Maximizing Parallelism in the Construction of BVHs, Octrees , and k -d Trees

1. Maximizing Parallelism in the Construction of BVHs, Octrees, and k-d Trees Tero Karras NVIDIA Research

2. Trees

3. Trees Better Faster Ageia  Collision detection Pharr & Humphreys NVIDIA Path tracing NVIDIA Real-time ray tracing Particle simulation Crassin et al. Amenta et al. Voxel-based global illumination Uchida Surface reconstruction Photon mapping

4. Outline • Fastest existing methods are sequential • Parallelize within each hierarchy level • But not between levels

5. Outline • Fastest existing methods are sequential • Parallelize within each hierarchy level • But not between levels • Lack of parallelism • Small workloads bottlenecked by top levels • Sub-linear scaling of performance

6. Outline • Novel way to build the entire tree in parallel • Two algorithmic “building blocks” • Fast, scalable

7. Outline • Novel way to build the entire tree in parallel • Two algorithmic “building blocks” • Fast, scalable • Main focus: BVHs • Point-based octrees and k-d trees also covered in the paper

8. Bounding volume hierarchy

9. Bounding volume hierarchy ?

10. LBVH - Lauterbach et al. [2009] • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes p 1 0 1 0 px = 0. py = 0. 0 1 1 1 1 1 0 0 pz = 0.

11. LBVH - Lauterbach et al. [2009] • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes p 1 0 1 1 0 0 1 0 px = 0. py = 0. 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 pz = 0.

12. LBVH - Lauterbach et al. [2009] • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes p 1 0 1 0 px = 0. py = 0. code = 1 0 1 0 1 1 1 1 0 0 1 0 1 1 0 0 pz = 0.

13. LBVH - Lauterbach et al. [2009] • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes

14. LBVH - Lauterbach et al. [2009] • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes

15. Binary radix tree 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

16. Binary radix tree 00 0 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

17. Binary radix tree 00 000 0 00 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

18. Binary radix tree 00 1 11 000 0010 1100 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

19. Binary radix tree 00 1 n-1 11 000 0010 1100 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110 n

20. Longest common prefix 00 1 11 000 0010 1100 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

21. Longest common prefix δ(5,6) = 4 00 δ(5,6) = 4 4 1100 4 0 1 2 3 4 5 5 6 6 7 00001 00010 00100 00101 10011 11000 11000 11001 11001 11110

22. Longest common prefix δ(0,3) = 2 δ(0,3) = 2 δ(0,3) = ? 00 2 δ(5,6) = 4 2 1100 4 0 0 1 2 3 3 4 5 6 7 00001 00001 00010 00100 00101 00101 10011 11000 11001 11110 

23. Garanzha et al. [2011] Level 0 1 node 0 Level 1 2 nodes 1 2 3 nodes Level 2 4 5 3 Level 3 1 node 6 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

24. Our method ? 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

25. Our method • Define a numbering scheme for the nodes • Gain some knowledge of their identity • Establish a connection with the keys

26. Our method • Define a numbering scheme for the nodes • Gain some knowledge of their identity • Establish a connection with the keys • Find the children of a given node • Only look at node index and nearby keys

27. Our method • Define a numbering scheme for the nodes • Gain some knowledge of their identity • Establish a connection with the keys • Find the children of a given node • Only look at node index and nearby keys • Do this for all nodes in parallel

28. Numbering scheme 0 3 4 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

29. Numbering scheme 0 3 4 5 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

30. Numbering scheme 0 0 3 3 4 4 1 1 2 2 5 5 6 6 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

31. Numbering scheme 0 3 4 1 2 5 6 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

32. Algorithm 0 3 4 1 2 5 6 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

33. Algorithm ?  δ(2,3) = 4 δ(3,4) = 0 δ(2,3) = 4 3 0 1 2 2 3 3 3 4 4 5 6 7 00001 00010 00100 00100 00101 00101 00101 10011 10011 11000 11001 11110

34. Algorithm  δ(1,3) = 2 δ(3,4) = 0 δ(0,3) = 2 δ(2,3) = 4 ? 3 0 1 2 3 3 4 4 5 6 7 00001 00010 00100 00101 00101 10011 10011 11000 11001 11110

35. Algorithm δ(0,3) = 2 3 ?   δ(2,3) = 4 δ(1,3) = 2 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

36. Algorithm δ(0,3) = 2 3 1 2 2 2 0 1 2 3 4 5 6 7 00001 00010 00100 00101 10011 11000 11001 11110

37. Algorithm For each node i=0..n-2 in parallel: • Determine direction of the range • Expand the range as far as possible • Find where to split the range • Identify children Binary search

38. Duplicate keys • The algorithm only works with unique keys • Duplicates are common in practice

39. Duplicate keys • The algorithm only works with unique keys • Duplicates are common in practice • Trick: Augment each key with its index • Distinguishes between duplicates • Keys are still in lexicographical order

40. Duplicate keys • The algorithm only works with unique keys • Duplicates are common in practice • Trick: Augment each key with its index • Distinguishes between duplicates • Keys are still in lexicographical order • Tie-break when evaluating δ(i,j)

41. LBVH • Assign Morton codes • Sort primitives • Generate hierarchy • Fit bounding boxes

42. Lauterbach et al. [2009]

43. Our method • Need a different approach • How many levels are there? • Which nodes are located on a given level?

44. Our method • Need a different approach • How many levels are there? • Which nodes are located on a given level? • Traverse paths in the tree in parallel • Start from leaves, advance toward the root • Terminate threads using per-node atomic flags

45. Our method             

46. Results • Evaluate performance on GTX 480 (Fermi) • CUDA, 30-bit Morton codes

47. Results • Evaluate performance on GTX 480 (Fermi) • CUDA, 30-bit Morton codes • Compare against Garanzha et al. [2011] • Identical tree (top-level SAH splits disabled)

48. Results • Evaluate performance on GTX 480 (Fermi) • CUDA, 30-bit Morton codes • Compare against Garanzha et al. [2011] • Identical tree (top-level SAH splits disabled) • Simulate large GPUs • N times as many cores • N times the memory bandwidth

49. Fairy Forest 174K triangles Results milliseconds 1 × cores 2 × 4 × Our method Garanzha et al.

50. Fairy Forest 174K triangles Results milliseconds 12.5 × 33.6 × 1.7 × 1.3 × 1.1 × 2.4 × Our method Garanzha et al.