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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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
Polyhedral Model December 15, 2009 4 / 35
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
Polyhedral Model December 15, 2009 4 / 35
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
⇒ need for compact representation ofiterations of a loop/elements of an array
⇒ + efficient to manipulate
Polyhedral Model December 15, 2009 4 / 35
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
⇒ 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 Model December 15, 2009 5 / 35
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
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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 }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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 }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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 }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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 }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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 }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 5 / 35
Polyhedral Representation Example: Iteration Domain
#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) }
Assumptions on sequential code:
iterators are integers
loops with affine bounds
affine conditions
affine index expressions
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
Polyhedral Model December 15, 2009 6 / 35
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
• • • • •
◦ ◦ ◦ ◦ ◦
Polyhedral Model December 15, 2009 6 / 35
Polyhedral Program Transformation Example
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])
• • • • •
◦ ◦ ◦ ◦ ◦
Polyhedral Model December 15, 2009 6 / 35
Polyhedral Program Transformation Example
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])
• • • • •
◦ ◦ ◦ ◦ ◦
•
•
•
•
•
◦
◦
◦
◦
◦
Polyhedral Model December 15, 2009 6 / 35
Polyhedral Program Transformation Example
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 Model December 15, 2009 7 / 35
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])
Polyhedral Model December 15, 2009 7 / 35
Polyhedral Program Analysis and Transformation Overview
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])
• • • • •
◦◦◦◦◦
Polyhedral Model December 15, 2009 7 / 35
Polyhedral Program Analysis and Transformation Overview
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])
• • • • •
◦◦◦◦◦
•
•
•
•
•
◦
◦
◦
◦
◦
Polyhedral Model December 15, 2009 7 / 35
Polyhedral Program Analysis and Transformation Overview
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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
isl Introduction and Representation December 15, 2009 9 / 35
Polyhedral Program Analysis and Transformation Overview
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 Introduction and Representation December 15, 2009 10 / 35
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
isl Introduction and Representation December 15, 2009 10 / 35
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-offconstraints based
◮ Omega◮ isl
isl Introduction and Representation December 15, 2009 10 / 35
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-offconstraints based
◮ Omega◮ isl
number decision diagrams◮ LASH◮ . . .
generating functions
isl Introduction and Representation December 15, 2009 10 / 35
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-offconstraints based
◮ Omega◮ isl
number decision diagrams◮ LASH◮ . . .
generating functions
isl Introduction and Representation December 15, 2009 11 / 35
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,
isl Introduction and Representation December 15, 2009 12 / 35
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}
isl Introduction and Representation December 15, 2009 12 / 35
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}
Especially integer divisions/remaindersE.g., i % 10 <= 6
i − 10
⌊
i
10
⌋
≤ 6
isl Introduction and Representation December 15, 2009 12 / 35
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}
Especially integer divisions/remaindersE.g., i % 10 <= 6
i − 10
⌊
i
10
⌋
≤ 6
i − 10α ≤ 6
with i − 9 ≤ 10α ≤ i
isl Introduction and Representation December 15, 2009 12 / 35
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}
Especially integer divisions/remaindersE.g., i % 10 <= 6
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
isl Introduction and Representation December 15, 2009 13 / 35
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
Polyhedral Program Analysis and Transformation Overview
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 Parametric Integer Programming December 15, 2009 15 / 35
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)
isl Parametric Integer Programming December 15, 2009 15 / 35
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
isl Parametric Integer Programming December 15, 2009 16 / 35
Parametric Integer Programming
Given a relation R with parameters s
R(s) = { (x1, x2) ∈ Zd1 × Zd2 | ∃z ∈ Ze : A1x1 + A2x2 + Bs + Dz ≥ c }
compute the lexicographic minimum of R:
lexminR = { (x1, x2) ∈ R | ∀x′2 ∈ R(s, x1) : x2 4 x′2 }
isl Parametric Integer Programming December 15, 2009 16 / 35
Parametric Integer Programming
Given a relation R with parameters s
R(s) = { (x1, x2) ∈ Zd1 × Zd2 | ∃z ∈ Ze : A1x1 + A2x2 + Bs + Dz ≥ c }
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
isl Parametric Integer Programming December 15, 2009 16 / 35
Parametric Integer Programming
Given a relation R with parameters s
R(s) = { (x1, x2) ∈ Zd1 × Zd2 | ∃z ∈ Ze : A1x1 + A2x2 + Bs + Dz ≥ c }
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
