Finding Bounds on Ehrhart Quasi-Polynomials

FACULTY OF ENGINEERING

Finding Bounds on Ehrhart Quasi-Polynomials

Harald Devos Jan Van Campenhout Dirk Stroobandt

ELIS/PARISGhent University

Seventh ACES SymposiumSeptember 17–18 2007, Edegem

Outline

1 IntroductionWhat are (Ehrhart) quasi-polynomials?Where do they arise?Why do we need bounds on quasi-polynomials?

2 How do we find bounds?Continuous versus discrete domain extrema of polynomialsConverting quasi-polynomials into polynomials

3 Conclusions and Future Work

H. Devos et al. (Ghent University) Finding Bounds on Ehrhart Quasi-Polynomials ACES’07, Edegem 2 / 19

Outline





Periodic Numbers

Example

0 1 2 3 4 5 6 7 8n

0123456u(n)

u(n) = [3, 1, 4]n


Periodic Numbers

DefinitionLet n be a discrete variable, i.e. n ∈ Z. A 1-dimensional periodic numberis a function that depends periodically on n.

u(n) = [u0, u1, . . . , ud−1]n =

u0 if n ≡ 0 (mod d)

u1 if n ≡ 1 (mod d)...

ud−1 if n ≡ d − 1 (mod d)

d is called the period.


Quasi-Polynomials

Example

f (n) = −[

12 , 1

3

]nn2 + 3n − [1, 2]n

=

{− 1

3n2 + 3n − 2 if n ≡ 0 (mod 2)

− 12n2 + 3n − 1 if n ≡ 1 (mod 2)

0 1 2 3 4 5 6 7n

−3

−2

−1

0

1

2

3

4

5

f(n

)=

−

[

1 2,

1 3

]

n

n2+

3n−

[1,2

] n

−

1

2n2 + 3n − 1

−

1

3n2 + 3n − 2


Quasi-Polynomials

DefinitionA polynomial in a variable x is a linear combination of powers of x :

f (x) =

g∑i=0

cixi

DefinitionA quasi-polynomial in a variable x is a polynomial expression withperiodic numbers as coefficients:

f (n) =

g∑i=0

ui (n)ni

with ui (n) periodic numbers.


Quasi-Polynomials

DefinitionA polynomial in a variable x is a linear combination of powers of x :

f (x) =

g∑i=0

cixi

DefinitionA quasi-polynomial in a variable x is a polynomial expression withperiodic numbers as coefficients:

f (n) =

g∑i=0

ui (n)ni

with ui (n) periodic numbers.


Outline





Where do Quasi-Polynomials arise?

Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =3

x + y ≤ p

p f (p)3 5

4 85 106 13

5

2p +

[−2,

−5

2

]p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =4

x + y ≤ p

p f (p)3 54 8

5 106 13

5

2p +

[−2,

−5

2

]p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =5

x + y ≤ p

p f (p)3 54 85 10

6 13

5

2p +

[−2,

−5

2

]p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =6

x + y ≤ p

p f (p)3 54 85 106 13

5

2p +

[−2,

−5

2

]p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =6

x + y ≤ p

p f (p)3 54 85 106 13

5

2p +

[−2,

−5

2

]p



The number of integer points in a parametric polytope Pp ofdimension n is expressed as a piecewise a quasi-polynomial of degreen in p (Clauss and Loechner).

More general polyhedral counting problems:Systems of linear inequalities combined with ∨,∧,¬,∀, or ∃(Presburger formulas).

Many problems in static program analysis can be expressed aspolyhedral counting problems.


Outline





Why do we need bounds on quasi-polynomials?

Some problems in static program analysis need bounds onquasi-polynomials.

Example

Number of live elements = quasi-polynomial⇓

Memory usage = maximum over all execution points


Outline





Continuous vs. Discrete domain extrema of polynomials

Discrete domain ⇒ evaluate in each pointNot possible for

parametric domains

large domains (NP-complete)


Continuous vs. Discrete domain extrema of polynomials

0 1 2 3 4n

0

1

2

3

4

5

6

f(n

)=

1 3n

3−

2n2+

5 3n

+5 F c

Fd

Fd

F c

The relative difference issmaller for

I larger intervalsI lower degree

⇒ Continuous-domain extremacan be used as approximationof discrete-domain extrema.


Outline





How: Mod Classes

Example

0 1 2 3 4 5 6 7n

−2

−1

0

1

2

3

4

5f(n)

f0 = −

1

2n2 + 3n − 1

f1 = −

1

3n2 + 3n − 2F c

Fd

F c = Fd

Good for

small period

large domains


How: Other Methods

Other methods

needed for large periods

offer trade-off betweenaccuracy andcomputation time

see poster


Conclusions and Future Work

Bounds on quasi-polynomials useful for static program analysis

Different methods fit different situations (period, degree, domainsize).

OutlookI A hybrid method should be constructed.I Parametric bounds on parameterized quasi-polynomials


Documents

Finding Bounds on Ehrhart Quasi-Polynomials