First of all …. Thanks to Janos

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

1 A Coloring of pts Definition of Conflict-Free Coloring is Conflict Free if: 4 Any (non-empty) disc contains a unique color

1 A Coloring of pts What the … is Conflict-Free Coloring? is Conflict Free if: 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 mobile clients: create links with base-stations within reception radius 1

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

Problem Statement (cont) Thm: f(n) > log n What is the minimum number f(n) s.t. any n points can be CF-colored with f(n) colors? Easy: n pts on a line! Discs = Intervals  log n colors n pts n/2n/2  n / 4

Points on a line (cont) log n colors suffice (in this special case) Divide & Conquer 132 Color median with 1 Recurse on right and left Reusing colors!

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

Proof of: f(n) = O(log n) Consider the Delauney Graph i.e., the “empty pairs” graph It is planar. Hence, By the four colors Thm  “large” independent set n pts

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

Proof of: f(n) = O(log n) (cont)  IS  P s.t. |IS|  n/4 and IS is independent! |P|=n 1.Color IS with 1 2. remove IS 3. Construct the new Delauney graph … and iterate (O(log n) times ) on remaining pts 22

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

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

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

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

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

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

Remarks: Algorithm is very easy to implement

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 How about axis-parallel rectangles?

Thm: O ( sqrt (n) ) colors always suffice CF-coloring pts w.r.t axis-parallel rectangles How small is an independent set in the “Delauney” graph ? I DON’T KNOW!

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