Ice Cream And Wedge Graph

Ice Cream And Wedge Graph

Eyal Ackerman Tsachik Gelander Rom Pinchasi

Table of content

• Introduction– Wedge-graph– Overview

• Main theorem– Ice-cream lemma– Intuition for the lemma– Explanation of the main theorem

• Proofs– main theorem– Ice-cream lemma

Wedge-graph

• Given 3 -directional antenna a

b

c

𝛼

a

b

c

We can position antenna a

𝛼𝛼

𝛼

a

b

c

We can position antenna b

𝛼𝛼

𝛼

a

b

c

We can position antenna c

𝛼𝛼

The wedge-graph we got:

a

b

c

Overview• [1] I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc and A. Wiese, Communication

in wireless networks with directional antennas, Proc. 20th Symp. on Parallelism in Algorithms and Architectures, 344{351, 2008. – formulated a different model of directed communication graph

• [2] P. Carmi, M.J. Katz, Z. Lotker, A. Rosen, Connectivity guarantees for wireless networks with directional antennas, Computational Geometry: Theory and Applications, to appear.

• [3] M. Damian and R.Y. Flatland, Spanning properties of graphs induced by directional antennas, Electronic Proc. 20th Fall Workshop on Computational Geometry, Stony Brook University, Stony Brook, NY, 2010.– formulated a different model of directed communication graph

• [4] S. Dobrev, E. Kranakis, D. Krizanc, J. Opatrny, O. Ponce, and L. Stacho, Strong connectivity in sensor networks with given number of directional antennae of bounded angle, Proc. 4th Int. Conf. on Combinatorial Optimization and Applications, 72{86, 2010.– formulated a different model of directed communication graph

Overview• [1] I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc and A. Wiese, Communication in

wireless networks with directional antennas, Proc. 20th Symp. on Parallelism in Algorithms and Architectures, 344{351, 2008.

• [2] P. Carmi, M.J. Katz, Z. Lotker, A. Rosen, Connectivity guarantees for wireless networks with directional antennas, Computational Geometry: Theory and Applications, to appear.– found the minimum so that no metter what finit set P of location, it is always posible to position

them so the wedge graph is connected• [3] M. Damian and R.Y. Flatland, Spanning properties of graphs induced by directional

antennas, Electronic Proc. 20th Fall Workshop on Computational Geometry, Stony Brook University, Stony Brook, NY, 2010.

• [4] S. Dobrev, E. Kranakis, D. Krizanc, J. Opatrny, O. Ponce, and L. Stacho, Strong connectivity in sensor networks with given number of directional antennae of bounded angle, Proc. 4th Int. Conf. on Combinatorial Optimization and Applications, 72{86, 2010.

Ice-cream lemma

• S is a compact convex set in the plane.• . • There exist a point O in the plane and 2 rays,

q and r.• q and r apex is O • q and r are touching S at X and Y exclusively• r and q satisfy • the angle bounded by r and q is

Intuition for the lemma

Let S be a Compact convex set and fix

S

Intuition for the lemma

There exist a point O

S

O

Intuition for the lemma

And 2 rays q and r

S

O

r

q

Intuition for the lemmaRays are touching S in X and Y and creating an angle of

S

O

r

q

X

Y

𝛼

Intuition for the lemmaO was selected to satisfy

S

O

r

q

X

Y

𝛼

Intuition for the lemmaThe ice-cream lemma

S

O

r

q

X

Y

𝛼

Main Theorem

• P is a set of n points in the plain (general position).• CH(p) is the convex hull of p.• h is the number of vertices in CH(p). • It takes O(n log h) - time to find n wedges (apexes

are in p) of angle • The wedge-graph is connected. • The wedge-graph has a path of length 2 and each

of the other vertices in the graph is connected by an edge to one of the three vertices of the path.

Explanation of the main theorem

• The angle is best possible as written by Carmi et al.

