Є-nets and applications

BasicsSet 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 Xe.g. area/volume is a uniform probability

A hits B means is not emptyBA

Є-nets is an Є-net for (X,F) if N hits all “large”

sets of FS 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?

XN

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 shatteringRestriction 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.

XY }:{| FSYSF Y

XA AAF 2|

VC dimension of (R2, 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 (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 !

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 functionShatter function:

Example: Given VC-dimension of (X,F) is d

More precisely

|||max)(||,

YmYXY

F Fm

mF

mF

m

m

2)(

2)(

if m<=d

otherwise

d

iF i

mm

0

)(

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 TheoremTheorem: 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.

XS

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 arrangementProblem: Point location in an arrangement

of n hyperplanes in Rd

in O(log n) timeusing O(nd+Є) preprocessing time and O(nd+Є)

query data structure.

Solution: Construct a tree like data structure for queriesThanks to Є-nets, the height of this tree is

O(log n)

Recursive construction of Query treeEach 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 d

3log d

Internal leaves of the query structureChoose 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.

Query tree and queriesNumber of children for each node=number of

simplices in the partitioning=(r log r)d

Height of tree = log n sinceRoot is associated with n hyperplanesLeaves are associated with n0 hyperplanesEach 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.