Efficient Partition Trees Jiri Matousek

1. Efficient Partition TreesJiriMatousek Presented By Benny Schlesinger Omer Tavori

2. Simplex Range Searching • Simplex range searching: We preprocess a set P of n points in so that, given any query region , the points in P  can be counted or reported efficiently. • Let's assume that the query region is a simple polygon; if it is not, we can always approximate it (by triangulation).

3. Simplex Range Searching • We have arrived at the triangular range searching problem: given a set P of n points in the plane, count the points from P lying in a query triangle delta. • Let’s first look at a simpler version of this problem, where the query triangle degenerates into a half plane (simply a query line). • Extending the simpler version to the original problem is just looking at 3 lines.

4. Half plane query example

5. How do we solve the simplex range searching problem? We will use Partitions Trees

6. Partitions Trees vs. Cutting Trees

7. Partitioning Point Sets • A simplicial partition for a set P of n points in the plane is a collection := where the Pi are disjoint subsets of P whose union is P and i is a triangle containing Pi. The subsets Pi are called classes. • We do not require the triangles to be disjoint.

8. Simplicial Partition

9. Simplicial Partition • We say that a line h crosses a triangle  if h intersects the interior of . • A crossing number of a line h with respect to  is the number of triangles crossed by h. • In our example the crossing number of h is 2.

10. Simplicial Partition • The crossing number of  is the maximum crossing number over all possible lines. In the last figure you can find lines that intersect four triangles, but no line intersects all five.

11. Fine Simplicial Partition • We say that a simplicial partition is fine if s |Pi|< 2s for a given s. In fine simplicial partitions none of the classes contains more than twice the average number of points of the classes. (If there are r classes and n points The average number of points in class is n/r = s)

12. Simplicial Partition • If a triangle , of the partition is not crossed by the line h, then its class p, either lies completely in h, or it is completely disjoint from h (i.e. either completely above or below the line) • In our example, if we queried with l+, the half-plane lying above l, we would have to recourse on two of the five classes.

13. The Partition Theorem • It is always possible to find a simplicial partition with crossing number • The partition Theorem: For a set P of n points in the plane, and a parameter r with 1  r  n , a fine simplicial partition of size r and crossing number can be constructed in time O(n).

14. -Cutting • cutting : A cutting  is a collection of closed triangles with disjoint interiors, whose union is the whole plane. The size of a cutting is the number of its triangles. • -cutting: Let H be a collection of n lines and let  be a cutting. For a triangle , let denote the collection of lines of H intersecting . A cutting  is an  -cutting for H provided that

15. -cutting for weighted collection of lines • A weighted collection of lines is a pair (H, w), where H is a collection of lines and is a weight function . • A Cutting  is an -cuttingfor (H,w), provided that, for every triangle :

16. The Partition Theorem • In order to prove the partition theorem will use two lemmas. • In the first lemma we will use the cutting theorem. Let us remind what is the cutting theorem.

17. The cutting theorem

18. Lemma 1 Let P be an n-point set in , let s be an integer parameter, 2  s < n, set r = n/s. and let H be a set of lines. Then there exists a simplicial partition  for P, whose classes Pi satisfy s  |Pi| < 2s for every i and that the crossing number of every line of H relative to  is

19. Lemma 1: Building the Partition • We inductively construct the disjoint sets Pi and i. • Suppose that P1,…Pi have already been constructed and set Pi'=Pi-(P1,…,  Pi) • If |Pi|' < 2s we set Pi+1 = Pi' , i+1 = R2. m = i+1 and := {(P1, 1),…,(Pm, m)} which finishes the construction. • Otherwise Let ni = |Pi'| >= 2s.

20. Lemma 1: Finding a cutting using weights How do we find Pi+1 and i+1? We will find them by building a cutting for weighted collection of lines. For a line hH let ki(h) denote the number of triangles among 1,…, i (the triangles that are already in the partition) that are crossed by h. For every hH let . we also define a weighted collection (H,wi).

21. Lemma 1: Finding a cutting using weights At the first iteration:  hH Notice that if a line h intersects i-1 at the next iteration:

22. Lemma 1: Finding a cutting using weights The intuition: While building the cutting we will try to avoid constructing triangles that are crossed by lines with “heavy” weights. (One of these triangles will be chosen to be a triangle in the simplicial partition.)

23. Lemma 1: Constructing A (1/ri)-cutting W (h) = 2

24. Lemma 1: Constructing A (1/ri)-cutting W (h) = 2

25. Lemma 1: Constructing A (1/ri)-cutting W (h) = 2

26. Lemma 1: Constructing A (1/ri)-cutting W (h) = 2

27. Lemma 1: Constructing A (1/ri)-cutting W (h) = 2

28. Lemma 1: choosing i W (h) = 2

29. Lemma 1: choosing i W (h) = 2

30. Lemma 1: choosing i W (h) = 2

31. Lemma 1: Updating The Weights W (h) = 4

32. Lemma 1: Constructing a new Cutting

33. Lemma 1: Constructing a new Cutting

34. Lemma 1: Constructing a new Cutting

35. Lemma 1: Finding Pi+1 and i+1 We want to find a triangle i+1 which will hold the class Pi+1. Don’t forget that we want at least s points in Pi+1. Currently we have ni points left.

36. Lemma 1: Finding Pi+1 and i+1 According to the cutting theorem we know that we have a cutting with triangles. By the pigeonhole principle, One of the triangles in the cutting will have at least = s points. This triangle will be i+1. We choose some s points from i+1 to be Pi+1 in the partition.

37. Creating The Partition Algorithm

38. Lemma 1: Finding The Crossing Number Of The Partition • We want to show that the crossing number of every line h in H relative to our partition is:

39. Lemma 1: Finding The Crossing Number Of The Partition

40. Lemma 1: Finding The Crossing Number Of The Partition

41. Lemma 1: Finding The Crossing Number Of The Partition

42. Lemma 1: Finding The Crossing Number Of The Partition

43. Lemma 2: The Test Set Lemma • For an n-point set P and a parameter r, there exists set H of at most r lines, s.t, for any simplicial partition for P satisfying |Pi| s for i, the following holds: if k0 is the maximum crossing number of lines of H relative to the partition, then the crossing number of the partition is bounded by:

44. The cutting theorem Now that we can use the two lemmas we can easily prove the cutting theorem. Given an n-point set P in the plane, an integer parameter 2 <= s < n , and r = n/s. In order to obtain the desired simplicial partition, we first use lemma 2. We get a set H of at most r lines.

45. The cutting theorem • Second we use lemma 1, obtaining a simplicial partition , whose classes have size between s and 2s, and such that the crossing number of any line of H is at most: • By the property of H guaranteed by the Test Set Lemma the crossing number of  is at most:

46. Lemma 2Test-Set Lemma

47. Test-Set Lemma (cont.) • Why Test-Set? We will show that we can literally find a “small” set of lines as stated, that is, a test-case of lines that ensures a certain crossing number for any partition whose classes’ sizes is no less than a certain bound, s.

48. Test-Set Lemma – proof

49. Test-Set Lemma (cont.) Primal plane – Dual plane

50. Test-Set Lemma (cont.)