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2 - 1
Probability and Delaunay triangulations 1 Randomized algorithms for - - PowerPoint PPT Presentation
Probability and Delaunay triangulations 1 Randomized algorithms for - - PowerPoint PPT Presentation
Probability and Delaunay triangulations 1 Randomized algorithms for Delaunay triangulations Poisson Delaunay triangulation 2 - 1 Randomized algorithms for Delaunay triangulations Randomized backward analysis of binary trees Randomized
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2 - 2
- Randomized backward analysis of binary trees
- Randomized incremental construction of Delaunay
- Jump and walk
- The Delaunay hierarchy
- Biased randomized incremental order
- Chew algorithm for convex polygon
Randomized algorithms for Delaunay triangulations Poisson Delaunay triangulation
- Poisson distribution
- Slivnyak-Mecke formula
- Blaschke-Petkanschin variables substitution
- Stupid analysis of the expected degree
- Straight walk expected analysis
- Catalog of properties
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3 - 1 1 1
Sorting
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3 - 2 1 1
Sorting
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4 - 1
Binary tree
8
Sorting
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4 - 2
Binary tree
8
Sorting
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4 - 3
Binary tree
8
8 > 8 Sorting
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4 - 4
Binary tree
8 4
Sorting
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4 - 5
Binary tree
8 4 7
Sorting
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4 - 6
Binary tree
8 4 7 14
Sorting
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4 - 7
Binary tree
8 4 7 14 12
Sorting
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4 - 8
Binary tree
8 4 7 14 12 1
Sorting
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4 - 9
Binary tree
8 4 7 14 12 1 11
Sorting
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4 - 10
Binary tree
8 4 7 14 12 1 11 ] 1, 1]
Sorting
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4 - 11
Binary tree
8 4 7 14 12 1 11 ] 1, 1] ]1, 4]
Sorting
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4 - 12
Binary tree
8 4 7 14 12 1 11 1 1 8 4 7 14 12 1 11
Sorting Sorting Sorting
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5 - 1 8 time new drawing
1
Sorting
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5 - 2 8 4 time new drawing
1 2
Sorting
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5 - 3 8 4 7 time new drawing
1 2 3
Sorting
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5 - 4 8 4 7 14 time new drawing
1 2 3 4
Sorting
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5 - 5 8 4 7 14 12 time new drawing
1 2 3 4 5
Sorting
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5 - 6 8 4 7 14 12 1 time new drawing
1 2 3 4 5 6
Sorting
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5 - 7 8 4 7 14 12 1 11 time new drawing
1 2 3 4 5 6 7
Sorting
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5 - 8 8 4 7 14 12 1 11 time new drawing
1 2 3 4 5 6 7
1 1 8 4 7 14 12 1 11
Sorting
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5 - 9 8 4 7 14 12 1 11 time new drawing
1 2 3 4 5 6 7
1 1 8 4 7 14 12 1 11
Sorting
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6 - 1 time
Sorting
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6 - 2 time 1 1
Sorting
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6 - 3 time 1 1
k
Sorting
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6 - 4 time 1 1
k
Sorting
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6 - 5 time 1 1
k n Localisation
Sorting
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6 - 6 time 1 1
k n Localisation
Sorting
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6 - 7 time 1 1
k n Localisation
Sorting
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6 - 8 time 1 1
k n Localisation
Sorting
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6 - 9 time 1 1
k n Localisation
Sorting
E []visited nodes] O( 2
k)
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6 - 10 time 1 1
k n Localisation
Total insertion : X
k
2 k ' 2 log n
Sorting
E []visited nodes] O( 2
k)
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6 - 11 time 1 1
k n Localisation
Total insertion : X
k
2 k ' 2 log n
Sorting
Total construction : X
k
2 log k ' 2n log n
E []visited nodes] O( 2
k)
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7 - 1 new drawing
] 1, 1[ conflict graph
Sorting
1 1 8 4 7 14 12 1 11
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7 - 2 8
] 1, 8[ ]8, 1[
Sorting
1 1 8 4 7 14 12 1 11
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7 - 3 8 4
] 1, 8[ ]8, 1[
14
] 1, 4[ ]4, 8[ ]8, 14[ ]14, 1[
Sorting
1 1 8 4 7 14 12 1 11
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8
Quicksort Unbalanced binary tree History graph Conflict graph Same analysis O(n log n) Backwards analysis Analyse last insertion and sum Last object is a random object
Sorting
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9
Randomization
Backwards analysis for Delaunay triangulation
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10 - 1
Delaunay triangulation ] of triangles during incremental construction?
