Transcript Geometry Agorithms - Tata Institute of Fundamental Research

Convexity of Point Set
Sandip Das ([email protected])
Indian Statistical Institute
Convex Set

A set C   d is convex if for every two
points a, b  C, the line segment joining a
and b is also contained in C.
Convex Set (Contd.)

A set C   d is convex if for every two
points a, b  C, the line segment joining a
and b is also contained in C.

A set C   d is convex if for every two
points a, b  C, and for every t  [0, 1],
the point t.a + (1- t).b belongs to C.
Are These Two Definitions Equivalent?
Convex hull

The convex hull CH (S) of a set S is the
smallest convex set that contains S.
Convex hull (Contd.)

Intersection of all convex set containing S.
Algebraic Observation:
A point a belongs to CH (S) iff there exist
points s1, s2, . . ., sn  S, and non-negative
n
real number t1, t2, . . ., tn with i =1 ti =1
n
such that a = i =1 ti.si
Convex hull: Application in optimization
Consider the following Database
Person
…
Queries:
income
…
expenditure
…
Find person having maximum income
Find person whose expenditure is minimum
Find person having maximum savings
expenditure
Application in optimization (Contd.)
5
2
3
8
7
1
income
Queries:
Find person having maximum income
Find person whose expenditure is minimum
Find person having maximum savings
Linear Programming
Maximizing
c1 x1+ c2 x2+ . . .+ cn xn
Subject to
a11 x1+ a12 x2+ . . .+ a1n xn ≤ b1
a21 x1+ a22 x2+ . . .+ a2n xn ≤ b2
…
an1 x1+ an2 x2+ . . .+ ann xn ≤ bn
Linear Programming (Contd.)
a11 x1+ a12 x2+ . . .+ a1n xn = b1
is a hyperplane in n dimensional plane
a11 x1+ a12 x2+ . . .+ a1n xn ≤ b1
Implies a halfplane bounded by this
hyperplane
Linear Programming (Contd.)
Set of constraints generate intersection of n
hyperplanes
Intersection of convex regions is convex
Linear Programming (Contd.)
Set of constraints generate intersection of n
hyperplanes
Intersection of convex regions is convex
Intersection region may be empty => no solution
Intersection region may be unbounded => it may
generate unbounded optimal solution
Center point
Looking for center point among points
arranged on a line.
Have a sense of center point but not clear
- Mean ?
-Median ?
Center point (Contd.)
Observe that
Median say x is such a point where
| # of points on left of x
-
# of points on right of x| ≤1
We want to extend this idea in 2D
n points in a plane.
Center point (Contd.)
n points in a plane.
Left and right is not well defined on plane.
We can define left and right with respect to a line
l
Left side
of l
right side
of l
Center point (Contd.)
Consider a point x in 2D
Draw a line l through x.
We can compute # of points on left with respect to l
Similarly # of points on right with respect to l
So,
| # of points on left of l - # of points on right of l |
varies as the line l rotate and passing through x
x
l
Center point (Contd.)
| # of points on left of l - # of points on right of l |
What is the maximum value of this difference for all line l
passing through x
x
l
Let us say that value as c(x)
Center point (Contd.)
The term c(x) may be considered as a measure of x for
being a center
Can you identify a point x such that c(x) is less than equal
to 1?
x
Center point (Contd.)
For any point set
Can you identify a point x such that c(x) is less than equal
to 1?
Does such a point always exist?
Center point (Contd.)
Example:
Let the point set be
Can you identify a point x such that
c(x) ≤1
x
x
x
Centerpoint Theorem
A point x Rd is called a centerpoint of a point set
if each closed halfspace containing x contains at
least n/(d+1) points of the point set.
Theorem: Each finite point set in Rd has at least
one centerpoint.
Follows from Helly’s theorem.
Helly’s Theorem
Let C1, C2, …, Cn be convex sets in 2D plane.
Suppose that the intersection of every 3 of these
sets is nonempty. Then the intersection of all the
Ci is nonempty.
Proof of Centerpoint Theorem
Consider any point set with n points. Take all convex
set containing at least 2n/3 points.
Number of such convex sets are finite.
Observe that intersection of any three of them is not
null
Hence, from Helly’s theorem, intersection of all such
convex hull is not null.
Any point on that intersection is the centerpoint.
Algorithm for finding centerpoint
Shreesh Maharaj et al. proposed an excellent algorithm in
O(n) time
Prune and search technique
T(n) = T(c.n) +O(n), 0< c <1
Generate a convex region such that centerpoint region of
point set including vertices of convex region is a
superset of earlier one
If some vertices of that convex region is discarded
centerpoint remains same.
Discard that fraction of boundary points, and continue the
process.
Convex independent set
A set S  Rd is convex independent if all points in S
lie on convex hull of S
That is for every x  S, x  conv{S\{x}}
Let P be a set of points and the points be in general
position.
Any three point subset is convex independent
But any subset of 4 points is not convex independent
Convex independent set (contd.)
Suppose the set P contains 5 point
May we always get a subset of size 4 that are convex
independent?
Size of convex hull will be either 3, 4 or 5
If the size of convex hull is 5, then … …
Ramsey Theorem
G(V, E) is a graph with |V|=6, then either G or Gc
must have a triangle.
So, R(3, 3) = 6
If the number of vertices is sufficiently large, there
always exist a k vertex subset Y such the all
hyperedge of 4 vertices is in G or in Gc.
Erdös-Szekeres Theorem
Given n points set, color a 4 tuple red if its 4
points are convex independent and blue
otherwise.
From Ramsey Theorem, there is a k point subset
such that all hyperedge is same color.
But for k ≥ 5, this color cannot be blue.
So, that k point subset is convex independent
Erdös-Szekeres Theorem (Contd.)
For every natural number k, there exist a number
n(k) such that any n(k) point set in the plane in
general position contains a k-point convex
independent subset.
2k-2 + 1 ≤ n(k) ≤ 2k-5Ck-2 - 2
K-Hole
Let X be a set of point. A k-point set Y is called a khole in the point set if Y is convex independent and
conv(Y) X = Y.
Erdös raised the question about the size of point set
for k-hole
3-hole?
4-hole?
5-hole?
6-hole?
7-hole … … Does not exist.
A lot of questions remain unanswered….