The document discusses advancements in geometric algorithms through dynamic determinant computation, emphasizing efficient handling of geometric predicates which rely on determinants as the dimensionality increases. It outlines concepts such as the convex hull problem, various computational techniques including the Sherman-Morrison formula, and performance comparisons of different implementations using statistical experiments. Additionally, it provides details on experiment results related to determinant computation time and memory consumption across various algorithms and input sizes.

1. Faster Geometric Algorithms via Dynamic Determinant Computation
Faster Geometric Algorithms via
Dynamic Determinant Computation
Vissarion Fisikopoulos
joint work with L. Pe˜aranda (now IMPA, Rio de Janeiro)
n
Department of Informatics, University of Athens
ESA, Ljubljana, 12.Sep.2012
V.Fisikopoulos | ESA’12
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2. Faster Geometric Algorithms via Dynamic Determinant Computation
Geometric algorithms and predicates
Setting
geometric algorithms → sequence of geometric predicates
many geometric predicates → determinants
as dimension grows predicates become more expensive
V.Fisikopoulos | ESA’12
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3. Faster Geometric Algorithms via Dynamic Determinant Computation
Geometric algorithms and predicates
Setting
geometric algorithms → sequence of geometric predicates
many geometric predicates → determinants
as dimension grows predicates become more expensive
Examples
b
Orientation: Does c
lie on, left or right
of ab?
ax ay 1
bx by 1
cx cy 1
V.Fisikopoulos | ESA’12
a
0
c
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4. Faster Geometric Algorithms via Dynamic Determinant Computation
Geometric algorithms and predicates
Setting
geometric algorithms → sequence of geometric predicates
many geometric predicates → determinants
as dimension grows predicates become more expensive
Examples
a
inCircle: Does d lie
on, inside or outside
of abc?
ax
bx
cx
dx
ay
by
cy
dy
V.Fisikopoulos | ESA’12
a2 + a2
x
y
b2 + b2
x
y
c2 + c2
x
y
d2 + d2
x
y
b
1
1
1
1
d
0
c
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5. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Outline
1 Dynamic determinant computation
2 Geometric algorithms: convex hull
3 Implementation - experiments
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6. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Outline
1 Dynamic determinant computation
2 Geometric algorithms: convex hull
3 Implementation - experiments
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7. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Determinant: Exact Computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω ), ω ∼ 2.7 [KaltofenVillard05]
Practice: Gaussian elimination, O(d3 )
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8. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Determinant: Exact Computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω ), ω ∼ 2.7 [KaltofenVillard05]
Practice: Gaussian elimination, O(d3 )
Division-free algorithms
Laplace (cofactor) expansion, O(d!)
[Rote01], O(d4 )
[Bird11], O(dω+1 )
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9. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Determinant: Exact Computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω ), ω ∼ 2.7 [KaltofenVillard05]
Practice: Gaussian elimination, O(d3 )
Division-free algorithms
Laplace (cofactor) expansion, O(d!)
[Rote01], O(d4 )
[Bird11], O(dω+1 )
Working on integers
combinatorics, lattice polyhedra, algebraic geometry
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10. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Dynamic Determinant Computations
One-column update problem
Given matrix A ⊆ Rd×d , answer queries for det(A) when i-th column of
A, (A)i , is replaced by u ⊆ Rd .
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11. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Dynamic Determinant Computations
One-column update problem
Given matrix A ⊆ Rd×d , answer queries for det(A) when i-th column of
A, (A)i , is replaced by u ⊆ Rd .
Solution: Sherman-Morrison formula (1950)
A −1 = A−1 −
(A−1 (u − (A)i )) (eT A−1 )
i
1 + eT A−1 (u − (A)i )
i
det(A ) = (1 + eT A−1 (u − (A)i )det(A)
i
Complexity: O(d2 )
V.Fisikopoulos | ESA’12
Algorithm: dyn inv
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12. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Dynamic Determinant Computations
One-column update problem
Given matrix A ⊆ Rd×d , answer queries for det(A) when i-th column of
A, (A)i , is replaced by u ⊆ Rd .
Variant Sherman-Morrison formulas
Q: What happens if we work over the integers?
V.Fisikopoulos | ESA’12
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13. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Dynamic Determinant Computations
One-column update problem
Given matrix A ⊆ Rd×d , answer queries for det(A) when i-th column of
A, (A)i , is replaced by u ⊆ Rd .
Variant Sherman-Morrison formulas
Q: What happens if we work over the integers?
A: Use the adjoint of A (Aadj ) → exact divisions
V.Fisikopoulos | ESA’12
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14. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation
Dynamic Determinant Computations
One-column update problem
Given matrix A ⊆ Rd×d , answer queries for det(A) when i-th column of
A, (A)i , is replaced by u ⊆ Rd .
Variant Sherman-Morrison formulas
Q: What happens if we work over the integers?
A: Use the adjoint of A (Aadj ) → exact divisions
Complexity: O(d2 )
V.Fisikopoulos | ESA’12
Algorithm: dyn adj
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15. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
Outline
1 Dynamic determinant computation
2 Geometric algorithms: convex hull
3 Implementation - experiments
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16. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Example
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17. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
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18. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
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19. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

20. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

21. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

22. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

23. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

24. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

25. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

26. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
V.Fisikopoulos | ESA’12
8 / 16

27. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem
Definition
The convex hull of A ⊆ Rd is the smallest convex set containing A.
Incremental convex hull construction
connect each outer point to the
visible segments
visibility test = orientation test
Similar problems: Delaunay, regular triangulations, point-location
V.Fisikopoulos | ESA’12
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28. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem: REVISITED
convex hull → sequence of similar orientation predicates
take advantage of computations done in previous steps
V.Fisikopoulos | ESA’12
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29. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem: REVISITED
convex hull → sequence of similar orientation predicates
take advantage of computations done in previous steps
Incremental convex hull construction → 1-column updates
p1
A=
p1 p2 p3
1
1
1
p2
p3
Orientation(p1 , p2 , p3 ) = det(A) in O(dω )
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30. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem: REVISITED
convex hull → sequence of similar orientation predicates
take advantage of computations done in previous steps
Incremental convex hull construction → 1-column updates
p1
A =
p1 p4 p3
p4
1
1
1
p2
p3
Orientation(p1 , p2 , p4 ) = det(A ) in O(d2 )
V.Fisikopoulos | ESA’12
9 / 16

31. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
The convex hull problem: REVISITED
convex hull → sequence of similar orientation predicates
take advantage of computations done in previous steps
Incremental convex hull construction → 1-column updates
p1
A =
p1 p4 p3
p4
1
1
1
p2
p3
Orientation(p1 , p2 , p4 ) = det(A ) in O(d2 )
Store det(A), A−1 + update det(A ), A −1 (Sherman-Morrison)
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32. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull
Dynamic determinants in geometric algorithms
Given A ⊆ Rd , n = |A| and t = the number of cells
Theorem
Orientation predicates in increm. convex hull: O(d2 ) (space: O(d2 t))
Proof: update det(A ), A −1
Corollary
Orientation predicates involved in point-location: O(d) (space: O(d2 t))
Proof: update only det(A )
Corollary
Incremental convex hull and volume comput.: O(dω+1 nt) → O(d3 nt)
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33. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Outline
1 Dynamic determinant computation
2 Geometric algorithms: convex hull
3 Implementation - experiments
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34. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Determinant implementation
Dynamic determinant computation
C++, GNU Multiple Precision arithmetic library (GMP)
Implement dyn inv & dyn adj
Exact determinant computation software
LU decomposition (CGAL)
optimized LU decomposition (Eigen)
asymptotically optimal algorithms (LinBox)
Maple’s default determinant algorithm (Maple 14)
Bird’s algorithm (our implementation)
Laplace (cofactor) expansion (our implementation)
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35. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Determinant experiments
Uniform distributed rational coefficients, matrix A ⊆ Qd×d .
d
3
4
5
6
7
8
9
10
spec:
Bird
.013
.046
.122
.268
.522
.930
1.520
2.380
CGAL
.021
.050
.110
.225
