First of all

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First of all . Thanks to Janos
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Conflict-free coloring problems
Shakhar Smorodinsky Tel Aviv University

Part of this work is joint with Guy Even, Zvi
Lotker and Dana Ron.
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Definition of Conflict-Free Coloring
Any (non-empty) disc contains a unique color
A Coloring of pts
is Conflict Free if
4
1
3
2
4
3
3
2
1
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What the is Conflict-Free Coloring?
A Coloring of pts
1
is Conflict Free if
1
3
2
4
3
3
2
1
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Problem Statement
What is the smallest number f(n) s.t. any n
points can be CF-colored with only f(n)
colors? Remark We can define a CF-coloring for a
general set system (X,A) where A?P(X) i.e., a
coloring of the elements of X s.t. each set
s?A contains an element with a unique color
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Motivation from Frequency Assignment in cellular
networks
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mobile clients create links with base-stations
within reception radius
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Frequency assignment
1
2
1
Power and location of clients cellular may vary
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Problem Statement (cont)
What is the minimum number f(n) s.t. any n points
can be CF-colored with f(n) colors?
Thm f(n) gt log n
Easy n pts on a line! Discs Intervals
3
1
2
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Points on a line (cont)
log n colors suffice (in this special
case) Divide Conquer
1
3
1
3
2
3
2
3
Color median with 1
Recurse on right and left Reusing colors!
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CF-coloring in general case
Divide Conquer doesnt work!
Thm f(n) O(log n)
n pts
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Proof of f(n) O(log n)
Consider the Delauney Graph i.e., the empty
pairs graph
n pts
It is planar. Hence, By the four colors Thm ?
large independent set
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent

Pn
1. Color IS with 1 2. Remove IS
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent!

Pn
1.Color IS with 1 2. remove IS
3. Construct the new Delauney graph and iterate
(O(log n) times) on remaining pts
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent!

Pn
1.Color IS with 1 2. remove IS
5
4
3. Iterate (O(log n) times) on remaining pts
3
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Proof of f(n) O(log n) (cont)
Algorithm is correct
Consider a non-empty disc
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maximal color is unique
5
1
1
4
3
1
2
maximal color 3
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Proof of f(n) O(log n) (cont)
maximal color i is unique
Proof Assume i is not unique and ignore colors lt
i
i
maximal color i
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Proof of f(n) O(log n) (cont)
maximal color i is unique
Assume i is not unique and ignore colors lt i
i
i
maximal color i
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Proof maximal color i is unique
Consider the ith iteration
independent
i
i
i
A third point exists
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Proof maximal color i is unique
Consider the ith iteration
Contradiction!
i
i
i
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Remarks
Algorithm is very easy to implement
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What about other ranges?
CF-coloring pts w/ respect to other
ranges? Previous proof works for homothetic
copies of a convex body
Thm O(sqrt (n)) colors always suffice
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CF-coloring pts w.r.t axis-parallel rectangles
Thm O(sqrt (n)) colors always suffice
How small is an independent set in the Delauney
graph ?
I DONT KNOW!
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CF-coloring pts w.r.t axis-parallel rectangles
Note