tools for the polyhedral model

December 15, 2009 1 / 35

Tools for the Polyhedral Model

Sven Verdoolaege

Katholieke Universiteit Leuven, Belgium

December 15, 2009

December 15, 2009 2 / 35

Outline

1 Polyhedral Model

2 isl

Introduction and RepresentationParametric Integer Programming

3 CLooG

4 barvinok

5 bernstein

6 Conclusions

Polyhedral Model December 15, 2009 3 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions


Polyhedral Program Analysis and Transformation

Observation:

Most expensive part of a multimedia or signal processing program:manipulation of arrays inside loops

⇒ most interesting part to optimize



Observation:



⇒ need for compact representation ofiterations of a loop/elements of an array

⇒ + efficient to manipulate



Observation:



⇒ need for compact representation ofiterations of a loop/elements of an array

⇒ + efficient to manipulate

⇒ Integer points in polyhedra (“polyhedral model”)⇒ More generally: sets of integers bounded by affine inequalities


Polyhedral Representation Example: Iteration Domain

#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

Assumptions on sequential code:

iterators are integers

loops with affine bounds

affine conditions

affine index expressions



#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

iIteration domain: P = { [i , j ] | i ≥ 1 }




affine conditions




#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

iIteration domain: P = { [i , j ] | i ≥ 1 ∧ i ≤ N }




affine conditions




#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

iIteration domain: P = { [i , j ] | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 }




affine conditions




#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

iIteration domain: P = { [i , j ] | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 ∧ j ≤ i }




affine conditions




#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

i

•

• •

• • •

• • • •

• • • • •

Iteration domain: P = { [i , j ] | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 ∧ j ≤ i }




affine conditions




#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

j

i

•

• •

• • •

• • • •

• • • • •

•

• •

Iteration domain: P = { [i , j ] | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 ∧ j ≤ i }

Alternative representation: P = conv.hull { (1, 1), (N, 1), (N, N) }




affine conditions



Polyhedral Program Transformation Example

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

Execution order: top-down, left-right

•, ◦ Statement iterationData flow dependence

• Executed statementData in memory



for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

•






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •

•






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •

• •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •

• • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •

• • • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

• • • • •

• • • • •






for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦




• • • • •

◦ ◦ ◦ ◦ ◦



for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[N-i] = f(a[i])

• • • • •

◦ ◦ ◦ ◦ ◦



for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[N-i] = f(a[i])

• • • • •

◦ ◦ ◦ ◦ ◦

•

•

•

•

•

◦

◦

◦

◦

◦



for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[N-i] = f(a[i])

• • • • •

◦ ◦ ◦ ◦ ◦

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

•

•

•

•

•

◦

◦

◦

◦

◦


Polyhedral Program Analysis and Transformation Overview

program code

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])



program code

parser

polyhedral model

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦



program code

parser

polyhedral model analysis andtransformation

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦



program code

parser

polyhedral model

polyhedralscanner

analysis andtransformation

for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦

isl December 15, 2009 8 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions

isl Introduction and Representation December 15, 2009 9 / 35


program code

parser

polyhedral model

polyhedralscanner


for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦


Representation and Manipulation of Sets of IntegersPossible representations:

double description based◮ PolyLib◮ PPL◮ polymake◮ . . .

Some operations can be performed more efficiently on explicitrepresentation, but

◮ Computing the dual can be costly◮ Double description requires more space

⇒ trade-off






⇒ trade-offconstraints based

◮ Omega◮ isl






⇒ trade-offconstraints based

◮ Omega◮ isl

number decision diagrams◮ LASH◮ . . .

generating functions


isl Overviewisl is an

LGPL

thread-safe

C

library for manipulating integer sets and relations

bounded by affine constraints

involving parameters and

existentially quantified variables

Supported operations include

basic operations such as intersection, union and set difference

integer projection

set coalescing

convex hull, integer affine hull

sampling and scanning using generalized basis reduction

computing the lexicographic minimum of a relation,


Why do we need existentially quantified variables?

