An Algorithm for Polytope Decomposition and Exact Computation of Multiple Integrals.
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Transcript of An Algorithm for Polytope Decomposition and Exact Computation of Multiple Integrals.
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