"Faster Geometric Algorithms via Dynamic Determinant Computation."
0 likes10 views
The document proposes faster algorithms for geometric problems by using dynamic determinant computation. Many geometric algorithms involve computing determinants of matrices to evaluate geometric predicates. Computing determinants directly is expensive, especially for high-dimensional problems. The document presents an algorithm for dynamically updating determinants when a column of the matrix is changed in O(d^2) time, faster than recomputing from scratch. This dynamic determinant computation can speed up algorithms that require repeated predicate evaluations, such as the incremental convex hull algorithm, by updating determinants instead of recomputing them.
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Motivation
Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12]
[YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein])
Respol: compute resultant, discriminant polytopes up to dim. 6
4-d resultant polytope
V.Fisikopoulos | ESA’12 5 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-7-320.jpg)
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Motivation
Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12]
[YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein])
Respol: compute resultant, discriminant polytopes up to dim. 6
Counting lattice points in polyhedra [Barvinock’99] [DeLoera et.al’04],
Integer convex optimization [Gr¨otschel,Lov´asz,Schrijver’88]
4-d resultant polytope
V.Fisikopoulos | ESA’12 5 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-8-320.jpg)
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Motivation
Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12]
[YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein])
Respol: compute resultant, discriminant polytopes up to dim. 6
Counting lattice points in polyhedra [Barvinock’99] [DeLoera et.al’04],
Integer convex optimization [Gr¨otschel,Lov´asz,Schrijver’88]
4-d resultant polytope
Focusing on . . .
Hi-dim CompGeom (3 < d < 10)
Computation over the integers
V.Fisikopoulos | ESA’12 5 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-9-320.jpg)
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Existing work
Determinant: exact computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90]
Practice: Gaussian elimination, O(d3)
V.Fisikopoulos | ESA’12 6 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-10-320.jpg)
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Existing work
Determinant: exact computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90]
Practice: Gaussian elimination, O(d3)
Division-free determinant algorithms
Laplace (cofactor) expansion, O(d!)
[Rote01], O(d4)
[Bird11], O(dω+1)
V.Fisikopoulos | ESA’12 6 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-11-320.jpg)
![Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work
Existing work
Determinant: exact computation
Given matrix A ⊆ Rd×d
Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90]
Practice: Gaussian elimination, O(d3)
Division-free determinant algorithms
Laplace (cofactor) expansion, O(d!)
[Rote01], O(d4)
[Bird11], O(dω+1)
Dynamic determinant computation
Dynamic transitive closure in graphs [Sankowski FOCS’04]
V.Fisikopoulos | ESA’12 6 / 19](https://image.slidesharecdn.com/fisikopoulosesa12slides-200316102044/85/Faster-Geometric-Algorithms-via-Dynamic-Determinant-Computation-12-320.jpg)
"Faster Geometric Algorithms via Dynamic Determinant Computation."
- 1. Faster Geometric Algorithms via Dynamic Determinant Computation Faster Geometric Algorithms via Dynamic Determinant Computation Vissarion Fisikopoulos joint work with L. Pe˜naranda (now IMPA, Rio de Janeiro) Department of Informatics, University of Athens, Greece ESA, Ljubljana, 14.Sep.2012 V.Fisikopoulos | ESA’12 1 / 19
- 2. Faster Geometric Algorithms via Dynamic Determinant Computation Geometric algorithms and predicates Setting geometric algorithms → sequence of geometric predicates many geometric predicates → determinants Hi-dim: as dimension grows predicates become more expensive V.Fisikopoulos | ESA’12 2 / 19
- 3. Faster Geometric Algorithms via Dynamic Determinant Computation Geometric algorithms and predicates Setting geometric algorithms → sequence of geometric predicates many geometric predicates → determinants Hi-dim: as dimension grows predicates become more expensive Examples Orientation: Does c lie on, left or right of ab? ax ay 1 bx by 1 cx cy 1 0 a b c V.Fisikopoulos | ESA’12 2 / 19
- 4. Faster Geometric Algorithms via Dynamic Determinant Computation Geometric algorithms and predicates Setting geometric algorithms → sequence of geometric predicates many geometric predicates → determinants Hi-dim: as dimension grows predicates become more expensive Examples inCircle: Does d lie on, inside or outside of abc? ax ay a2 x + a2 y 1 bx by b2 x + b2 y 1 cx cy c2 x + c2 y 1 dx dy d2 x + d2 y 1 0 a b c d V.Fisikopoulos | ESA’12 2 / 19
