First of all …. Thanks to Janos

First of all …. Thanks to Janos

Conflict-free coloring problems Shakhar Smorodinsky Tel Aviv University Part of this work is joint with Guy Even, Zvi Lotker and Dana Ron.

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

What the … is Conflict-Free Coloring? A Coloring of pts 1 is Conflict Free if: 1 3 2 4 3 3 2 1

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

Motivation from Frequency Assignment in cellular networks 1 mobile clients: create links with base-stations within reception radius

If 2 is unique Frequency assignment 1 2 1 Power and location of clients’ cellular may vary

log n colors n pts n/2  n/4 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) > log n Easy: n pts on a line! Discs = Intervals 3 1 2

Points on a line (cont) log n colors suffice (in this special case) Divide & Conquer 1 3 3 2 1 3 2 3 Color median with 1 Recurse on right and left Reusing colors!

CF-coloring in general case Divide & Conquer doesn’t work! Thm: f(n) = O(log n) n pts

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

1 1 1 Proof of: f(n) = O(log n) (cont) • ISP s.t. |IS| n/4 and IS isindependent |P|=n 1. Color IS with 1 2. Remove IS

2 2 Proof of: f(n) = O(log n) (cont) • ISP s.t. |IS| n/4 and IS isindependent! |P|=n 1.Color IS with 1 2. remove IS 3. Construct the new Delauney graph … anditerate (O(log n) times)on remaining pts

Proof of: f(n) = O(log n) (cont) • ISP s.t. |IS| n/4 and IS isindependent! |P|=n 1.Color IS with 1 2. remove IS 5 4 3. Iterate (O(log n) times)on remaining pts 3

Proof of: f(n) = O(log n) (cont) Algorithm is correct Consider a non-empty disc 2 “maximal” color is unique 5 1 1 4 3 1 2 “maximal” color 3

Proof of: f(n) = O(log n) (cont) “maximal” color i is unique Proof: Assume i is not unique and ignore colors < i i “maximal” color i

Proof of: f(n) = O(log n) (cont) “maximal” color i is unique Assume i is not unique and ignore colors < i i i “maximal” color i

Proof: maximal color iis unique Consider the i’th iteration independent i i i A third point exists

Proof: maximal color iis unique Consider the i’th iteration Contradiction! i i i

Remarks: Algorithm is very easy to implement

How about axis-parallel rectangles? 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

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 DON’T KNOW!

CF-coloring pts w.r.t axis-parallel rectangles Note:

First of all …. Thanks to Janos

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