Modeling some problemsWhich array elements are accessed in this loop?

for (j = 1; j <= s; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;

S(s) = {l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ s ∧ 1 ≤ i ≤ 8}




for (j = 1; j <= s; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;

S(s) = {l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ s ∧ 1 ≤ i ≤ 8}

Especially integer divisions/remaindersE.g., i % 10 <= 6

i − 10

⌊

i

10

⌋

≤ 6




for (j = 1; j <= s; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;

S(s) = {l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ s ∧ 1 ≤ i ≤ 8}


i − 10

⌊

i

10

⌋

≤ 6

i − 10α ≤ 6

with i − 9 ≤ 10α ≤ i




for (j = 1; j <= s; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;

S(s) = {l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ s ∧ 1 ≤ i ≤ 8}


i − 10

⌊

i

10

⌋

≤ 6

i − 10α ≤ 6

with i − 9 ≤ 10α ≤ i

◮ May appear in original code◮ May be introduced by (PIP-based) dependence analysis


Internal Representation

Polyhedral sets and relations

S(s) = { x ∈ Zd | ∃z ∈ Ze : Ax + Bs + Dz ≥ c }

R(s) = { (x1, x2) ∈ Zd1 × Zd2 | ∃z ∈ Ze : A1x1 + A2x2 + Bs + Dz ≥ c }

“basic” types: “convex” sets and maps (relations)

◮ equality + inequality constraints◮ parameters s◮ (optional) explicit representation of existentially quantified variables as

integer divisions

⇒ useful for aligning dimensions when performing set operations (e.g., setdifference)

⇒ can be computed using PIP⇒ already available if obtained from PIP-based dependence analysis

union types: sets and maps

⇒ (disjoint) unions of basic sets/maps

isl Parametric Integer Programming December 15, 2009 14 / 35


program code

parser

polyhedral model

polyhedralscanner


for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦


Lexicographical Order

#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

S = { (i , j) | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 ∧ j ≤ i }

Execution order: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3),(4,4) (5,1), (5,2), (5,3), (5,4), (5,5)


Lexicographical Order

#define N 5

for (i = 1; i <= N; ++i)

for (j = 1; j <= i; ++j)

a[i][j]=

S = { (i , j) | i ≥ 1 ∧ i ≤ N ∧ j ≥ 1 ∧ j ≤ i }

Execution order: (1,1), (2,1), (2,2), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3),(4,4) (5,1), (5,2), (5,3), (5,4), (5,5)

Lexicographical order:

a ≺ b ≡n

∨

i=1

ai < bi ∧i−1∧

j=1

aj = bj

⇒ smaller in first position where vectors differ


Parametric Integer Programming

Given a relation R with parameters s


compute the lexicographic minimum of R:

lexminR = { (x1, x2) ∈ R | ∀x′2 ∈ R(s, x1) : x2 4 x′2 }







Parametric integer programming is useful for

dependence analysis

(enumeration of) integer projections







Parametric integer programming is useful for

dependence analysis

(enumeration of) integer projections

Technique: dual simplex + Gomory cuts


Dataflow Analysis

Given a read from an array element, what was the last write to

the same array element before the read?

Simple case: array written through a single access

for (i = 0; i < N; ++i)

for (j = 0; j < N - i; ++j)

a[i+j] = f(a[i+j]);

for (i = 0; i < N; ++i)

Write(a[i]);


Dataflow Analysis




for (i = 0; i < N; ++i)

for (j = 0; j < N - i; ++j)

a[i+j] = f(a[i+j]);

for (i = 0; i < N; ++i)

Write(a[i]);

Access relations:A1 = {(i , j) → (i + j) | 0 ≤ i < N ∧ 0 ≤ j < N − i}A2 = {(i) → (i) | 0 ≤ i < N}


Dataflow Analysis




for (i = 0; i < N; ++i)

for (j = 0; j < N - i; ++j)

a[i+j] = f(a[i+j]);

for (i = 0; i < N; ++i)

Write(a[i]);


Map to all writes: R ′ = A1−1 ◦ A2 = {(i) → (i ′, i − i ′) | 0 ≤ i ′ ≤ i < N}


Dataflow Analysis




for (i = 0; i < N; ++i)

for (j = 0; j < N - i; ++j)

a[i+j] = f(a[i+j]);

for (i = 0; i < N; ++i)

Write(a[i]);