- 5. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Outline 1 Motivation and existing work 2 Dynamic determinant computation 3 Geometric algorithms: convex hull 4 Implementation - experiments V.Fisikopoulos | ESA’12 3 / 19
- 6. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Outline 1 Motivation and existing work 2 Dynamic determinant computation 3 Geometric algorithms: convex hull 4 Implementation - experiments V.Fisikopoulos | ESA’12 4 / 19
- 7. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Motivation Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12] [YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein]) Respol: compute resultant, discriminant polytopes up to dim. 6 4-d resultant polytope V.Fisikopoulos | ESA’12 5 / 19
- 8. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Motivation Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12] [YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein]) Respol: compute resultant, discriminant polytopes up to dim. 6 Counting lattice points in polyhedra [Barvinock’99] [DeLoera et.al’04], Integer convex optimization [Gr¨otschel,Lov´asz,Schrijver’88] 4-d resultant polytope V.Fisikopoulos | ESA’12 5 / 19
- 9. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Motivation Algorithms for resultant polytopes [Emiris,F,Konaxis,Pe˜naranda SoCG’12] [YuJensen’12] (discriminant polytopes [Emiris, F, Dickenstein]) Respol: compute resultant, discriminant polytopes up to dim. 6 Counting lattice points in polyhedra [Barvinock’99] [DeLoera et.al’04], Integer convex optimization [Gr¨otschel,Lov´asz,Schrijver’88] 4-d resultant polytope Focusing on . . . Hi-dim CompGeom (3 < d < 10) Computation over the integers V.Fisikopoulos | ESA’12 5 / 19
- 10. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Existing work Determinant: exact computation Given matrix A ⊆ Rd×d Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90] Practice: Gaussian elimination, O(d3) V.Fisikopoulos | ESA’12 6 / 19
- 11. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Existing work Determinant: exact computation Given matrix A ⊆ Rd×d Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90] Practice: Gaussian elimination, O(d3) Division-free determinant algorithms Laplace (cofactor) expansion, O(d!) [Rote01], O(d4) [Bird11], O(dω+1) V.Fisikopoulos | ESA’12 6 / 19
- 12. Faster Geometric Algorithms via Dynamic Determinant Computation | Motivation and existing work Existing work Determinant: exact computation Given matrix A ⊆ Rd×d Theory: State-of-the-art O(dω), ω ∼ 2.37 [CoppersmithWinograd90] Practice: Gaussian elimination, O(d3) Division-free determinant algorithms Laplace (cofactor) expansion, O(d!) [Rote01], O(d4) [Bird11], O(dω+1) Dynamic determinant computation Dynamic transitive closure in graphs [Sankowski FOCS’04] V.Fisikopoulos | ESA’12 6 / 19
- 13. Faster Geometric Algorithms via Dynamic Determinant Computation | Dynamic determinant computation Outline 1 Motivation and existing work 2 Dynamic determinant computation 3 Geometric algorithms: convex hull 4 Implementation - experiments V.Fisikopoulos | ESA’12 7 / 19
- 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. V.Fisikopoulos | ESA’12 8 / 19
- 15. 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 i A−1) 1 + eT i A−1(u − (A)i) det(A ) = (1 + eT i A−1 (u − (A)i)det(A) Only vector×vector, vector×matrix → Complexity: O(d2) Algorithm: dyn inv V.Fisikopoulos | ESA’12 8 / 19
- 16. 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 8 / 19
- 17. 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 8 / 19
- 18. 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) Algorithm: dyn adj V.Fisikopoulos | ESA’12 8 / 19
- 19. Faster Geometric Algorithms via Dynamic Determinant Computation | Geometric algorithms: convex hull Outline 1 Motivation and existing work 2 Dynamic determinant computation 3 Geometric algorithms: convex hull 4 Implementation - experiments V.Fisikopoulos | ESA’12 9 / 19
- 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. Example V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 28. 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 29. 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 30. 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test V.Fisikopoulos | ESA’12 10 / 19
- 31. 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 - Beneath-and-Beyond connect each outer point to the visible segments visibility test = orientation test Similar problems: Delaunay, regular triangulations, point-location V.Fisikopoulos | ESA’12 10 / 19
- 32. 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 11 / 19
- 33. 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 p2 p5 p2 p4 p5 1 1 1 A = p1 p3 p4 Orientation(p2, p4, p5) = det(A) V.Fisikopoulos | ESA’12 11 / 19
- 34. 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 p2 p5 p6 p4 p5 1 1 1 A = p1 p3 p4 p6 Orientation(p6, p4, p5) = det(A ) in O(d2) V.Fisikopoulos | ESA’12 11 / 19
- 35. 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 p2 p5 p6 p4 p5 1 1 1 A = p1 p3 p4 p6 Orientation(p6, p4, p5) = det(A ) in O(d2) Store det(A), A−1 + update det(A ), A −1 (Sherman-Morrison) V.Fisikopoulos | ESA’12 11 / 19
