An Algorithm for Polytope Decomposition and Exact Computation of Multiple

Integrals

Overview

Definitions Background Some algorithmic problems in polytope

theory Repetitive decomposition of a polyhedron

Calculating multiple integrals (and volumes) Uniformly repetitive decomposition of

polyhedra finding distribution functions

Definition

Polyhedron Examples of bounded and

unbounded polyhedra

.bAxdxP : R

x

y

z

}0,0,0{ zyxP

Polytope

A Polytope is a bounded polyhedron.

H-V representation of polytope

1V

2V

3V

4V

1H 2H

V-Representation

3H

4H

H-Representation

Simplex

1V

1V

2V

3V

2V

3V

4V

A Simplex has vertices. )1( d

Triangulation

(a) A triangulation using 6-simplices (b) A triangulation using 5-simplices

Boundary triangulation

P

c

b

a

d

e

P

Signed Decomposition Methods

iiP

P P

a

d

b

c

e

ade

cdecbe

abe

Volume of simplex

.!

),...,,( 00201

d

vvvvvvDetV d

be the vertices of

dvvv ,...,, 21 dLet a -simplex

The volume of the simplex is:

Some algorithmic problems in polytope theory

Number of vertices Input: Polytope in -

representation Output: Number of vertices of Status (general): -complete Status (fixed dim.): Polynomial time

P#

HP

P

Some algorithmic problems in polytope theory (cont.)

Minimum triangulation Input: Polytope in -

representation, positive integer k Output: “Yes” if has a triangulation

of size k or less, “No” otherwise Status (general): -complete Status (fixed dim.): -complete

NP

NP

HP

P

Minimum Triangulation

2V3V

4V

5V6V

1V

A polygon has simplices minimum 2n

P

Minimum Triangulation

(a) A triangulation using 6-simplices (b) A triangulation using 5-simplices

5323 dn

Some algorithmic problems in polytope theory (cont.)

Volume Input: Polytope in -

representation, Output: Volume of P Status (general): -complete Status (fixed dim.): Polynomial time

P#

P H

Repetitive decomposition of a

polyhedron

Definition of a repetitive polyhedron

A polytope is repetitive if it may be represented in the form

for appropriate and linear functions

}...,

,,:),...,,{(

1111

11211121

ddddd

d

xfxxf

xfxxfbxaxxxP

dP R

Rba,

.11,:, diff iii RR

Example of a repetitive polyhedron

y

x

1 1

1

.1

,10

,11

yx

y

x

P

Theorem 1Theorem 1:

Any polyhedron P is effectively decomposable into a union of finitely many repetitive polyhedra, the intersection of any two of which is contained in a

-dimensional polytope. 1d

)2(

0,...,,

0,...,,

21

211

dm

d

xxxf

xxxf

Proof of Theorem 1.

Proof of Theorem 1.(cont.)

Proof of Theorem 1.(cont.)

Proof of Theorem 1.(cont.)

Decomposition into repetitive polytopes

Y

X

xg 1

xg 2

xg 3

xg 4

xg 5

2a1a

.

,

,

3

2

1

gy

gy

gy

.

,

,

,

14

54

31

21

4,1

gg

gg

gg

gg

Q

.

,

5

4

gy

gy

Decomposition into repetitive polytopes (cont.)

Y

X

xg 1

xg 2

xg 3

xg 4

xg 5

2a 3a 4a1a5a

5435

4325

3224

2114

,

,

,

,

axagyg

axagyg

axagyg

axagyg

P

Decomposition into repetitive polytopes (cont.)

Y

X

xg 1

xgk

xgk1 xgl

d

d

m

m

4

2

a b

Decomposition into repetitive polyhedra

2a 3a4a1a x

y

.,

,

,

,

435

4334

3224

2114

xagyg

axagyg

axagyg

axagyg

P1g

2g

3g

5g

4g

Multiple integral for repetitive polyhedron

b

a

xf

xf

xf

xf

dd

dd

dd

dd

dxdxdxxxxf

dxdxdxxxxfIP

11

11

11

11

1221

1221

...),...,,(...

...),...,,(...

Multiple integral

v

i

b

a

xf

xf

xf

xf

dd

dd

dd

dd

dxdxdxxxxf

dxdxdxxxxfIP

11221

1221

11

11

11

11

...),...,,(...

...),...,,(...

Volume of a repetitive polytope

b

a

xf

xf

xf

xf

d

dd

dd

dxdxdxPVol11

11

11

11

12......)(

Uniformly repetitive decomposition of

polyhedra

Background

Let be a -dimensional random variable, uniformly distributed in the polytope . That is, the probability of

to assume a value in some set is .

),...,,( 21 dXXX d

P

),...,,( 21 dXXX

PA )(

)(

PVol

AVol

Background(cont.)

