Conflict-Free Coloring
of Shallow Discs
Shakhar Smorodinsky
Courant Institute (NYU)
Joint Work with
Noga Alon
Hope you didn’t eat too much…
So you will stay awake
What is Conflict-Free Coloring?
A Coloring of n
regions
Any point in the union is
contained in at least
one region whose color
is ‘unique’
is Conflict Free (CF) if:
1
2
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
Some History
[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).
•
[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) )
Thm: |R|= n and 1 ≤ k ≤ n. Each disc intersects ≤ k discs.
Then R can be CF-colored with O(log3 k) colors.
Def: Depth d(p) of a point p, is # of discs in R covering p
Note: p d(p) ≤ k+1
(maximum depth is ≤ k+1 )
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))
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 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)
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:
Depth d(p) in Ri
≤ log k
Otherwise:
By Lemma 2 p is
Conflict-Free in Ri
P Ri+1
“maximal”
i: p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!!!