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Shakhar Smorodinsky Courant Institute, NYU

Online conflict-free coloring. Shakhar Smorodinsky Courant Institute, NYU. work with Amos Fiat, Meital Levy, Jiri Matousek, Elchanan Mossel, Janos Pach, Micha Sharir, Uli Wagner, Emo Welzl,. Background Conflict-Free Coloring of Points w.r.t Discs.

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Shakhar Smorodinsky Courant Institute, NYU

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

  2. BackgroundConflict-FreeColoring ofPoints w.r.t Discs Any (non-empty) disc contains a unique color AColoringof pts isConflict Free (CF) if: 4 1 3 2 4 3 3 2 1

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

  4. So, what are the problems? For example: What is the minimum numberf(n)s.t. anynpoints can be CF-colored (w.r.t discs)withf(n)colors?

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

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

  7. Points on a line: Upper Bound (cont) log ncolors suffice(when pts colinear) 3 1 1 2 3 1 2 1 Color every other point withi Remove colored points; i = i+1 Iterate until no points remain

  8. Previous work • There are 2 previous papers on offlineCF 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…

  9. 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 3 2 2 1

  10. i < i < i 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 x

  11. A simple algorithm (Cont) Give each newly inserted point the lowest color that it does not see x 2 1 2 1 3

  12. A simple algorithm (Cont) Give each newly inserted point the lowest color that it does not see x 2 1 2 1 3 1 This alg maintains the stronger property that the maximum is unique

  13. 4 2 1 1 2 2 1 1 1 1 1 3 3 1 Example 2 O(log n) for “extreme ends” insertion sequence

  14. 2 1 2 2 1 3 3 1 1 4 Is this algorithm good for general insertion sequences ?

  15. 2 1 1 1 2 2 2 1 1 3 3 1 …… …… …… k-1 k 1 1 4 Is this algorithm good for general insertion sequences ? For this sequence the simple algorithm uses Ω(n) colors

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

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

  18. New level

  19. 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

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

  21. 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

  22. 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

  23. 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

  24. How many levels can we get? x 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

  25. A forest describes thecharging 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 2i descendants So we have at most log(n) levels

  26. Summary Thm: The algorithm produces a CF coloring with O(log2(n)) colors

  27. 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.

  28. More open problems • Is there a deterministic algorithm that uses o(log2(n)) colors ? • Is there a randomized algorithm that uses o(log n log log n) colors ? • Ω(log n) lower bound

  29. 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?

  30. Recent result [Kaplan-Sharir] A randomized algorithm for online CF coloring in the plane with respect to unitdisks with O(log3(n)) colors w.h.p. (also works for halfplanes and nearly equal axis-parallel rectangles)

  31. I guess now there is a conflictwith time… Thank You!

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