i==N+1 && j>=1 && j<=N) b[j][1] = f( a[N-j+1][j], a[N-j][j] ); if (i>=N+2 && i<=2*N+1 && j>=N+1 && j<=N+k) b[i-N-1][j-N+1] = g (b[i-N-1][j-N]); j i 1 2*N+1 12*N Initial solution having a common iteration space: Bad locality Bad regularity Requires 2N memory locations Many dummy iterations Ordering vector

@HC 5KK70 Platform-based Design11 Cost function needed for automation Regularity Equal direction for dependency vectors Avoid that dependency vectors cross each other Good for storage size Temporal locality Equal length of all dependency vectors Good for storage size Good for data reuse

@HC 5KK70 Platform-based Design12 Regularity Regular Irregular

@HC 5KK70 Platform-based Design13 Bad regularity limits the ordering freedom j i 1 2*N+1 12*N Ordering freedom = 90 degrees

@HC 5KK70 Platform-based Design14 Locality estimates P C C C C P C C C C P = production C = consumption P C C C C C Dependency vector length is measure for locality Q: Which length is the best estimate? Sum{d i } Max {d i }Spanning tree didi

@HC 5KK70 Platform-based Design15 1.Affine loop transformations 1. Only geometric information is available during placement 2. Rotation, skewing, interchange, reverse 2.Polytope placement 1. Only geometric information is available during placement 2. Translation 3.Choose ordering vector Three step approach for loop transformation tool Combined transformation:

@HC 5KK70 Platform-based Design16 A: (i: 1..N):: (j: 1.. N-i+1):: a[i][j] = in[i][j] + a[i-1][j]; C: (k: 1..N):: (l: 1..k):: b[N-k+1][l+1] = g( b[N-k+1][l] ); B: (p: 1..N):: b[p][1] = f( a[N-p+1][p], a[N-p][p] ); Affine loop transformations Polytope placement Choose ordering vector Three step approach for loop transformation tool

@HC 5KK70 Platform-based Design17 Three step approach for loop transformation tool Affine loop transformations Polytope placement Choose ordering vector

@HC 5KK70 Platform-based Design18 Three step approach for loop transformation tool Affine loop transformations Polytope placement = merging loops Choose ordering vector

@HC 5KK70 Platform-based Design19 Choose optimal ordering vector Ordering Vector 1 Ordering Vector 2

@HC 5KK70 Platform-based Design20 From the Polyhedral model back to C for (j=1; j<=N; ++j) { for (i=1; i<=N-j+1; ++i) a[i][j] = in[i][j] + a[i-1][j]; b[j][1] = f( a[N-j+1][j], a[N-j][j] ); for (l=1; l<=j; ++l) b[j][l+1] = g( b[j][l] ); } Affine loop transformations Polytope placement Choose ordering vector Optimized solution having a common iteration space: Optimal locality Optimal regularity Requires 2 memory locations

@HC 5KK70 Platform-based Design21 Scanner Loop trafo - cavity detection Gauss Blur y Gauss Blur x N x M X-Y Loop Interchange N x M From N x M to N x (2GB+1) buffer size X Y N x M

@HC 5KK70 Platform-based Design22 Loop trafo- cavity (1) 1 Transform: interchange 2 Translate: merge 3 Order

@HC 5KK70 Platform-based Design23 Loop trafo- cavity (2) 1 Transform: interchange 2 Translate: merge 3 Order x-blur filter:

@HC 5KK70 Platform-based Design24 Scanner Loop trafo - cavity detection Gauss Blur y Gauss Blur x N x M · X-Y Loop Interchange N x M From N x M to N x (2GB+1) buffer size X Y N x M

@HC 5KK70 Platform-based Design25 Loop trafo- cavity (3) 2 Translate 1: 2 Translate 2: 3 Comparing different translations

@HC 5KK70 Platform-based Design26 Loop trafo- cavity (4) 3 3 Order += Combining (merging) multiple polytopes

@HC 5KK70 Platform-based Design27 Result on gauss filter for (y=0; y<M+GB; ++y) { for (x=0; x<N+GB; ++x) { if (x>=GB && x =GB && y<=M-1-GB) { gauss_x_compute = 0; for (k=-GB; k<=GB; ++k) gauss_x_compute += image_in[x+k][y]*Gauss[abs(k)]; gauss_x_image[x][y] = gauss_x_compute/tot; } else if (x<N && y<M) gauss_x_image[x][y] = 0; if (x>=GB && x =GB && (y-GB)<=M-1-GB) { gauss_xy_compute = 0; for (k=-GB; k<=GB; ++k) gauss_xy_compute += gauss_x_image[x][y-GB+k]* Gauss[abs(k)]; gauss_xy_image[x][y-GB] = gauss_xy_compute/tot; } else if (x =0 && (y-GB)<M) gauss_xy_image[x][y-GB] = 0;

@HC 5KK70 Platform-based Design28 Intermezzo Before we continue with data reuse, have a look at other loop transformations

