1. Hierarchical Design of Parallel Architectures for Signal Processing Applications Patrice Quinton, Tanguy Risset IRISA - COSI http://www.irisa.fr/cosi/ALPHA

2. July 2001Samos - 20012 Outline  What is MMAlpha?  Example of design flow  The Alpha language  Structured scheduling  Performance  Conclusion

3. July 2001Samos - 20013 What is MMAlpha?  A public domain silicon compiler for loop nests FPGAASIC VHDL for i = 1 to n do for k = 1 to m do y[i,k] = y[i,k-1] + w[i,k]*x[i-k]

4. July 2001Samos - 20014 What is MMAlpha?  User controlled design process Uniformization Scheduling/Mapping HDL derivation MMAlpha Vhdl1 Alpha Vhdl2Vhdl3  Design script easily reused

5. July 2001Samos - 20015 Target applications  Data intensive applications  Fir  Adaptive LMS  Kalman filtering  Signal processing  Motion estimators  2D-filters  Multimedia  DNA sequencing  Bio-Informatics

6. July 2001Samos - 20016 MMAlpha highlights  Compilation of loop nests to parallel circuits  by means of the polyhedral model for i = 1 to n do for k = 1 to m do y[i,k] = y[i,k-1] + w[i,k]*x[i-k]

7. July 2001Samos - 20017 MMAlpha highlights  Semi-automatic design space exploration Uniformization Scheduling/Mapping1 HDL derivation MMAlpha Alpha Scheduling/Mapping2

8. July 2001Samos - 20018 MMAlpha highlights  Hierarchical design methodology Uniformization Scheduling/Mapping HDL derivation MMAlpha Alpha

9. July 2001Samos - 20019 MMAlpha highlights  Multi-target output for codesign VHDL for i = 1 to n do for k = 1 to m do y[i,k] = y[i,k-1] + w[i,k]*x[i-k] C

10. July 2001Samos - 200110 Outline  What is MMAlpha?  Example of design flow  The Alpha language  Structured scheduling  Performance  Conclusion

11. July 2001Samos - 200111 The algorithm

12. July 2001Samos - 200112 The algorithm Alignment matrix

13. July 2001Samos - 200113 Equations Recurrence for one point of the matrix

14. July 2001Samos - 200114 Iteration space

15. July 2001Samos - 200115 Uniformization

16. July 2001Samos - 200116 Uniformization

17. July 2001Samos - 200117 Scheduling

18. July 2001Samos - 200118 Mapping

19. July 2001Samos - 200119 Architecture

20. July 2001Samos - 200120 Architecture

21. July 2001Samos - 200121 Control

22. July 2001Samos - 200122 Control

23. July 2001Samos - 200123 Outline  What is MMAlpha?  Example of design flow  The Alpha language  Structured scheduling  Performance  Conclusion

24. July 2001Samos - 200124 Alpha code for dot product b(2) b(1) b(4) b(3) a(2) a(1) a(4) a(3) k 0 c system dot:{N|1<=N} (a,b:{k|1<=k<=N} of integer) returns (c: integer); var Acc:{k|0<=k<=N} of integer; let Acc[k] = case {|k=0}:0[]; {|k>0}:Acc[k-1]+a[k]*b[k]; esac; c[]=Acc[N]; tel;

25. July 2001Samos - 200125 Matrix product using dot product {i,j,k|1<=i,j,k<=N}: A[i,j,k] = A[i,j-1,k-1]; {i,j,k|1<=i,j,k<=N}: B[i,j,k] = B[i-1,j,k-1]; use {i,j|1<=i<=N} dot[N](A,B) returns (C);

26. July 2001Samos - 200126 Outline  What is MMAlpha?  Example of design flow  The Alpha language  Structured scheduling  Performance  Conclusion

27. July 2001Samos - 200127 Classical linear schedule in MMAlpha design flow 1 4 2 6 10 14 Duration

28. July 2001Samos - 200128 Classical linear schedule in MMAlpha design flow 2 AB C F D E 14 1 2 0 1 0 1 0 2 14 10 6

29. July 2001Samos - 200129 Classical linear schedule in MMAlpha design flow schedule[ scheduleType -> sameLinearPart, durations -> {0,0,1,4,2,6,10,14,0,0}, addConstraints -> {/* linear constraints */} ]

30. July 2001Samos - 200130 Multi-dimensional scheduling  Virtual clock counter is a vector (hours, minutes,….) t1 t2 N N

31. July 2001Samos - 200131 Multi-dimensional scheduling  Useful for Fast prototyping of parallelism in complex applications  SVD (S. Robert, 1997)  Kalman filtering (A. Mozipo, 1998) Efficient code generation  Quilleré 1999 Structured scheduling

32. July 2001Samos - 200132 Extension to structured scheduling  Structured systems of recurrence equations (Dinechin97)  Example: matrix product can be expressed as N 2 independent dot products.  Question: Provided we have a layout for the dot product, can we use it for matrix product?

