1 / 23

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.

gadolph
Download Presentation

First of all …. Thanks to Janos

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. First of all …. Thanks to Janos

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

  3. 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

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

  5. 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

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

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

  8. 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

  9. 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!

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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

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

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

  20. Remarks: Algorithm is very easy to implement

  21. 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

  22. 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!

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

More Related