2. Shakhar Smorodinsky Courant Institute (NYU) Joint Work with Noga Alon Conflict-Free Coloring of Shallow Discs

3. Hope you didn’t eat too much… So you will stay awake

4. 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

5. Motivation for CF-colorings Frequency Assignment in cellular networks 1 1 2

6. Goal: Minimize the total number of frequencies

7. More motivations: RFID-tags network RFID tag: No battery needed. Can be triggered by a reader to trasmit data (e.g., its ID)

8. 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

9. RFID-tags network (cont) Tags and …Readers Goal: Assign time slots to readers from {1,..,t} such that all tags are read. Minimize t

10. [ 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).

11. .(Algorithmic) Online version: [FLMMPSSWW, SODA 2005] pts arrive online on a line; CF-color w.r.t intervals: O(log 2 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(log 3 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

12. 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 …..

13. 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(log 3 k) colors (Compare with O(log n) )

14. 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(log 3 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(log 2 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

15. 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(log 2 k) colors s.t. faces of depth O(log k) are Conflict-Free.

16. Lemma 2:  i one can color a set R of n discs with O(i 2 ) colors s.t. every p with d(p) ≤ i is Conflict-Free

17. Algorithm: 1.Find a subset R 1 (as in Lemma 1) and color it with O(log 2 k) colors As in Lemma 2 1.Iterate on R\R’ until max depth ≤ O(log k) Correctness: “ maximal ” i: p  R i Depth d(p) in R i ≤ log k Otherwise: P  R i+1 By Lemma 2 p is Conflict-Free in R i

18. Remark: Proof works for regions with linear union complexity (e.g., pseudo-discs have linear union complexity [KLPS 86] )

19. 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

20. THANK YOU WAKE UP!!!