1 / 37

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.

schaff
Télécharger la présentation

Shakhar Smorodinsky Courant Institute, NYU

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. 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!

More Related