.412
.710
1.140
1.740
Eigen
.014
.033
.072
.137
.243
.390
.630
.940
Laplace
.008
.020
.056
.141
.993
–
–
–
Maple
.058
.105
.288
.597
.824
1.176
1.732
2.380
dyn inv
.046
.108
.213
.376
.613
.920
1.330
1.830
dyn adj
.023
.042
.067
.102
.148
.210
.310
.430
Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux
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36. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Determinant experiments
Uniform distributed integer coefficients, matrix A ⊆ Zd×d .
d
3
4
5
6
7
8
9
10
11
12
13
spec:
Bird
.002
.012
.032
.072
.138
.244
.408
.646
.956
1.379
1.957
CGAL
.021
.041
.080
.155
.253
.439
.689
1.031
1.485
2.091
2.779
Eigen
.013
.028
.048
.092
.149
.247
.376
.568
.800
1.139
1.485
Laplace
.002
.005
.016
.040
.277
–
–
–
–
–
–
Linbox
.872
1.010
1.103
1.232
1.435
1.626
1.862
2.160
10.127
13.101
–
Maple
.045
.094
.214
.602
.716
.791
.906
1.014
1.113
1.280
1.399
dyn inv
.030
.058
.119
.197
.322
.486
.700
.982
1.291
1.731
2.078
dyn adj
.008
.015
.023
.033
.046
.068
.085
.107
.133
.160
.184
Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux
V.Fisikopoulos | ESA’12
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37. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Convex hull implementation
Hashed dynamic determinants
dyn inv & dyn adj + hash table (dets & matrices)
geometric software → geometric predicates
Convex hull software
randomized incremental (triangulation/CGAL)
beneath-and-beyond (bb) (polymake)
double description method (cdd)
gift-wrapping with reverse search (lrs)
triangulation + hashed dynamic determinants = hdch z or hdch q
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38. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Convex hull experiments
6-dim points, integer coeffs uniformly distributed inside a 6-ball
70
hdch_q
hdch_z
triang
lrs
bb
cdd
60
Time (sec)
50
hdch q (hdch z)
rational (integer)
arithmetic
drawback: memory
consuption
40
30
20
10
0
100
spec:
150
200
250
300
350
400
Number of input points
450
500
500 6D
triang
hdch z
hdch q
points
0.1GB
2.8GB
3.8GB
Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux
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39. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Point location experiments
triangulation of A ⊆ Zd points uniformly distibuted on a d-ball surface
1K and 1000K query points uniformly distibuted on a d-cube
d
hdch z
triang
hdch z
triang
hdch z
triang
hdch z
triang
|A|
8
8
9
9
10
10
11
11
120
120
70
70
50
50
39
39
preprocess
time (sec)
45.20
156.55
45.69
176.62
43.45
188.68
38.82
181.35
ds mem
(MB)
6913
134
6826
143
6355
127
5964
122
# cells
triang
319438
319438
265874
265874
207190
207190
148846
148846
query time (sec)
1K
1000K
0.41
392.55
14.42 14012.60
0.28
276.90
13.80 13520.43
0.27
217.45
14.40 14453.46
0.18
189.56
14.41 14828.67
up to 78 times faster using up to 50 times more memory
spec:
Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux
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40. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Conclusions and Future work
The code
http://hdch.sourceforge.net
Future work
Delaunay triangulations (inSphere predicate)
gift-wrapping convex hull algorithms
overcome large memory consumption (clean periodically the hash
table)
filtered computations
CGAL submission
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41. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments
Conclusions and Future work
The code
http://hdch.sourceforge.net
Future work
Delaunay triangulations (inSphere predicate)
gift-wrapping convex hull algorithms
overcome large memory consumption (clean periodically the hash
table)
filtered computations
CGAL submission
THANK YOU !!!
V.Fisikopoulos | ESA’12
17 / 16