Experiences with Enumeration of Integer Projections of Parametric Polytopes Sven Verdoolaege, Kristof Beyls, Maurice Bruynooghe, Francky Catthoor Compiler Construction

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 and 1 <= 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 P = 3 Answer: Not a polytope!

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 PIP (Feautrier’88) Ehrhart (Clauss’96) Solution: closed form Ehrhart quasi-polynomial 3 Heuristics (novel) Barvinok (Verdoolaege’04) Boulet(1998): Worst-case exponential exec. time, even for fixed number of variables Novel method: Worst-case polynomial exec. time, for fixed number of variables

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 1. Unique existential variables “thickness” always smaller than 1: treat existential variable as regular variable 2. Redundant existential variables “thickness” always larger than 1: project polytopes, and count project. Legal, since there are no “holes”. (=Omega test). 3. 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 Test 1: Matrix multiply, matrix size multiple of cache line size.  PIP vs. Heuristics PIP (Feautrier’88) 3 Heuristics (novel) Barvinok (Verdoolaege’04)

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

25 Other applications. a) communication in HPF [Boulet98]  Computation of communication volume (HPF, Boulet98): EhrhartBarvinok 8x8 processors 713s/0.04s0.01s 64x64 processors 6855s/1.43s0.01s

26 Other applications b) Memory size estimation [Balasa95] EhrhartBarvinok 4 memory references 1.38s/0.01s/ 1.41s/1.41s 0.06s/0.01s/ 0.07s/0.04s Memory accessed by 4 different references in a motion estimation loop kernel, with symbolic loop bounds

27 Other applications c) 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. PIP+ Ehrhart PIP+ Barvinok Heuristics+ Barvinok symbolic cache miss counting > 15 h449.39s434.47s

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 at