Experiences with Enumeration of Integer Projections of Parametric Polytopes

1. Experiences with Enumeration of Integer Projections of Parametric Polytopes Sven Verdoolaege, Kristof Beyls, Maurice Bruynooghe, Francky CatthoorCompiler Construction - 2005

2. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

3. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

4. Introduction • Counting problems in compiler: • How many executed calculations? • How many data addresses accessed? • How many cache misses? • How many dynamically allocated bytes? • How many live array elements at a symbolic iteration (i,j)? • How much communication between parallel processes? • … • Often, answering these questions lead to counting the number integer solutions to a system of linear inequalities: • when the code consists of loops with linear loop bounds. • when the array index expressions have a linear form.

5. Example 1: Counting solutions to systems of linear inequalities void s(int N, int M) { int i,j; for(i=max(0,N-M); i<=N-M+3; i++) for(j=0; j<=N-2*i; j++) S1; } How many times is statement S1 executed? Equals the number of elements in the set: linear inequalities defining a bounded domain polytope parameters  parametric

6. Geometric representation: parametric integer polytope

7. Solution: counting the number of integer points in a parametric polytope • Algorithm see CASES2004:“Analytical Computation of Ehrhart Polynomials: Enabling more Compiler Analyses and Optimizations”. • Solution:

8. Contribution: Extension to include existential variables • Goal: count the solution in parameterized sets of the form: • CASES2004: • CC2005:

9. l = 6i+9j-7 1 <= j <= P and1 <= i <= 8 Example: How many array elements accessed in following loop? for j:= 1 to P do for i:= 1 to 8 do a(6i+9j-7) += 5

10. Geometric representation: Integer projection of parametric polytope. for j:= 1 to P do for i:= 1 to 8 do a(6i+9j-7) += 5 Answer: Not a polytope! P = 3

11. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

12. Examples of compiler analyses benefiting from our work • Data placement while taking into account real-time constraints • Anantharaman et al., RTSS 1998 • Memory size estimation of loop nests after translation to VLSI designs • Balasa et al., IEEE T.VLSI 1995 • Zhao et al., IEEE T.VLSI 2000 • Compilation to parallel FPGA / VLSI • Bednara et al., Samos 2002 • Hannig et al., PaCT 2001 • Calculating Cache Behavior • Beyls et al., JSA 2005 • Chatterjee et al., PLDI 2001 • Computing communication in distributed memory computers (HPF) • Boulet et al., Euro-Par 1998 • Heine et al., Euro-Par 2000 • Su et al., ICS 1995 • Low-Power Compilation • D’Alberto et al., COLP 2001

13. Usefulness • In many of the above papers, the authors spent most of the paper discussing estimation methods to get approximate answers to the question: How many elements in S? • In this paper, we answer this question exactly.

14. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

15. Overall idea: PIP/heuristics + Barvinok 3 Heuristics(novel) PIP(Feautrier’88) Novel method:Worst-case polynomialexec. time, for fixednumber of variables Boulet(1998):Worst-case exponentialexec. time, even for fixednumber of variables Ehrhart(Clauss’96) Barvinok(Verdoolaege’04) Solution: closed form Ehrhart quasi-polynomial

16. PIP(Feautrier’88) Parametric Integer Programming (PIP) • PIP allows to compute the lexicographical minimal element of a parametric polytope • Compute the lexicographical minimum of all points in S that are projected onto the same point in S’. (Worst-case exponential time)

17. 3 Heuristics(novel) 3 Polynomial-time Heuristics • Unique existential variables • “thickness” always smaller than 1: treat existential variable as regular variable • Redundant existential variables • “thickness” always larger than 1: project polytopes, and count project. Legal, since there are no “holes”. (=Omega test). • Independent splits • If none of the above applies, try to split polytope in multiple polytopes, for which one of the above rules applies

18. 3 Heuristics(novel) 3 Heuristics: example “thickness” ≥1 “thickness” ≤1

19. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

20. Experiments • Reuse Distance Calculation [Beyls05] • Communication volume computation in HPF [Boulet98] • Memory Size Estimation [Balasa95] • Parametric Cache Miss Calculation [Chatterjee01]

21. Reuse Distance Calculation • Computes the number of data locations accessed between two consecutive reuses of the same data. • Parameters: iteration point where reuse occurs + program parameters.

22. 3 Heuristics(novel) PIP(Feautrier’88) Barvinok(Verdoolaege’04) Test 1: Matrix multiply, matrix size multiple of cache line size. • PIP vs. Heuristics

23. Test 2: Matrix multiply, matrix size 19 and 41, cache line size 4 • Heuristics: 2 sets couldn’t be computed in one hour of time.(vertex calculation during change of basis) • PIP: 4 sets couldn’t be computed in an hour of time. • Conclusion: There are sets for which neither method can compute the solution in reasonable time.

24. PIP(Feautrier’88) Ehrhart(Clauss’96) Barvinok(Verdoolaege’04) Test 3: Ehrhart vs. Barvinok

25. Other applications.a) communication in HPF [Boulet98] • Computation of communication volume (HPF, Boulet98):

26. Other applicationsb) Memory size estimation [Balasa95] Memory accessed by 4 different references in a motion estimation loop kernel, with symbolic loop bounds

27. Other applicationsc) Cache miss analysis [Chatterjee01] • Computes the number of cache misses in a two-way set-associative cache, for matrix-vector multiplication with symbolic loop bounds.

28. Overview • Explaining the title. • Useful in which Compiler Analyses? • High-level Algorithm Overview • Experiments • Conclusion

29. Conclusions • Many compiler analyses and optimization require the enumeration of integer projections of parametric polytopes. • Can be done by reduction to enumeration of parametric polytopes. • No clear performance difference between PIP and heuristics. • Can solve many problems that were previously considered very difficult or unsolvable. • Software available athttp://freshmeat.net/projects/barvinok