Last write: R = lexmaxR ′ = {(i) → (i , 0) | 0 ≤ i < N}


Dataflow Analysis




for (i = 0; i < N; ++i)

for (j = 0; j < N - i; ++j)

a[i+j] = f(a[i+j]);

for (i = 0; i < N; ++i)

Write(a[i]);



Last write: R = lexmaxR ′ = {(i) → (i , 0) | 0 ≤ i < N}

In general: impose lexicographical order on shared iterators

CLooG December 15, 2009 18 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions



program code

parser

polyhedral model

polyhedralscanner


for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦


CLooG ExampleCLooG generates code for scanning integer points in polyhedra (iterationdomains)

S1: {(i , j) | 1 ≤ i ≤ n ≤ m ∧ j = i}S2: {(i , j) | 1 ≤ i ≤ n ≤ m ∧ i ≤ j ≤ n}S3: {(i , j) | 1 ≤ i ≤ m ∧ j = n ≤ m}

i

j


CLooG ExampleCLooG generates code for scanning integer points in polyhedra (iterationdomains)

S1: {(i , j) | 1 ≤ i ≤ n ≤ m ∧ j = i}S2: {(i , j) | 1 ≤ i ≤ n ≤ m ∧ i ≤ j ≤ n}S3: {(i , j) | 1 ≤ i ≤ m ∧ j = n ≤ m}

i

jfor (i=1;i<=m;i++) {

if (i <= n) {

S1(i,i);

}

for (j=i;j<=n;j++) {

S2(i);

}

S3(i,n);

}

barvinok December 15, 2009 21 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions



program code

parser

polyhedral model

polyhedralscanner


for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦


Counting Example 1

How many times is S1 executed ?

for (i=1; i<=N; i++)

for (j=1; j<=i; j++)

S1;


Counting Example 1


for (i=1; i<=N; i++)

for (j=1; j<=i; j++)

S1;

N = 4

i

j(0, 0)

••••

•••

•• •

#{ (i , j) ∈ Z2 | 1 ≤ i ≤ N ∧ 1 ≤ j ≤ i } =N(N + 1)

2


Counting Example 1


for (i=1; i<=N; i++)

for (j=1; j<=i; j++)

S1;

N = 4

i

j(0, 0)

••••

•••

•• •

#{ (i , j) ∈ Z2 | 1 ≤ i ≤ N ∧ 1 ≤ j ≤ i } =N(N + 1)

2

= #

[

i

j

]

∈ Z2 |

1 0−1 00 11 −1

[

i

j

]

≥

0−100

[

N]

+

1010


Counting Example 2


for (i=max(0,N-M); i<=N-M+3; i++)

for (j=0; j<=N-2*i; j++)

S1;


Counting Example 2


for (i=max(0,N-M); i<=N-M+3; i++)

for (j=0; j<=N-2*i; j++)

S1;

Equal to the number of elements in

P( NM ) =

[

i

j

]

∈ Z2 |

1 01 0−1 00 1−2 −1

[

i

j

]

≥

0 01 −1−1 10 0−1 0

[

N

M

]

+

00−300


Counting Example 2

#P( NM ) =

−4N + 8M − 8 if M ≤ N ≤ 2M − 6

MN − 2N − M2 + 6M − 8

if N ≤ M ≤ N + 3 ∧ N ≤ 2M − 6

N2

4 + 34N + 1

2

⌊

N2

⌋

+ 1 if 0 ≤ N ≤ M ∧ 2M ≤ N + 6

N2

4 − MN − 54N + M2 + 2M + 1

2

⌊

N2

⌋

+ 1

if M ≤ N ≤ 2M ≤ N + 6


Counting Example 2

#P( NM ) =

−4N + 8M − 8 if M ≤ N ≤ 2M − 6

MN − 2N − M2 + 6M − 8

if N ≤ M ≤ N + 3 ∧ N ≤ 2M − 6

N2

4 + 34N + 1

2

⌊

N2

⌋

+ 1 if 0 ≤ N ≤ M ∧ 2M ≤ N + 6

N2

4 − MN − 54N + M2 + 2M + 1

2

⌊

N2

⌋

+ 1

if M ≤ N ≤ 2M ≤ N + 6

Chambers:

N

M


Counting Example 3

How many elements of array a are accessed ?