33. July 2001Samos - 200133 Example: Matrix-Matrix product {i,j,k|1<=i,j,k<=N}: A[i,j,k] = A[i,j-1,k-1]; {i,j,k|1<=i,j,k<=N}: B[i,j,k] = B[i-1,j,k-1]; {i,j,k|1<=i,j,k<=N}: C[i,j,k] = C[i,j,k-1]*A[i,j,k]*B[i,j,k]; {i,j|1<=i,j<=N}: c[i,j] = C[i,j,N];

34. July 2001Samos - 200134 Example: Matrix-Matrix product

35. July 2001Samos - 200135 Alpha code for dot product b(2) b(1) b(4) b(3) a(2) a(1) a(4) a(3) k 0 c system dot:{N|1<=N} (a,b:{k|1<=k<=N} of integer) returns (c: integer); var Acc:{k|0<=k<=N} of integer; let Acc[k] = case {|k=0}:0[]; {|k>0}:Acc[k-1]+a[k]*b[k]; esac; c[]=Acc[N]; tel;

36. July 2001Samos - 200136 Matrix product using dot product {i,j,k|1<=i,j,k<=N}: A[i,j,k] = A[i,j-1,k-1]; {i,j,k|1<=i,j,k<=N}: B[i,j,k] = B[i-1,j,k-1]; use {i,j|1<=i<=N} dot[N](A,B) returns (C);

37. July 2001Samos - 200137 Structured dependence graph AB C AB ACC C SDG: DG:

38. July 2001Samos - 200138 What is a structured scheduling?  Schedule each computations such that dependencies are respected Timing functions are positive  All instances of a given subsystem refer to the same schedule  Schedule is built from the structured dependence graph.

39. July 2001Samos - 200139 Necessary form of a structured scheduling  This form can be imposed in term of linear constraints.

40. July 2001Samos - 200140 Schedule of the dot product b(2) b(1) b(4) b(3) a(2) a(1) a(4) a(3) k 0 c

41. July 2001Samos - 200141 Structured 1D schedule use {i,j|1<=i<=N} dot[N](A,B) returns (C);

42. July 2001Samos - 200142 2D schedule use {i,j|1<=i<=N} dot[N](A,B) returns (C);

43. July 2001Samos - 200143 How to interpret this? B(1,1,2) B(1,1,1) B(1,1,4) B(1,1,3) A(1,1,2) A(1,1,1) A(1,1,4) A(1,1,3) 0 c[1,1] i k j

44. July 2001Samos - 200144 j=1 B(1,1,2) B(1,1,1) B(1,1,4) B(1,1,3) A(1,1,2) A(1,1,1) A(1,1,4) A(1,1,3) 0 c[1,1] i k j

45. July 2001Samos - 200145 j=1 B(1,1,2) B(1,1,1) B(1,1,4) B(1,1,3) A(1,1,2) A(1,1,1) A(1,1,4) A(1,1,3) 0 c[1,1] i k j

46. July 2001Samos - 200146 j=1 B(1,1,2) B(1,1,1) B(1,1,4) B(1,1,3) A(1,1,2) A(1,1,1) A(1,1,4) A(1,1,3) 0 c[1,1] i k j

47. July 2001Samos - 200147 j=1 B(1,1,2) B(1,1,1) B(1,1,4) B(1,1,3) A(1,1,2) A(1,1,1) A(1,1,4) A(1,1,3) 0 c[1,1] i k j

48. July 2001Samos - 200148 j=2 i k j

49. July 2001Samos - 200149 j=3 i k j

50. July 2001Samos - 200150 Structured schedules for MM  Dot product:  Matrix product: Or

51. July 2001Samos - 200151 Matrix product re-using hardware A 1,* A 3,* C 3,* A 2,* C 2,* C 1,* B *,* Acc + * A B C Dot product Matrix product

52. July 2001Samos - 200152 Advantage of structured scheduling  Preserves designer’s structuring  Re-uses hardware  Constraints are linear ”uses a classical schedule tool Reduces the schedule computation complexity Improves readability of the schedule information

53. July 2001Samos - 200153 Outline  What is MMAlpha?  Example of design flow  The Alpha language  Structured scheduling  Performance  Conclusion

54. July 2001Samos - 200154 Experiments  Vertex method: Simplex of Mathematica  Farkas method: Pip software  Evaluation of structured vs « flat » scheduling

55. July 2001Samos - 200155 Test set (vertex method)

56. July 2001Samos - 200156 #Constraints vs #Variables

57. July 2001Samos - 200157 Schedule time vs #constraints

58. July 2001Samos - 200158 #Constraints vs #Variables (Farkas method)

59. July 2001Samos - 200159 Schedule time vs #constraints

60. July 2001Samos - 200160 Flat vs structured schedule