Shakhar Smorodinsky Courant Institute (NYU) Joint Work with Noga Alon

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

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

3. 1 2 1 1 What is Conflict-Free Coloring? A Coloring of nregions Any point in the union is contained in at least one region whose color is ‘unique’ is Conflict Free (CF) if:

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

5. Goal: Minimize the total number of frequencies

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

7. More motivations: RFID-tags network [H. Gupta] Readers Tags and … A tag can be read at a given time only if one reader is triggering a read action

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

9. Some History • [Even, Lotker, Ron,S, FOCS 2002] • Anyndiscs can be CF-colored withO(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).

10. .(Algorithmic) Online version: • [FLMMPSSWW, SODA2005] • pts arrive online on a line; CF-color w.r.t intervals: • O(log2n) colors. O(log n log log n) w.h.p • [Bar-Noy, Chilliaris, S, SPAA2006] • O(log n) colors deterministic… weaker adversary • [Kaplan, Sharir, 2004] • pts arrive online in the plane colorw.r.t unit discs: • O(log3n) 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

11. 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 …..

12. 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))

13. Thm: |R|= n and 1≤k ≤ n. Each disc intersects ≤k discs. Then R can be CF-colored with O(log3 k) colors. Def: Depthd(p) of a point p, is # of discs in R covering p Note: p d(p)≤ k+1 (maximum depth is ≤ k+1 ) • Sketch: • We discard a subset R’ R s.t. max depth in R\R’ is ≤ (2/3)k • We color R’ with O(log2 k) colors s.t. faces of depth O(log k) are Conflict-Free. • Repeat until all faces are shallow (Depth≤ O(log k))

14. Sketch: • We will discard a subset R’ R s.t. max depth of R\R’ is ≤ (2/3)k • We color R’ with O(log2 k) colors s.t. faces of depth O(log k) are Conflict-Free. • Lemma 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)

15. Lemma 2: • i one can color a setRof n discs with O(i2)colors s.t. everyp with d(p)≤ i is Conflict-Free

16. P Ri+1 • Algorithm: • Find a subset R1 (as in Lemma 1) and color it with O(log2 k) colors • As in Lemma 2 • Iterate on R\R’ until max depth ≤ O(log k) • Correctness: “maximal” i: pRi Depth d(p) in Ri ≤ log k Otherwise: By Lemma 2 p is Conflict-Free in Ri

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

18. Open Problems • Can we use O(log k) colors. • Can we use polylog(k) colors for discs with max depth k • 2 => 1 but not vice versa

19. THANK YOU WAKE UP!!!