Є -nets and applications

1. Є-nets and applications

2. Basics • Set systems: (X,F) where F is a collection of subsets of X. • e.g. (R2, set of half-planes) • µ: a probability measure on X • e.g. area/volume is a uniform probability • A hits B means is not empty

3. Є-nets • is an Є-net for (X,F) if N hits all “large” sets of F • S is large if µ(S) >= Є • Main theorem: Set systems with finite VC-dimension have a (1/r)-net of size at most Cr log r. • Size of Є-net is independent of the size of X or F! • Applications: Point location, segment intersection searching, range searching. Approximation algorithms?

4. VC-dimension and shattering • Connections between VC-dimension and Є-nets • Proof of Є-net Theorem • Application of Є-net in point location

5. VC dimension and shattering • Restriction of F on is the set • A subset of is shattered by F if • VC-dimension of (X,F) is the size of the largest subset of X that is shattered by F. • If F can shatter arbitrarily large subsets of X, then F has an infinite VC-dimension.

6. VC dimension of (R2, set of halfplanes) • Can a set of 3 points be shattered? • Can a set of 4 points be shattered?

7. Why VC-dimension? • VC-dimension of (Rd, set of half spaces) • d+1 • VC-dimension of (Rd, set of convex polytopes) • infinite • VC-dimension of (X, 2X) (X is finite in this case) • infinite • (½)-net for (X, 2X) must have size at least |X|/2 !

8. VC-dimension and shattering • Connections between VC-dimension and Є-nets • Proof of Є-net Theorem • Application of Є-net in point location

9. VC-dimension and shatter function • Shatter function: • Example:Given VC-dimension of (X,F) is d • More precisely if m<=d otherwise

10. VC-dimension and shattering • Connections between VC-dimension and Є-nets • Proof of Є-net Theorem • Application of Є-net in point location

11. Є-net Theorem Theorem: Given a set system (X, F) with dim(F) ≤ d, such that d ≥ 2 and r ≥ 2 is a parameter, there exists a (1/r)-net for (X, F) of size at most Cdr log r, where C is an absolute constant. Idea of the proof: Two steps: • Randomly choose . If S does not hit some (1/r)-large set A, then chose another |S| elements from X. • Choose 2|S| elements from X at random.

12. VC-dimension and shattering • Connections between VC-dimension and Є-nets • Proof of Є-net Theorem • Application of Є-net in point location

13. Point Location in an arrangement • Problem: Point location in an arrangement of n hyperplanes in Rd • in O(log n) time • using O(nd+Є) preprocessing time and O(nd+Є) query data structure. • Solution: Construct a tree like data structure for queries • Thanks to Є-nets, the height of this tree is O(log n)

14. Recursive construction of Query tree • Each node v is associated with a subset Γ(v) of H. Root is associated with the whole of H. • If v has less than n0 associated hyperplanes it is a leaf • For other nodes v, consider the set system • The above set system has a VC dimension less than d3log d

15. Internal leaves of the query structure • Choose a (1/r)-net R(v) for the given set system at v. • Construct a simplex partitioning of Rd using the hyperplanes in R(v) • Any such simplex δ, is not intersected by more than| Γ(v) |/r hyperplanes in Γ(v) • Since, no hyperplane in R(v) intersects δ and R(v) is a (1/r)-net of Γ(v). • For each simplex in the simplex partitioning above, create a child node of v. With each of these child nodes associate the hyperplanes that intersect the corresponding simplex.

16. Query tree and queries • Number of children for each node=number of simplices in the partitioning=(r log r)d • Height of tree = log n since • Root is associated with n hyperplanes • Leaves are associated with n0hyperplanes • Each interior node is associated with less than (1/r)thof the hyperplanes associated with its parent. • Point location: • At each node locate the child simplex in which the query point lies and then recurse.