Shakhar SmorodinskyCourant Institute (NYU)

Joint Work with Noga Alon

Conflict-Free Coloring of Shallow Discs

Hope you didn’t eat too much…

So you will stay awake

A Coloring of n regions

What is Conflict-Free Coloring?

is Conflict Free (CF) if:

Any point in the union is contained in at least one region whose color is ‘unique’

2

1

1

1

Motivation for CF-colorings

Frequency Assignment in cellular networks

1

1

2

Goal: Minimize the total number of frequencies

More motivations: RFID-tags network

RFID tag: No battery needed. Can be triggered by a reader to trasmit data (e.g., its ID)

More motivations: RFID-tags network [H. Gupta]

Tags and …

Readers

A tag can be read at a given time only if one reader is triggering a read action

RFID-tags network (cont)

Tags and …

Readers

Goal: Assign time slots to readers from {1,..,t} such that all tags are read. Minimize t

• [Even, Lotker, Ron, S, FOCS 2002]

Any n discs can be CF-colored with O(log n) colors. Tight!

Finding optimal coloring is NP-HARD even for congruent discs. (some approximation algorithms provided)

• [Har-Peled, S, SOCG 2003]

Extensions, randomized framework for general ``nice’’ regions

(i.e., low union complexity).

Some History

• [S, SODA 2006]

Deterministic framework ``nice’’ regions (low union complexity).

.(Algorithmic) Online version:

• [FLMMPSSWW, SODA 2005]

pts arrive online on a line; CF-color w.r.t intervals:

O(log2 n) colors. O(log n log log n) w.h.p

• [Bar-Noy, Chilliaris, S, SPAA 2006]

O(log n) colors deterministic… weaker adversary

[Kaplan, Sharir, 2004]

pts arrive online in the plane color w.r.t unit discs:

O(log3 n) colors w.h.p

• [Chen SOCG 2006 (just few mins…)]

O(log n) colors w.h.p

• [Bar-Noy, Chilliaris, S, 2006]

O(log n) colors w.h.p for general hypergraphs with `nice’ properties

CF-coloring Discs (in the worst case)

Lower Bound[Even, Lotker, Ron, S 2002]

Sometimes:

(log n ) colors are necessary!

However, in this case there are discs that intersect all other discs

In view of the motivation …..

CF-coloring (Shallow) Discs

…. a natural question arise:

Suppose |R|= n discs and each disc intersects

At most k other discs where k << n

Our result:

We can always CF-color R with O(log3 k) colors

(Compare with O(log n) )

Note: p d(p) ≤ k+1

(maximum depth is ≤ k+1 )

Thm: |R|= n and 1 ≤ k ≤ n. Each disc intersects ≤ k discs. Then R can be CF-colored with O(log3 k) colors.

Sketch:

1. We discard a subset R’ R s.t. max depth in R\R’ is ≤ (2/3)k

2. We color R’ with O(log2 k) colors s.t. faces of depth O(log k) are Conflict-Free.

3. Repeat until all faces are shallow (Depth ≤ O(log k))

Def: Depth d(p) of a point p, is # of discs in R covering p

Lemma 1:

1. One can color R with two colors Red and Blue s.t. :

p with d(p) >> log k # b(p) of blue discs covering p obeys:

(1/3) d(p) < b(p) < (2/3) d(p)

(a random coloring will do it…

Chernoff bound + Lovasz’ Local Lemma

here we use the assumption on max intersections)

Sketch:

1. We will discard a subset R’ R s.t. max depth of R\R’ is ≤ (2/3)k

2. We color R’ with O(log2 k) colors s.t. faces of depth O(log k) are Conflict-Free.

Lemma 2: i one can color a set R of n discs with O(i2) colors

s.t.

every p with d(p) ≤ i is Conflict-Free

Algorithm:

1. Find a subset R1 (as in Lemma 1) and color it with O(log2 k) colors

As in Lemma 2

1. Iterate on R\R’ until max depth ≤ O(log k)

Correctness:“maximal” i: pRi

Depth d(p) in Ri

≤ log k

Otherwise:

P Ri+1

By Lemma 2 p is Conflict-Free in Ri

Remark:

• Proof works for regions with linear union complexity

(e.g., pseudo-discs have linear union complexity [KLPS 86] )

Open Problems

1. Can we use O(log k) colors.

2. Can we use polylog(k) colors for discs with max depth k

2 => 1 but not vice versa

THANK YOU

WAKE UP!!!