Consider a 1-dimensional random variable of the form for some constants . Then the value of the distribution function at any point t is

.

dd XcXcXcT ...2211

dccc ,...,, 21

tFT

)(

)...:( 2211

PVol

txcxcxcxPVol ddd R

Classical example

Let , where is uniformly distributed in the d -dimensional cube . That is, is the sum of d independent variables distributed uniformly in .For example,

t

dd XXXS ...21

dS d1,0

dd

i

id it

i

d

dtF

0

1!

1

2

ttt

1

1

),...,,( 21 dXXX

1,0 )(1 tF

Classical example

x

.2,1

,21,122

1

,10,2

1

,0,0

22

11

2

1

2

2

2222

t

ttt

tt

t

ttttF

t

Example

Let

We would like to express as a function of .

1,0,,:,, 3 zyxzyxRzyxP

tFT t

x

z

y

ZYXZYXLT 32),,(

Example(cont.)

1,0,,:,, 3 zyxzyxRzyxP

}32:),,{( tzyxzyxPtPL

032

1

0

0

0

zyxt

zyx

z

y

x

tPL

)(

)32:),,((

PVol

tzyxzyxPVoltFT

Definition of uniformly repetitive polyhedra

Let be a family of polyhedra, where is some interval (finite or infinite). The family is uniformly repetitive if there exist linear functions

, such that

(where some of the functions or may be replaced by or ).

}:{ ItPt

},,...,

,,,,:{

1111

10210010

ddddd

dt

xtfxxtf

xtfxxtftfxtfRP x

11,:, diff iii RR

if

if

I

Example of decomposition into uniformly repetitive families(cont.)

1,0,,:,, 3 zyxzyxRzyxP

}32:),,{( tzyxzyxPtPL

032

1

0

0

0

zyxt

zyx

z

y

x

tPL

)(

)32:),,((

PVol

tzyxzyxPVoltFT

Example of decomposition into uniformly repetitive families

}101010{

}101232

30{

}101012

3{

}3

20230

2

32{

}3

20

2020{

}101232

32{

}3

20

201{

,7

,6

,5

,4

,3

,2

,1

yxzxyxP

yxzxyxtt

xP

yxzxyxt

P

yxtzxty

txtP

yxtz

xtyxP

yxzxyxtt

xtP

yxtz

xtytxP

t

t

t

t

t

t

t

Result of decomposition

.3,

,32,

,21,

,10,

,0Ø,

,7

,6,5

,5,4,3,2

,1

tP

tPP

tPPPP

tP

t

tP

t

tt

tttt

t

L

Theorem 2:( )

Let be a polyhedron and a linear function. Then we can effectively find a decomposition of , say , into a union of finitely many (finite and infinite ) intervals, and uniformly repetitive families

, such that

R k

j jI1

R

}:{ ,, jtij ItP

71,1

,, kjItPtP j

l

itijL

j

dP R dL RR :

})(:{ tLPtP dL xRx

Proof of Theorem 2.

Proof of Theorem 2.(cont.)

Proof of Theorem 2.(cont.)

Example of decomposition into uniformly repetitive families(cont.)

1,0,,:,, 3 zyxzyxRzyxP

}32:),,{( tzyxzyxPtPL

032

1

0

0

0

zyxt

zyx

z

y

x

tPL

)(

)32:),,((

PVol

tzyxzyxPVoltFT

Example of decomposition into uniformly repetitive families

}101010{

}101232

30{

}101012

3{

}3

20230

2

32{

}3

20

2020{

}101232

32{

}3

20

201{

,7

,6

,5

,4

,3

,2

,1

yxzxyxP

yxzxyxtt

xP

yxzxyxt

P

yxtzxty

txtP

yxtz

xtyxP

yxzxyxtt

xtP

yxtz

xtytxP

t

t

t

t

t

t

t

Result of decomposition

.3,

,32,

,21,

,10,

,0Ø,

,7

,6,5

,5,4,3,2

,1

tP

tPP

tPPPP

tP

t

tP

t

tt

tttt

t

L

EXAMPLE (CONT.) Distribution function

.3,1

,32,6/2/32/92/7

,21,2/2/32/32/1

,10,3/

,00,

32

32

t

tttt

tttt

tt

t

tFL

Theorem 3:

Let be a -dimensional random variable, uniformly distributed in a polytope of positive volume in . Given any constants

, the distribution function of the 1-dimensional random variable

is a continuous piecewise polynomial function of the degree at most , and can be effectively computed.

d

dXXX ,...,, 21

dR

dd XcXcXcT ...2211

dccc ,...,, 21

d

Distribution function

.,)(: ttLPtP dL xRx

.),...,,(,...)( 212211d

ddd xxxxcxcxcL Rxx

.,)(

))(( t

PVol

tPVoltF L

T

Polytope decomposition

Questions & Answers