@HC 5KK70 Platform-based Design29 DM methodology Dataflow Transformations Analysis/Preprocessing Loop/control-flow transformations Data Reuse Storage Cycle Budget Distribution Memory Allocation and Assignment Memory Layout organisation C-out C-in Address optimization

@HC 5KK70 Platform-based Design30 Layer 1 Layer 2 Layer 3 Data paths Memory hierarchy and Data reuse 1. Determines reuse candidates 2. Combine reuse candidates into reuse chains 3. If multiple access statements/array combine into reuse trees 4. Determine number of layers (if architecture is not fixed) 5. Select candidates and assign to memory layers 6. Add extra transfers between the different memory layers (for scratchpad RAM; not for caches)

@HC 5KK70 Platform-based Design31 TI example platform Register file + Core 4Kx16 dual 32x Total256Kb 1 elem in 1 cycle 16Kx16 ROM Offchip MAX: 8MBx16 SRAM/EPROM/ SDRAM/SBSRAM Vdd= 1.5 V P = unknown 8x Total64Kb 2 elem in 1 cycle 4Kx16 dual 4Kx16 dual 4Kx16 sing 4Kx16 sing 4Kx16 sing ROM (Data/program/DMA) first 3 cycles, next 2 cycles It seems this can be in parallel with the 256Kb memory Bandwidth 100M words/S Bandwidth 400M words/s Size 32kB Size 320kB ROM partition Variable size RAM partition Bandwidth 50M words/s Size 16 MB Fixed size RAM partition Bandwidth 4.8Gwords/s Size 2x16 registers Processor partition BW: 50M Word/s single port L2 L0 L1 BW: 400M Word/s dual port

@HC 5KK70 Platform-based Design32 M P = 1 Exploiting Memory Hierarchy for reduced Power: principle Processor Data Paths Register File Processor Data Paths Register File A P = 1 #A = 100% P total (before) = 100%

@HC 5KK70 Platform-based Design33 P total (before) = 100% M P = 1 A A’ P = % 5% Exploiting Memory Hierarchy for reduced Power: principle P total (after) = 100%x %x0.1+1%x1 = 3% M P = 1 A A’ P = 0.1 A’’ P = % 1% 10% Processor Data Paths Register File Processor Data Paths Register File

@HC 5KK70 Platform-based Design34 M Data reuse decision and memory hierarchy: principle Processor Data Paths Register File Processor Data Paths Register File BABA A’A’’ customized connections Customized connections in the memory subsystem to bypass the memory hierarchy and avoid the overhead.

@HC 5KK70 Platform-based Design35 Step 1: identify arrays with data reuse potential for (i=0; i<4; i++) for (j=0; j<3; j++) for (k=0; k<6; k++) … = A[i*4+k]; time copy3 copy4 copy1 copy2 Time frame 1Time frame 2Time frame 3Time frame 4 array index intra-copy reuse inter-copy reuse

@HC 5KK70 Platform-based Design36 Importance of high level cost estimate for (i=0; i<4; i++) for (j=0; j<3; j++) for (k=0; k<6; k++) … = A[i*4+k]; time copy3 copy4 copy1 copy2 Time frame 1Time frame 2Time frame 3Time frame 4 array index 6 Mk Array copies are stored in-place!

@HC 5KK70 Platform-based Design37 Step 1: determine gains Intra-copy reuse factor for (i=0; i<4; i++) for (j=0; j<3; j++) for (k=0; k<6; k++) … = A[i*4+k]; time copy3 copy4 copy1 copy2 Time frame 1Time frame 2Time frame 3Time frame 4 array index 6 Mk intra-copy reuse factor= 3 j iterator =not present so intra-copy reuse 3

@HC 5KK70 Platform-based Design38 Step 1: determine gains Inter-copy reuse factor time copy3 copy4 copy1 copy2 Time frame 1Time frame 2Time frame 3Time frame 4 array index inter-copy reuse factor = 1/(1-1/3)=3/2 6 Mk for (i=0; i<n; i++) for (j=0; j<3; j++) for (k=0; k<6; k++) … = A[i*4+k]; for (i=0; i<4; i++) for (j=0; j<3; j++) for (k=0; k<6; k++) … = A[i*4+k]; i iterator has smaller weight than k range so inter-copy reuse

@HC 5KK70 Platform-based Design39 5 Mm tf 1tf 2tf 3tf 4tf 5tf 6tf 7tf 8tf 9 Possibility for multi-level hierarchy array index time for (i=0; i<10; i++) for (j=0; j<2; j++) for (k=0; k<3; k++) for (l=0; l<3; l++) for (m=0; m<5; m++) … = A[i*15+k*5+m]; Mk 15 time frame 1time frame 2 5 Mm tf 1.1tf 1.2tf 1.3tf 1.4tf 1.5tf 1.6tf 2.1tf 2.2tf 2.3

@HC 5KK70 Platform-based Design40 Step 2: determine data reuse chains for each memory access R1(A) A A’ R1(A) A A’ R1(A) A A’ A’’ Many reuse possibilities Cost estimate needed Prune for promising ones R1(A) A

