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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