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

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?

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

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.

VC dimension of (R 2, set of halfplanes) Can a set of 3 points be shattered? Can a set of 4 points be shattered?

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

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

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

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

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.

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

Point Location in an arrangement Problem: Point location in an arrangement of n hyperplanes in R d in O(log n) time using O(n d+Є ) preprocessing time and O(n d+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)

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 n 0 associated hyperplanes it is a leaf For other nodes v, consider the set system The above set system has a VC dimension less than d 3 log d

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 R d 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.

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 n 0 hyperplanes Each interior node is associated with less than (1/r) th of the hyperplanes associated with its parent. Point location: At each node locate the child simplex in which the query point lies and then recurse.