- 36. 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(d2t)) Proof: update det(A ), A −1 Corollary Orientation predicates involved in point-location: O(d) (space: O(d2t)) Proof: query point never enters data-set → update only det(A ) Corollary Incremental convex hull and volume comput.: O(dω+1nt) → O(d3nt) V.Fisikopoulos | ESA’12 12 / 19
- 37. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Outline 1 Motivation and existing work 2 Dynamic determinant computation 3 Geometric algorithms: convex hull 4 Implementation - experiments V.Fisikopoulos | ESA’12 13 / 19
- 38. 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) V.Fisikopoulos | ESA’12 14 / 19
- 39. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Determinant experiments 1-column updates in A ⊆ Qd×d (uniform distr. rational coefficients) d Bird CGAL Eigen Laplace Maple dyn inv dyn adj 3 .013 .021 .014 .008 .058 .046 .023 4 .046 .050 .033 .020 .105 .108 .042 5 .122 .110 .072 .056 .288 .213 .067 6 .268 .225 .137 .141 .597 .376 .102 7 .522 .412 .243 .993 .824 .613 .148 8 .930 .710 .390 – 1.176 .920 .210 9 1.520 1.140 .630 – 1.732 1.330 .310 10 2.380 1.740 .940 – 2.380 1.830 .430 spec: Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux V.Fisikopoulos | ESA’12 15 / 19
- 40. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Determinant experiments 1-column updates in A ⊆ Qd×d (uniform distr. integer coefficients) d Bird CGAL Eigen Laplace Linbox Maple dyn inv dyn adj 3 .002 .021 .013 .002 .872 .045 .030 .008 4 .012 .041 .028 .005 1.010 .094 .058 .015 5 .032 .080 .048 .016 1.103 .214 .119 .023 6 .072 .155 .092 .040 1.232 .602 .197 .033 7 .138 .253 .149 .277 1.435 .716 .322 .046 8 .244 .439 .247 – 1.626 .791 .486 .068 9 .408 .689 .376 – 1.862 .906 .700 .085 10 .646 1.031 .568 – 2.160 1.014 .982 .107 11 .956 1.485 .800 – 10.127 1.113 1.291 .133 12 1.379 2.091 1.139 – 13.101 1.280 1.731 .160 13 1.957 2.779 1.485 – – 1.399 2.078 .184 spec: Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux V.Fisikopoulos | ESA’12 15 / 19
- 41. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Convex hull implementation Hashed dynamic determinants dyn inv & dyn adj + hash table (dets & matrices) plug into geometric software → geometric predicates Convex hull software randomized incremental (triang/CGAL) beneath-and-beyond (bb) (polymake) double description method (cdd) gift-wrapping with reverse search (lrs) triang/CGAL + hashed dynamic determinants = hdch z or hdch q V.Fisikopoulos | ESA’12 16 / 19
- 42. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Convex hull experiments 6-dim points, integer coeffs uniformly distributed inside a 6-ball 0 10 20 30 40 50 60 70 100 150 200 250 300 350 400 450 500 Time(sec) Number of input points hdch_q hdch_z triang lrs bb cdd hdch q (hdch z) rational (integer) arithmetic drawback: memory consumption 500 6D points triang 0.1GB hdch z 2.8GB hdch q 3.8GB spec: Intel Core i5-2400 3.1GHz, 6MB L2 cache, 8GB RAM, 64-bit Debian GNU/Linux V.Fisikopoulos | ESA’12 17 / 19
- 43. 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 |A| preprocess ds mem # cells query time (sec) time (sec) (MB) triang 1K 1000K hdch z 8 120 45.20 6913 319438 0.41 392.55 triang 8 120 156.55 134 319438 14.42 14012.60 hdch z 9 70 45.69 6826 265874 0.28 276.90 triang 9 70 176.62 143 265874 13.80 13520.43 hdch z 10 50 43.45 6355 207190 0.27 217.45 triang 10 50 188.68 127 207190 14.40 14453.46 hdch z 11 39 38.82 5964 148846 0.18 189.56 triang 11 39 181.35 122 148846 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 V.Fisikopoulos | ESA’12 18 / 19
- 44. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Conclusions and Future work Orientation predicates. CH: O(d2), point location: O(d) Efficient (division free) dynamic determinant implementation More efficient CH implementation, 78× speed-up in point location The code: http://hdch.sourceforge.net V.Fisikopoulos | ESA’12 19 / 19
- 45. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Conclusions and Future work Orientation predicates. CH: O(d2), point location: O(d) Efficient (division free) dynamic determinant implementation More efficient CH implementation, 78× speed-up in point location The code: http://hdch.sourceforge.net Future work Delaunay triangulations (inSphere predicate) overcome large memory consumption (hash table: clean periodically) filtered computations CGAL submission V.Fisikopoulos | ESA’12 19 / 19
- 46. Faster Geometric Algorithms via Dynamic Determinant Computation | Implementation - experiments Conclusions and Future work Orientation predicates. CH: O(d2), point location: O(d) Efficient (division free) dynamic determinant implementation More efficient CH implementation, 78× speed-up in point location The code: http://hdch.sourceforge.net Future work Delaunay triangulations (inSphere predicate) overcome large memory consumption (hash table: clean periodically) filtered computations CGAL submission THANK YOU !!! V.Fisikopoulos | ESA’12 19 / 19