@HC 5KK70 Platform-based Design41 Cost function needs both size and number of accesses to intermediate array for (i=0; i<10; i++) for (j=0; j<2; j++) for (k=0; k<3; k++) for (l=0; l<3; l++) for (m=0; m<5; m++) … = A[i*15+k*5+m]; Gk 15 5 Gm estimate #misses from different levels for one iteration of i R1(A) 2*3*3*5 =90 A’ 3*5 =15 A’ 2*3*5 =30 estimate size # elements #misses

@HC 5KK70 Platform-based Design42 R1(A) A A’ R1(A) A A’ R1(A) A A’ A’’ R1(A) A Very simplistic power and area estimation for different data-reuse versions x y z accesses size energy

@HC 5KK70 Platform-based Design43 R1(A) A A’ A’’ for (i=0; i<10; i++) for (j=0; j<2; j++) for (k=0; k<3; k++) for (l=0; l<3; l++) for (m=0; m<5; m++) … = A[i*15+k*5+m]; Step 3: determine data reuse trees for multiple accesses R2(A) A A’ for (x=0; x<8; x++) for (y=0; y<5; y++) … = A[i*5+y];

@HC 5KK70 Platform-based Design44 R1(A) A A’ A’’ R2(A) A A’ Reuse tree A R1(A) A’ A’’ R2(A) A’ Step 3: determine data reuse trees for multiple accesses

@HC 5KK70 Platform-based Design45 Assign all data reuse trees (multiple arrays) to memory hierarchy A R1(A) A’ A’’ R2(A) A’ R1(B) B B’ B’’ B’’’ Layer 1 Layer 2 Layer 3 A R1(A) A’ A’’ R2(A) A’ R1(B) B B’ B’’’

@HC 5KK70 Platform-based Design46 Step 4: Determine number of layers Data reuse trees A Data reuse trees B Hierarchy layers Layer1 Layer2 Layer3 Foreground mem. Datapath

@HC 5KK70 Platform-based Design47 Step 5: Select and assign reuse candidates Data reuse trees Hierarchy layers hierarchy assignments FG A A 4 A 5 all

@HC 5KK70 Platform-based Design48 Step 5: All freedom in array to memory hierarchy Data reuse trees A Hierarchy layers Data reuse trees B

@HC 5KK70 Platform-based Design49 Step 5: Prune reuse graph (platform independent) Hierarchy layers Full freedom Hierarchy layers Pruned Quite some solutions never make sense

@HC 5KK70 Platform-based Design50 Step 5: Prune reuse graph further (platform dependent) Hierarchy layers Pruned FG Final solution 4 layer platform A B B' A' FG Final solution 4 layer platform

@HC 5KK70 Platform-based Design51 int in[H][W+8], out[H][W]; const int c[] = {1,0,1,2,2,1,0,1}; for (r=0; r < H; r++) for (c=0; c < W; c++) for (dc=0; dc < 8; dc++) out[r][c] += in[r][c+dc]*c[dc]; int in[H][W+8], out[H][W], buf[8]; const int c[] = {1,0,1,2,2,1,0,1}; for (r=0; r < H; r++) for (i=0; i<7; i++) buf[i]=in[r][i]; for (c=0; c < W; c++) buf[(c+7)%8] = in[r][c+7]; for (dc=0; dc < 8; dc++) out[r][c] += buf[(c+dc)%8]*c[dc]; Introducing 1D reuse buffer Reuse Factor =7intermediate level decl. additional copyinitial copyreread from buffer

@HC 5KK70 Platform-based Design52 Data Reuse on 1D horizontal convolution How to make explicit copies? init buffer reuse data new data Image NxM, traversed row order

@HC 5KK70 Platform-based Design53 Introducing line buffers for vertical filtering whole image size[N][M] set of lines [2GB+1] Why keep the whole image in that case? [N]

@HC 5KK70 Platform-based Design54 Simplified “reuse script” 1. Identify arrays with sufficient reuse potential 2. Determine reuse chains and prune these (for every array read) 3. Determine reuse trees and prune these (for every array) 4. Determine reuse graph including bypasses and prune (for entire application) 5. Determine memory hierarchy layout assignment incorporating given background memory restrictions (layers) and real-time constraints 6. Introduce copies in code: init, update, use code For scratchpad memories only For caches we need a different approach

@HC 5KK70 Platform-based Design55 Data re-use trees: cavity detector N*M N*1 3*1 image_in N*3 1*3 gauss_x N*3 3*3 gauss_xy/comp_edge N*3 1*1 N*M*3 N*M N*M*3 N*M image_out 0 N*M*8 ¸ CPU Array reads: Array write:

@HC 5KK70 Platform-based Design56 Memory hierarchy assignment: cavity detector N*M 3*1 image_in N*3 gauss_x gauss_xycomp_edgeimage_out 3*3 1*1 3*3 1*1 L2 N*M N*M*3 N*M 0 N*M*3 N*M N*M*3N*M*8 N*3 L3 L1 1MB SDRAM 16KB Cache 128 B RegFile ¸

@HC 5KK70 Platform-based Design57 Data reuse & memory hierarchy