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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 lemma
The 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