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10 - 2
Delaunay triangulation ] of triangles during incremental construction?
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10 - 3
Delaunay triangulation ] triangles created/incident to last point? ] of triangles during incremental construction?
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10 - 4
Delaunay triangulation ] triangles created/incident to last point?
Last point?
] of triangles during incremental construction?
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10 - 5
1 n n
X
i=1
d(pi) 6
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10 - 6
1 n n
X
i=1
d(pi) 6
P 6 = 6n
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11 - 1 Alternative analysis Triangle ∆ with j stoppers
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11 - 2 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
if 2 j 1
α
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11 - 3 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
Size of the triangulation of the sample
=
n
X
j=0
P[∆withj stoppersisthere] ⇥ ]∆withj stoppers
if 2 j 1
α
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11 - 4 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
Size of the triangulation of the sample
=
n
X
j=0
P[∆withj stoppersisthere] ⇥ ]∆withj stoppers
- 1/α
X
j=0
↵3 4 ⇥ ]∆ with j stoppers if 2 j 1
α
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11 - 5 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
Size of the triangulation of the sample
=
n
X
j=0
P[∆withj stoppersisthere] ⇥ ]∆withj stoppers
- 1/α
X
j=0
↵3 4 ⇥ ]∆ with j stoppers= ↵3]∆ with 1
α stoppers
if 2 j 1
α
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11 - 6 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
Size of the triangulation of the sample
=
n
X
j=0
P[∆withj stoppersisthere] ⇥ ]∆withj stoppers
- 1/α
X
j=0
↵3 4 ⇥ ]∆ with j stoppers= ↵3]∆ with 1
α stoppers
= O(↵n) if 2 j 1
α
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11 - 7 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists in the triangulation of a sample of size ↵n
' ↵3(1 ↵)j ↵3(1 ↵)
1 α 1
4↵3
Size of the triangulation of the sample
=
n
X
j=0
P[∆withj stoppersisthere] ⇥ ]∆withj stoppers
- 1/α
X
j=0
↵3 4 ⇥ ]∆ with j stoppers= ↵3]∆ with 1
α stoppers
= O(↵n) Size (order k Voronoi) αn
α3 = nk2
if 2 j 1
α
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11 - 8 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
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11 - 9 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of created triangles
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11 - 10 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of created triangles
=
n
X
j=0
(P[∆ with j] P[∆ with j + 1]) ⇥ ]∆ with j stoppers
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11 - 11 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of created triangles
=
n
X
j=0
(P[∆ with j] P[∆ with j + 1]) ⇥ ]∆ with j stoppers '
n
X
j=0
18 j4 ⇥ nj2 = O(n X 1 j2 ) = O(n)
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11 - 12 Alternative analysis Triangle ∆ with j stoppers
It remains to analyze conflict location Conflict graph / History graph
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11 - 13 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
j ⇥ P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of conflicts occuring
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11 - 14 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
j ⇥ P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of conflicts occuring
=
n
X
j=0
j ⇥ (P[∆ with j] P[∆ with j + 1]) ⇥ ]∆ with j stoppers
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11 - 15 Alternative analysis Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
j ⇥ P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of conflicts occuring
=
n
X
j=0
j ⇥ (P[∆ with j] P[∆ with j + 1]) ⇥ ]∆ with j stoppers '
n
X
j=0
j ⇥ 18 j4 ⇥ nj2 = O(n X 1 j ) = O(nlog n)
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12 - 1
History graph
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12 - 2
History graph
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12 - 3
History graph
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12 - 4
Father Stepfather History graph
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13 - 1
Father Stepfather (Delaunay tree) History graph
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13 - 2
Father Stepfather (Delaunay tree) if conflict there was a conflict with the father
- r the stepfather
- r both
History graph
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14 - 1 Conflict graph
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14 - 2 Conflict graph
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14 - 3 Conflict graph
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14 - 4 Conflict graph Insert
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14 - 5 Conflict graph
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14 - 6 Conflict graph
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14 - 7 Conflict graph
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15 - 1 Walk
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15 - 2 Walk
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15 - 3 Walk
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15 - 4 Walk Complexity O(n)
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15 - 5 Walk Complexity O(n) T e a s e r p r
- b
a b i l i t y l e c t u r e Better bounds for random points
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16 - 1 Jump and walk
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16 - 2 Jump and walk
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16 - 3 Jump and walk
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16 - 4 Jump and walk Hopefully shorter walk Designed for random points O( 3 pn) expected location time
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17 - 1 Jump and walk (no distribution hypothesis)
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17 - 2 Jump and walk (no distribution hypothesis) E [] of in ] = n
k
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17 - 3 Jump and walk (no distribution hypothesis) choose k =
2
pn Walk length = O n
k
- E [] of
in ] = n
k
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17 - 4 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy Walk length = O n
k
- E [] of
in ] = n
k
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17 - 5 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1
Walk length = O n
k
- E [] of
in ] = n
k
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17 - 6 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1 + k1 k2
Walk length = O n
k
- E [] of
in ] = n
k
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17 - 7 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1 + k1 k2
Walk length = O n
k
- E [] of
in ] = n
k
+ k2
k3 + . . .
