Well Separated Pair Decomposition

1. Well Separated Pair Decomposition 16 GigaYears and 3.3 Million Light Years Playing

2. Applications • Astronomy • Molecular Dynamics • Fluid Dynamics • Plasma Physics • And Guess What: Surface Reconstruction

3. N-Body Simulation How do I do it Provably Fast?? Trees!! Trees!! Trees!! A Simple, Provable and Powerful Algorithm (Callahan-Kosaraju, FMM)

4. Andromeda from Earth

5. The Galactic Empire Enough of Sci-Fi 

6. Well Separated Pair Decomposition A Decomposition of the complete Eucledian Graph into O(n) pairs. The Curse of Dimensionality/Constants Can be used to do Surface Reconstruction In O(nlogn) theoretically! Practically????

7. Well Separated Pair Dumbbells !!

8. An Example of WSPD 100 points On a Circle

9. Another Picture An Output of my implementation that computes D-dimensional WSPD’s

10. The Fair Split Tree The tree on which the Well Separated Pairs Are computed. An adaptation of QuadTrees to give theoretical Guarantees Notion: R(P) minimum bounding hyper-rectangle That encloses the point set P The FST, breaks R(P) in the longest dimension

11. FST Algorithm CreateTree(vector<points> P){ if( P.size() == 1) return Leaf(P); Cut = Plane that cuts the longest dim of R(P) P1 = Points on left of Cut P2 = Points on right of Cut Node-> left = CreateTree(P1) Node->right= CreateTree(P2) }

12. The Fair Split Tree Guarantees that at least one point is split O( ) by Trivial Algorithm O( ) by pre-sorting in every dimension For O(nlogn) uses Partial Fair Split Tree Computation.<Double Sided pointer chasing>

13. Partial Fair Split Tree • Keep points d- list<points> sorted in every dimension • In selected dimension, walk from both sides • Remove cut elements and recurse till the largest • size of the set is less than (2/3)n • Create subsets and make recursive call

14. WSPD Algorithm • On Each Internal node of FST • Call Find_Pairs(Node->Left,Node->Right)

15. Find_Pairs(N1,N2) If Well-Separate(N1,N2) return pair<> If Lmax(N1) > Lmax(N2) Find_Pairs(N1->left,N2) Find_Pairs(N1->right,N2) Else Find_Pairs(N1,N2->left) Find_Pairs(N1,N2->right) endif

16. Time for a pic

17. Existence Does a WSPD for a point set always Exist?? Yes!!! Just take all pairs of points in P, they are well separated by definition What about Linear Size?? O(n) pairs?

18. Another Picture  Not too many disjoint “good/big” rectangles can intersect a cube

19. Packing cubes Lemma: A hypercube C which overlaps with a set S of disjoint hypercubes of length l’ follows:

20. Proof by picture!!!

21. Canonical Realization If A,B are well separated (Notation for Lmax(A) < Lmax(B))

22. Lemma 4.1 If C be a d-cube, and S={Ai} be a set of disjoint nodes in T and Lmax(p(Ai))>=l(C)/c and R(Ai) overlaps C, then: If u recall here l(C)/l’ is replaced by 3c here

23. Where did that 3 come from? Hint for Imagination: Think in 1D , use Induction for d-dimension

24. Bounding the size of Canonical Realization Lemma: Let P={p1,p2,…,pn} with FST T , For any node A in T , there are O(1) canonical pairs of the form (A,B) Proof: We know that p(A) and B are not well separated , so

25. Rewriting Intuitively, means a constant factor enlargement of R(p(A)) intersects with all B well separated from A.

26. Bounding the size of Canonical Realization We know that Lmax(p(B)) > Lmax(p(A)) We know that all B are disjoint We know that B overlaps an enlargement of R(p(A)) We know that |SB| is bounded in these conditions by a constant K(c,d) Implies there can only be O(1) B’s well separated from A’s, which in turn implies there are only O(n) WS-Pairs!!! 

27. Closest Pair Theorem: Suppose that there is a k-nearest neighbor pair (a,b) such that

28. Closest Pair By the previous theorem, putting k = 1 the Closest pair : {{a},{b}} is a well separated pair in the WSPD. Hence Given the WSPD, we can find the closest Pair of a d-dimensional point set in O(n) time.

29. K-nearest neighbors Put Yao’s cones on every point, and uses inheritance on the FST in these cones. In one cone, only k-neighbors needs to be maintained. Then a selection algorithm picks the nearest k-neighbors in O(nk) time.

30. General Idea of Applications Any application where one object affects every other object in a set of objects, is susceptible to WSPD paradigm. N-Body simulation is a glaring example. If there is a lower bound of O(n^2) for the problem, it might be susceptible to an Approximation algorithm using WSPD paradigm.

31. Applications II Nearest Neighbour queries Answering what is close in a r-radius ball Clustering Applications Dynamic Nearest Neighbor queries Approximate Voronoi diagrams Mesh Generation Combinatorics MST Approximations/Spanners N-Body Simulation…

32. Voronoi Diagrams

33. Approximate Voronoi • May not be a perfect Voronoi Diagram but each cell is some kind of Approximation • May not be a decomposition of the entire space. • Of Theoretical Interest • Till date no Implementation exists!

34. Another Nice Picture Courtesy Sariel Har Pelad

35. Open Problems • Is there a way to sort the edges of a complete • graph in O(n^2) time? • Is there a way to compute LFS(p) on a manifold • in O(nlogn) using WSPD’s? • How practical is it for k-nearest neighbour queries? • How about dimensions >= 5?? • Compute Medial Axis in O(nlogn)

36. Medial Axis Approximation Lfs Approximation isnt so easy!!

37. That’s All Folks!  --Piyush