for (j = 1; j <= p; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;


Counting Example 3


for (j = 1; j <= p; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;


Sp = { l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ p ∧ 1 ≤ i ≤ 8 }


Counting Example 3


for (j = 1; j <= p; ++j)

for (i = 1; i <= 8; ++i)

a[6i+9j-7] = a[6i+9j-7] + 5;


Sp = { l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ p ∧ 1 ≤ i ≤ 8 }

#Sp =

{

8 if p = 1

3p + 10 if p ≥ 2


Weighted Counting Example

p = a;

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p += j * ((j-i)/4);

*p = 0;

}

Can we parallelize this code?



p = a;

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p += j * ((j-i)/4);

*p = 0;

}


⇒ No, (false) dependency through p

⇒ Compute closed formula for p

p = a+∑

(i ′,j ′)∈S

(i ′,j ′)4(i ,j)

j ′⌊

j ′ − i ′

4

⌋

with S = { (i ′, j ′) ∈ Z2 | 0 ≤ i ′ < N ∧ i ′ ≤ j ′ < N }



p = a;

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p += j * ((j-i)/4);

*p = 0;

}




p = a+∑

(i ′,j ′)∈S

(i ′,j ′)4(i ,j)

j ′⌊

j ′ − i ′

4

⌋

= a+∑

(i ′,j ′)∈S

i ′<i

j ′⌊

j ′ − i ′

4

⌋

+∑

(i ′,j ′)∈S

i ′=i ,j ′≤j

j ′⌊

j ′ − i ′

4

⌋

with S = { (i ′, j ′) ∈ Z2 | 0 ≤ i ′ < N ∧ i ′ ≤ j ′ < N }



p = a;

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p += j * ((j-i)/4);

*p = 0;

}




p = a+∑

(i ′,j ′)∈S

(i ′,j ′)4(i ,j)

j ′⌊

j ′ − i ′

4

⌋

= a+∑

(i ′,j ′)∈S

i ′<i

j ′⌊

j ′ − i ′

4

⌋

+∑

(i ′,j ′)∈S

i ′=i ,j ′≤j

j ′⌊

j ′ − i ′

4

⌋

with S = { (i ′, j ′) ∈ Z2 | 0 ≤ i ′ < N ∧ i ′ ≤ j ′ < N }

⇒ weighted counting



p = a;

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p += j * ((j-i)/4);

*p = 0;

}

N − 1

i − 1

j ′

i ′

•••

••••

•••••

••••••




p = a+∑

(i ′,j ′)∈S

(i ′,j ′)4(i ,j)

j ′⌊

j ′ − i ′

4

⌋

= a+∑

(i ′,j ′)∈S

i ′<i

j ′⌊

j ′ − i ′

4

⌋

+∑

(i ′,j ′)∈S

i ′=i ,j ′≤j

j ′⌊

j ′ − i ′

4

⌋

with S = { (i ′, j ′) ∈ Z2 | 0 ≤ i ′ < N ∧ i ′ ≤ j ′ < N }

⇒ weighted counting


Parametric Polytopes and Piecewise Step-polynomials

Parametric polytope:

P(s) : s 7→ {t ∈ Qd | As + Bt + c ≥ 0}

(N, i , j) → { (i ′, j ′) | 0 ≤ i ′ < i∧i ′ ≤ j ′ < N }

N − 1

i − 1

j ′

i ′

•••

••••

•••••

••••••




P(s) : s 7→ {t ∈ Qd | As + Bt + c ≥ 0}

(N, i , j) → { (i ′, j ′) | 0 ≤ i ′ < i∧i ′ ≤ j ′ < N }

N − 1

i − 1

j ′

i ′

•••

••••

•••••

••••••

Step-polynomial: q(s, t) ∈ Q [⌊Q[s, t]≤1⌋] j ′⌊

j ′−i ′

4

⌋

Piecewise step-polynomial: (s, t) 7→

q1(s, t) if t ∈ P1(s)

. . .

qk(s, t) if t ∈ Pk(s)




