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,

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

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

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?

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

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  log n colors n pts n/2n/2  n / 4

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

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…

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

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

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

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

Example O(log n) for “extreme ends” insertion sequence 311

Is this algorithm good for general insertion sequences ?

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

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

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

New level

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

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

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

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

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

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

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

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

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.

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

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?

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)

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