Technique: dual simplex + Gomory cuts
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
Access relations:A1 = {(i , j) → (i + j) | 0 ≤ i < N ∧ 0 ≤ j < N − i}A2 = {(i) → (i) | 0 ≤ i < N}
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
Access relations:A1 = {(i , j) → (i + j) | 0 ≤ i < N ∧ 0 ≤ j < N − i}A2 = {(i) → (i) | 0 ≤ i < N}
Map to all writes: R ′ = A1−1 ◦ A2 = {(i) → (i ′, i − i ′) | 0 ≤ i ′ ≤ i < N}
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
Access relations:A1 = {(i , j) → (i + j) | 0 ≤ i < N ∧ 0 ≤ j < N − i}A2 = {(i) → (i) | 0 ≤ i < N}
Map to all writes: R ′ = A1−1 ◦ A2 = {(i) → (i ′, i − i ′) | 0 ≤ i ′ ≤ i < N}
Last write: R = lexmaxR ′ = {(i) → (i , 0) | 0 ≤ i < N}
isl Parametric Integer Programming December 15, 2009 17 / 35
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]);
Access relations:A1 = {(i , j) → (i + j) | 0 ≤ i < N ∧ 0 ≤ j < N − i}A2 = {(i) → (i) | 0 ≤ i < N}
Map to all writes: R ′ = A1−1 ◦ A2 = {(i) → (i ′, i − i ′) | 0 ≤ i ′ ≤ i < N}
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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
CLooG December 15, 2009 19 / 35
Polyhedral Program Analysis and Transformation Overview
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])
}
• • • • •
◦◦◦◦◦
•
•
•
•
•
◦
◦
◦
◦
◦
CLooG December 15, 2009 20 / 35
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 December 15, 2009 20 / 35
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 December 15, 2009 20 / 35
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 December 15, 2009 20 / 35
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);
}
CLooG December 15, 2009 20 / 35
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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
barvinok December 15, 2009 22 / 35
Polyhedral Program Analysis and Transformation Overview
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])
}
• • • • •
◦◦◦◦◦
•
•
•
•
•
◦
◦
◦
◦
◦
barvinok December 15, 2009 23 / 35
Counting Example 1
How many times is S1 executed ?
for (i=1; i<=N; i++)
for (j=1; j<=i; j++)
S1;
barvinok December 15, 2009 23 / 35
Counting Example 1
How many times is S1 executed ?
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
barvinok December 15, 2009 23 / 35
Counting Example 1
How many times is S1 executed ?
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
barvinok December 15, 2009 24 / 35
Counting Example 2
How many times is S1 executed ?
for (i=max(0,N-M); i<=N-M+3; i++)
for (j=0; j<=N-2*i; j++)
S1;
barvinok December 15, 2009 24 / 35
Counting Example 2
How many times is S1 executed ?
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
barvinok December 15, 2009 25 / 35
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
barvinok December 15, 2009 25 / 35
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
barvinok December 15, 2009 26 / 35
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;
barvinok December 15, 2009 26 / 35
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;
Equal to the number of elements in
Sp = { l ∈ Z | ∃i , j ∈ Z : l = 6i + 9j − 7 ∧ 1 ≤ j ≤ p ∧ 1 ≤ i ≤ 8 }
barvinok December 15, 2009 26 / 35
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;
Equal to the number of elements in
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
barvinok December 15, 2009 27 / 35
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?
barvinok December 15, 2009 27 / 35
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?
⇒ 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 }
barvinok December 15, 2009 27 / 35
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?
⇒ No, (false) dependency through p
⇒ Compute closed formula for p
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 }
barvinok December 15, 2009 27 / 35
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?
⇒ No, (false) dependency through p
⇒ Compute closed formula for p
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
barvinok December 15, 2009 27 / 35
Weighted Counting Example
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 ′
•••
••••
•••••
••••••
Can we parallelize this code?
⇒ No, (false) dependency through p
⇒ Compute closed formula for p
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
barvinok December 15, 2009 28 / 35
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 ′
•••
••••
•••••
••••••
barvinok December 15, 2009 28 / 35
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 ′
•••
••••
•••••
••••••
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)
barvinok December 15, 2009 28 / 35
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 ′
•••
••••
•••••
••••••
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 December 15, 2009 29 / 35
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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
bernstein December 15, 2009 31 / 35
Polyhedral Program Analysis and Transformation Overview
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])
}
• • • • •
◦◦◦◦◦
•
•
•
•
•
◦
◦
◦
◦
◦
bernstein December 15, 2009 32 / 35
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 December 15, 2009 32 / 35
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?
m(N) = max(i ,j)∈S
(ij + i − N + 1)
with S = { (i , j) ∈ Z2 | 0 ≤ i < N ∧ i ≤ j < N }
bernstein December 15, 2009 32 / 35
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?
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
bernstein December 15, 2009 33 / 35
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
Introduction and RepresentationParametric Integer Programming
3 CLooG
4 barvinok
5 bernstein
6 Conclusions
Conclusions December 15, 2009 35 / 35
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
Conclusions December 15, 2009 35 / 35
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
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