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17 - 8 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1 + k1 k2
Walk length = O n
k
- E [] of
in ] = n
k
+ k2
k3 + . . .
choose
ki ki+1 = ↵
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17 - 9 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1 + k1 k2
Walk length = O n
k
- E [] of
in ] = n
k
+ k2
k3 + . . .
choose
ki ki+1 = ↵
point location in O (↵ logα n)
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17 - 10 Jump and walk (no distribution hypothesis) choose k =
2
pn Delaunay hierarchy
n k1 + k1 k2
Walk length = O n
k
- E [] of
in ] = n
k
+ k2
k3 + . . .
choose
ki ki+1 = ↵
point location in O (↵ logα n) point location in O (p↵ logα n)
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18 - 1 Technical detail Walk length = O ] of in
( ) = O
n
k
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18 - 2 Technical detail Walk length = O ] of in
( ) = O
n
k
- random point
not a random point
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18 - 3 Technical detail Walk length = O ] of in
( ) = O
n
k
- random point
not a random point E [d ] = 1 n X
q
dNN(q) = 1 n X
q
X
v=NN(q)
dv = 1 n X
v
X
q;v=NN(q)
dv 1 n X
v
6dv 36
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18 - 4 Technical detail Walk length = O ] of in
( ) = O
n
k
- random point
not a random point E [d ] = 1 n X
q
dNN(q) = 1 n X
q
X
v=NN(q)
dv = 1 n X
v
X
q;v=NN(q)
dv 1 n X
v
6dv 36 P d
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19 How many randomness is necessary? If the data are not known in advance shuffle locally
Randomization
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20
Drawbacks of random order non locality of memory access data structure for point location Hilbert sort
Randomization
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21 - 1
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21 - 2
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21 - 3
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21 - 4
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22
Drawbacks of random order non locality of memory access data structure for point location Hilbert sort Last point is not at all a random point Walk should be fast no control of degree of last point
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23 - 1
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23 - 2
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23 - 3
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23 - 4
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23 - 5
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23 - 6
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24 - 1 Triangle ∆ with j stoppers
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24 - 2 Triangle ∆ with j stoppers Size (order k Voronoi) αn
α3 = nk2
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24 - 3 Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
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24 - 4 Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
remains Θ(j3)
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24 - 5 Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of created triangles
' O( X nj2 j4 ) = O(n)
remains Θ(j3)
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24 - 6 Triangle ∆ with j stoppers
Probability that it exists during the construction
=
3 j+3 2 j+2 1 j+1
=
n
X
j=0
j ⇥ P[∆ with j stoppers appears] ⇥ ]∆ with j stoppers
] of conflicts occuring
' O( X j nj2 j4 ) = O(nlog n)
remains Θ(j3)
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25 - 1 random order (visibility walk) x-order Hilbert order Biased order (Spatial sorting) locate using Delaunay hierarchy
0.7 seconds 157 seconds 3 seconds 0.8 seconds 6 seconds
Delaunay 2D 1M random points
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25 - 2 random order (visibility walk) x-order Hilbert order Biased order (Spatial sorting) locate using Delaunay hierarchy
Delaunay 2D 100K parabola points
128 seconds 632 seconds 46 seconds 0.3 seconds 0.3 seconds
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