P(s) : s 7→ {t ∈ Qd | As + Bt + c ≥ 0}

(N, i , j) → { (i ′, j ′) | 0 ≤ i ′ < i∧i ′ ≤ j ′ < N }

N − 1

i − 1

j ′

i ′

•••

••••

•••••

••••••

Step-polynomial: q(s, t) ∈ Q [⌊Q[s, t]≤1⌋] j ′⌊

j ′−i ′

4

⌋

Piecewise step-polynomial: (s, t) 7→

q1(s, t) if t ∈ P1(s)

. . .

qk(s, t) if t ∈ Pk(s)

Weighted counting:

Zn → Q : s 7→∑

i

∑

t∈Zd

[t ∈ Pi (s)] qi (s, t)


barvinok Overview

Main functionality: (weighted) counting of elements in polyhedral set

Input:

polyhedral set, or

piecewise step-polynomial

Output:

piecewise step-polynomial c(s)

⇒ number of elements in set⇒ weighted number of elements in each cell

or

generating function C (x) =∑

s c(s) xs

bernstein December 15, 2009 30 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions



program code

parser

polyhedral model

polyhedralscanner


for (i = 0; i <= N; ++i)

a[i] = ...

for (i = 0; i <= N; ++i)

b[i] = f(a[N-i])

for (i = 0; i <= N; ++i) {

a[i] = ...

b[N-i] = f(a[i])

}

• • • • •

◦◦◦◦◦

•

•

•

•

•

◦

◦

◦

◦

◦


bernstein example

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p = malloc(i * j + i - N + 1);

/* ... */

free(p);

}

How much memory is needed?


bernstein example

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p = malloc(i * j + i - N + 1);

/* ... */

free(p);

}


m(N) = max(i ,j)∈S

(ij + i − N + 1)

with S = { (i , j) ∈ Z2 | 0 ≤ i < N ∧ i ≤ j < N }


bernstein example

for (i = 0; i < N; ++i)

for (j = i; j < N; ++j) {

p = malloc(i * j + i - N + 1);

/* ... */

free(p);

}


m(N) = max(i ,j)∈S

(ij + i − N + 1)

with S = { (i , j) ∈ Z2 | 0 ≤ i < N ∧ i ≤ j < N }

Typical application: maximal number of live elements

⇒ count the number of live elements at each execution point

⇒ compute upper bound over all execution points


Computing Parametric Upper Bounds

Maximal memory requirements:

u(N) ≥ m(N) = max0≤i<N∧i≤j<N

(ij + i − N + 1)

m(N) can be approximated using Bernstein expansion ⇒ u(N)

Ideally:◮ maximum over◮ integer points

Bernstein expansion gives◮ upper bound over◮ rational points

In practice very accurate

Conclusions December 15, 2009 34 / 35

Outline

1 Polyhedral Model

2 isl


3 CLooG

4 barvinok

5 bernstein

6 Conclusions


ConclusionSeveral tools have been developed within the context of the polyhedralmodel, including,

isl (http://freshmeat.net/projects/isl/)Represents and manipulates sets and relations of integersSupports several operations including parametric integer programmingCLooG (http://www.cloog.org/)Generates code for scanning the elements of polytopesbarvinok (http://freshmeat.net/projects/barvinok/)Counts the number of elements in a polyhedral set, possibly weightedby a step-polynomialbernstein (included in barvinok)Computes bounds on polynomials over parametric polytopes

http://freshmeat.net/projects/isl/

http://www.cloog.org/

http://freshmeat.net/projects/barvinok/


ConclusionSeveral tools have been developed within the context of the polyhedralmodel, including,

isl (http://freshmeat.net/projects/isl/)Represents and manipulates sets and relations of integersSupports several operations including parametric integer programmingCLooG (http://www.cloog.org/)Generates code for scanning the elements of polytopesbarvinok (http://freshmeat.net/projects/barvinok/)Counts the number of elements in a polyhedral set, possibly weightedby a step-polynomialbernstein (included in barvinok)Computes bounds on polynomials over parametric polytopes

Future Work: port barvinok to isl; now uses

PolyLib: GPLv2 only, . . .⇒ isl needs chamber decomposition and parametric verticesNTL: not thread-safe, C++⇒ isl needs LLL

http://freshmeat.net/projects/isl/

http://www.cloog.org/

http://freshmeat.net/projects/barvinok/

tools for the polyhedral model

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