Online conflictfree coloring 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 Any (non-empty) disc contains a unique color A Coloring of pts is Conflict Free (CF) if: 4 1 3 3 2 2 4 3 1

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

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

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

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 Bound f(n) > log n Easy: n pts on a line! Discs => Intervals n pts log n colors n/2 n/ 4 1 3 2

Points on a line: Upper Bound (cont) log n colors suffice (when pts colinear) 3 1 2 1 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 1 3 2 1 2

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 i <i <i

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

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

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

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

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

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? 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

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) 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) 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!