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:
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)
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 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 2i descendants
So we have at most log(n) levels
Summary
Thm: The algorithm produces a CF
coloring with O(log2(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(log2(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(log3(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!