2. Shakhar Smorodinsky Courant Institute, NYU Online conflict-free coloring work with Amos Fiat, Meital Levy, Jiri Matousek, Elchanan Mossel, Janos Pach, Micha Sharir, Uli Wagner, Emo Welzl,

3. 1 A Coloring of pts Background Conflict-Free Coloring of Points w.r.t Discs 2 1 2 3 3 3 4 is Conflict Free (CF) if : 4 Any (non-empty) disc contains a unique color

4. 1 What is Conflict-Free Coloring of pts w.r.t Discs? 2 1 2 3 3 3 4 is Conflict Free if: 1 Any (non-empty) disc contains a unique color A Coloring of pts

5. So, what are the problems? For example: What is the minimum number f(n) s.t. any n points can be CF-colored (w.r.t discs) with f(n) colors?

6. Motivation [Even et al.]: From Frequency Assignment in cellular networks 1 1 2

7. Problem Statement for points (w.r.t discs) Lower Bound f(n) > log n What is the minimum number f(n) s.t. any n points can be CF-colored (w.r.t discs) with f(n) colors? Easy: n pts on a line! Discs => Intervals 1 3 2  log n colors n pts n/2n/2  n / 4

8. Points on a line: Upper Bound (cont) log n colors suffice (when pts colinear) 312 Color every other point with i Remove colored points ; i = i+1 Iterate until no points remain 12 1 1 3

9. Previous work There are 2 previous papers on offline CF coloring Even, Lotker, Ron, Smorodinsky (SICOMP 03) Approximation algs + bounds for discs. Har Peled and Smorodinsky (D&CG 05) Extended to different ranges, higher dimensions, relaxed colorings, VC-dim, etc…

10. Our result:Online CF-coloring for intervals: Points arrive online When a point arrives you need to give it a color Conflict free at any time: Any interval should contain a color that appears there exactly once 1 23 1 2

11. A simple algorithm Def: A point x sees color i, if there is a point y colored i, such that all points between x and y are colored < i < i i x

12. A simple algorithm (Cont) Give each newly inserted point the lowest color that it does not see 3 21 12 x

13. A simple algorithm (Cont) Give each newly inserted point the lowest color that it does not see 3 21 12 x 1 This alg maintains the stronger property that the maximum is unique

14. Example 2 1 214 21 31211 O(log n) for “extreme ends” insertion sequence 311

15. Is this algorithm good for general insertion sequences ? 21 121334 1 2

16. 21 121334 1 2 For this sequence the simple algorithm uses Ω(  n) colors 1k …… 1k-1 …… 1 1 2

17. Open problem #1 Is there a nontrivial upper bound on the number of colors used by this simple algorithm ?

18. Can we do it with fewer colors ? (using another algorithm)

19. New level

21. A new point gets into the lowest level at which it can extend a basic block either to the right or to the left It splits any basic block of lower level that surrounds it

27. Within a basic block we use the simple algorithm, with a separate set of colors for each level

28. Why is the coloring CF ? Any interval I intersects only one basic block of the highest level (of points in I) Use validity of the simple algorithm for this level

29. Analysis Within a level we use only O ( log (maximum block size) ) colors Because we are promised that points are always inserted in the extreme ends of a block

30. How many levels can we get? Def: Partition each basic block into atomic intervals: i i < i Each point closes exactly one atomic interval when it is inserted We associate each interval with the point that closed it

31. How many levels can we get? When we insert a point x at level i, it breaks atomic intervals of level 1,2,…i-1 Charge x to the closing points of those atomic intervals x

32. A forest describes the charging history These are binomial trees: A node of level i has a child of each level i-1,i-2,….,1 Such a node has 2 i descendants So we have at most log(n) levels

33. Summary Thm: The algorithm produces a CF coloring with O(log 2 (n)) colors

34. An improvement using randomization Use a bit more levels but fewer colors per level Make the basic blocks in each level short: O(log n) The result: a CF coloring with O(log n log log n) colors w.h.p.

35. More open problems Is there a deterministic algorithm that uses o(log 2 (n)) colors ? Is there a randomized algorithm that uses o(log n log log n) colors ? Ω(log n) lower bound

36. Online CF coloring in 2-D So what is really interesting are points in the plane, and online CF coloring with respect to disks For arbitrary disks, we show a lower bound n : Every point gets a new color Unit disks ? Halfplanes?

37. Recent result [Kaplan-Sharir] A randomized algorithm for online CF coloring in the plane with respect to unit disks with O(log 3 (n)) colors w.h.p. (also works for halfplanes and nearly equal axis-parallel rectangles)

38. I guess now there is a conflict with time… Thank You!