This document discusses a new algorithm for computing merge trees and contour trees in parallel using task-based parallelism. The algorithm uses Fibonacci heaps for efficient local computations and divides the work into asynchronous tasks for dynamic load balancing. It achieves good parallel speedups compared to other methods while maintaining the advantages of an augmented tree representation.

1. Task-based Augmented Merge trees
with Fibonacci Heaps
Charles Gueunet, Kitware & UPMC
Pierre Fortin, UPMC
Julien Jomier, Kitware
Julien Tierny, CNRS-UPMC

2. Topological analysis
• Topological abstractions
• Segmentation
[Bock et al. 2017]
[Favelier et al. 2016]

3. Large data sets
● [512³]
● 2.4 GB
● Compute power through parallelism

4. Related work
• Sequential
• Augmented tree
[Carr et al. 2000]

5. Related work
• Sequential
• Augmented tree
• Monotone path
[Chiang et al. 2005]

6. Related work
• Parallel
• Partitions:
• Load imbalance
• Redundant work
Partitions
V X
X
Shared memory:
● [Pascucci04]
● [Gueunet16]
Distributed:
● [Morozov13]
● [Landge14]
V
Monotonepath

7. Related work
• Parallel
• Partitions:
• Load imbalance
• Redundant work
• Monotone path (MP):
• Not augmented
Partitions
V X
X
Shared memory:
● [Pascucci04]
● [Gueunet16]
Distributed:
● [Morozov13]
● [Landge14]
V ● [Natarajan15]
● [Maadasamy12]
● [Carr16]
Monotonepath

8. Related work
• Parallel
• Partitions:
• Load imbalance
• Redundant work
• Monotone path (MP):
• Not augmented
Partitions
V X
X
Shared memory:
● [Pascucci04]
● [Gueunet16]
Distributed:
● [Morozov13]
● [Landge14]
Us!
V ● [Natarajan15]
● [Maadasamy12]
● [Carr16]
Monotonepath

9. • A local algorithm based on Fibonacci heaps
Contributions

10. • A local algorithm based on Fibonacci heaps
• Task-based parallelism:
• Augmented merge tree
• Augmented contour tree
Contributions

11. • A local algorithm based on Fibonacci heaps
• Task-based parallelism:
• Augmented merge tree
• Augmented contour tree
• Ready to use implementation
• Generic input (VTU/VTI, 2D/3D)
• Generic output
• Open source (TTK)
Contributions

12. Preliminaries

13. Mesh
• Generic input
• Piecewise linear scalar field
• Regular grid
• Implicit triangulation (TTK)

14. • Level set
•
Scalars

15. • Level set
•
• Sublevel set
•
Scalars

16. Merge tree
• Track merge of sublevel sets CC
• 1 Arc <-> 1 CC

17. Contour tree
• Track merge of level sets CC
• 1 Arc <-> 1 CC

18. Overview
• One local growth per arc region

19. Merge tree computation
Sequential

20. Leaf search
• Extract minima (JT)

21. • Start at minima
Leaf growth

22. Saddle stopping condition
• Vertex lower than current

23. • Fibonacci heap
• Constant merge time
Merging two regions
1 2 3 4

24. Saddle growth
• New growths:
• On saddles completed

25. Saddle growth
• New growths:
• On saddles completed
• Until 1 growth remain

26. Trunk
• Monotone path until root

27. Trunk
• Monotone path until root
• Vertex to arc: using scalars

28. Trunk
• Monotone path until root
• Vertex to arc: using scalars

29. Parallel merge tree computation

30. Tasks
• Asynchronous work unit
• Dynamic load balancing
• Runtimes:
• OpenMP
• Intel TBB
• Intel Cilk Plus
[R. van der Pas, IWOMP 2009]

31. Parallel leaf search
• Local operations
• Embarrassingly parallel

32. Parallel leaf growth
• Use tasks

33. Parallel saddle stopping condition
• No change

34. • Local only
Parallel merging regions
1 2 3 4

35. Parallel trunk detect
• Atomic counter on active tasks
• Initialized: number of leaves

36. Parallel trunk process
• Chunk of vertices
• Using scalar value

37. Parallel contour tree computation

38. Post-processing
• Post-processing
• Arc: Regular vertex list

39. Post-processing
• Post-processing
• Arc: Regular vertex list
• Nodes: Report in other tree

40. Contour tree computation
• Post-processing
• Arc: Regular vertex list
• Nodes: Report in other tree
• Combination

41. Results
Intel Xeon E5-2630 v3 CPUs (2.4 GHz, 2x8 cores, 2x16 threads)
64 GB of RAM
C++ (GCC-5.4.0), VTK, OpenMP 4.0 (additional material + TTK)

42. Merge tree computation
• 512³ regular grids
Join tree
Split tree
Time in seconds

43. Join tree
Split tree
Time in seconds
Merge tree computation
• Trunk not predictable

44. Join tree
Split tree
Time in seconds
Merge tree computation
• Most time-consuming

45. Join tree
Split tree
Times in seconds
Merge tree computation
• Average speedup: 10/16

46. Detailed speedups

47. Sequential comparison
LT: Libtourtre, CF: Contour Forests, FTM: Our implementation. 256³ grid
Time in seconds

48. Parallel comparison
Time in seconds
LT: Libtourtre, CF: Contour Forests, FTM: Our implementation. 256³ grid

49. Contour tree computation
Time in seconds
• 512³ regular grids
• Average speedup 7/16
`

50. Sequential compare (CT)
Time in seconds
LT: Libtourtre, CF: Contour Forests, FTM: Our implementation. 256³ grid

51. Parallel compare (CT)
LT: Libtourtre, CF: Contour Forests, FTM: Our implementation. 256³ grid
Time in seconds

52. Limitations

53. Application

54. Application
Segmentation

55. Application
Segmentation

56. Conclusion

57. Take home message
• New algorithm:
• Fast in sequential
• Good parallel performances
• Dynamic load balancing
thanks to tasks
• No redundancy
• Augmented tree

58. Take home message
• VTK-based implementation
• Generic input
• VTU/VTI
• 2D/3D
• Generic output
• Open source (TTK)
• Ready-to-use
• New algorithm:
• Fast in sequential
• Good parallel performances
• Dynamic load balancing
thanks to tasks
• No redundancy
• Augmented tree

59. Perspective
• Improve arc growth scalability
• Parallel combination
• Improve memory usage
• Study in-situ visualization capabilities

61. Unpredictable best scheduling:
Sequentially: How to maximize trunk ?
Appendix
E
A
B
C
D
J
I
F
G
H
K
L

62. Title Annotation
• Interesting fact