Experiences with Enumeration of Integer Projections of Parametric Polytopes

Sven Verdoolaege, Kristof Beyls, Maurice Bruynooghe, Francky Catthoor

Compiler Construction - 2005

2Overview

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

3Overview

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

4Introduction

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.

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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

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Geometric representation: parametric integer polytope

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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:

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Contribution: Extension to include existential variables

Goal: count the solution in parameterized sets of the form:

CASES2004:

CC2005:

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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

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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!

11Overview

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

13Usefulness

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.

14Overview

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

15Overall 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 exponentialexec. time, even for fixednumber of variables

Novel method:Worst-case polynomialexec. time, for fixednumber of variables

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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)

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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

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3 Heuristics(novel)

3 Heuristics: example

“thickness” ≥1

“thickness” ≤1

19Overview

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

20Experiments

Reuse Distance Calculation [Beyls05]

Communication volume computation in HPF [Boulet98]

Memory Size Estimation [Balasa95] Parametric Cache Miss Calculation

[Chatterjee01]

21Reuse 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.

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Test 1: Matrix multiply, matrix size multiple of cache line size.

PIP vs. Heuristics

PIP(Feautrier’88)

3 Heuristics(novel)

Barvinok(Verdoolaege’04)

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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.

24Test 3: Ehrhart vs. Barvinok

PIP(Feautrier’88)

Ehrhart(Clauss’96)

Barvinok(Verdoolaege’04)

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Other applications.a) communication in HPF [Boulet98]

Computation of communication volume (HPF, Boulet98):

Ehrhart Barvinok

8x8 processors

713s/0.04s 0.01s

64x64 processors

6855s/1.43s 0.01s

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Other applicationsb) Memory size estimation [Balasa95]

Ehrhart Barvinok

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

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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.

PIP+Ehrhart

PIP+Barvinok

Heuristics+Barvinok

symbolic cache miss counting

> 15 h 449.39s 434.47s

28Overview

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

29Conclusions

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