Say we have 3 points on the plane

a

b

c

𝜋3

𝜋3

𝜋3

m

m m

A specific case for the example, an equilateral triangle. We’ll try to position the –directional antennas with

a

b

c

𝛼

𝛼<𝜋3

a

b

c

We can position antenna a

𝛼

𝛼<𝜋3

a

b

c𝛼

And antenna b

𝛼

𝛼<𝜋3

a

b

c

Antenna c will never be connected in the wedge-graph of a,b and c

𝛼

The wedge-graph we got in any setting of antenna c is:

a

b

c

proof of the main theorem

• Let p be:

• CH(p) is:

This can be found in o(nlogh)

proof of the main theorem

• <X,Y> Is a “good pair” if exists O- point, and rays q,r

X

Y

O

q

r

𝛼

proof of the main theorem

proof of the main theorem

• Given CH(P) and 2 vretices X,Y, it takes O(1)-time to check whether X and Y are a “good pair”.

• for any pair of points, there are only 2 possible location for O, and you only need test whether their neighbors are in

• The ice cream lemma guarantees that CH(P) has a good pair

• How to find a good pair:– For any edge (x,x’) of CH(P) we can find in O(logn)

time (binary search) the Y point that satisfies this:

proof of the main theorem

X

Y

q

r O𝛼

X’

• How to find a good pair:– For any edge (x,x’) and point Y in CH(P) we can

test in constant time if (X,Y) or (X’,Y) are “good pair”:

proof of the main theorem

X

Y

q

r

O𝛼

X’

• How to find a good pair:– We got that a good pair can be found in O(nlogh)

time

proof of the main theorem

X

Y

q

r

O𝛼

X’

• is a line creating angle of with q and r• There is a point on • A and B are the points of the intersections

of ,q and r• and the triangle ABO is covering p– Z Can be found in O(logh)

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼

• X’ is a point on such that is equilateral and Y’ is a point on such that is equilateral.– there are 2 general cases for this:

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’X’Y’ X’

case1 case2

• Case 1:– Z is in and in

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’ X’

• Case 1:– .

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’ X’

• Case 1:– is a wedge of containing X and Y

– Note that the wedge graph of X,Y and Z is connected

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’ X’

𝛼

• Case 2:– The general case where Z is not in – We’ll find Z’ to be the point on r, and is equilateral

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’X’

r q

Z’

𝛼

• Case 2:– is

– Note that the wedge graph of X,Y and Z is connected

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’X’

r q

Z’

𝛼

• Finally we can see that the wedge graph contains 2-path on X,Y,Z.

• , and covere , which means that any other point in p is connected to X,Y or Z in the wedge-graph

proof of the main theorem

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’X’

r q

Z’

𝛼

𝛼

Z

X Y

O

𝑙 A B𝛼𝛼Y’ X’

𝛼

• Consider the point O such that the 2 tangents of S through O create and angle of

• And the area of is maximum.• From compactness -> such point exists

proof of the ice-cream lemma

S

O

r

q

X

Y

𝛼

• We’ll show that

proof of the ice-cream lemma

S

O

r

q

X

Y

𝛼

• X’ can be equal to X and Y’ can be equal to Y.• and

proof of the ice-cream lemma

S

O

r

q

X

Y

X’

Y’

𝛼

• Claim:– and

• This means that :– –

proof of the ice-cream lemma

S

O

r

q

X

Y

X’

Y’

𝛼

• Proof of the claim:– Assuming to the contrary that – Let small positive number

proof of the ice-cream lemma

S

O

r

q

X

Y

X’

Y’

𝛼

• Proof of the claim:– obtained from XO’ and is from Y’O’– Let small positive angle of rotation

proof of the ice-cream lemma

S O

r

q

X

Y

𝛼

X’

Y’

𝑙1

𝛼 O’𝛿

𝑙2

• Proof of the claim:– The area of is greater than the area of ,– The difference is .– This contradict the choice of the point O.

proof of the ice-cream lemma

S O

r

q

X

Y

𝛼

X’

Y’

𝑙1

𝛼 O’𝛿

𝑙2