Shakhar Smorodinsky Courant Institute, New-York University (NYU)

1. On the Chromatic Number of Some Geometric Hypergraphs Shakhar Smorodinsky Courant Institute, New-York University(NYU)

2. Hypergraph Coloring (definition) A Hypergraph H=(V,E) : V  1,…,k is a proper coloring if no hyperedge is monochromatic Chromatic number (H) = min #colors needed for proper coloringH

3. Example: R={1,2,3,4}, H(R) = (R,E), E ={ {1}, {2}, {3}, {4},{1,2}, {2,4},{2,3}, {1,3}, {1,2,3} {2,3,4}, {3,4} } 1 4 2 3

4. Conflict-Free Colorings A Hypergraph H=(V,E) : V  1,…,k is a Conflict-Free coloring (CF) if every hyperedge contains some unique color CF-chromatic number CF(H) = min #colors needed to CF-ColorH

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

6. Goal: Minimize the total number of frequencies

7. A CF-Coloring Framework for R 1. Find a proper coloring of R

8. 1 1 1 2. Color regions in largest color class with 1 and remove them

9. 3. Recurse on remaining regions

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12. 3 4

13. 1 1 1 3 2 2 4

14. New Framework for CF-coloring Summary • CF-coloring a finite family of regions R: • i =0 • While (R ) do { • ii+1 • Find a Proper ColoringofH(R) with ``few’’ colors • R’ largest color class of  ; R’ i • RR\R’ • }

15. “maximal” colori Anotheri Framework for CF-coloring (cont) • i=0 • While (R ) do { • i i+1 • Find a Coloring ofH(R) with ``few’’ colors • R’ largest color class of  ; R’ i • R  R \R’ • } Framework is correct! In fact, maximal color of any hyperedge is unique

16. Anotheri Framework for CF-coloring (cont) • i=0 • While (R ) do { • i i+1 • Find a Coloring ofH(R) with ``few’’ colors • R’ largest color class of  ; R’ i • R  R \R’ • } Framework is correct! In fact, maximal color of any hyperedge is unique “maximal” colori ith iteration Notmonochromatic Not discard ati’th iteration

17. New Framework (cont) • CF-coloring a finite family of regions R: • i =0 • While (R ) do { • ii+1 • Find a Coloring ofH(R) with ``few’’ colors • R’ largest color class of  ; R’ i • RR\R’ • } Key question: Can we make use only ``few” colors?

18. Our Results on Proper Colorings 1. D = finite family of discs. (H(D))≤ 4 (tight!) In fact, equivalent to the Four-Color Theorem. 2. R:axis-parallel rectangles. (H(R))≤8log |R| Asymptotically tight! [Pach,Tardos 05] provided matching lower bound. 3.R :Jordan regions with ``low’’ ``union complexity’’ Then (H(R)) is ``small’’ (patience….) For example: c s.t. (H(pseudo-discs))≤ c

19. Chromatic number of H(R): Definition: Union Complexity 1 4 2 Union complexity:= #vertices on boundary

20. Thm: R: Regions s.t. any nhave union complexity bounded by u(n)then (H(R)) = o(u(n)/n) Example: pseudo-discs

21. Coloring pseudo-discs Thm[Kedem, Livne, Pach, Sharir 86]: Thecomplexityof theunionof any npseudo-discs is ≤ 6n-12 Hence,u(n)/nisaconstant. By above Thm, its chromatic number is O(1)

22. How about axis-parallel rectangles? Union complexity could be quadratic !!!

23. Coloring axis-parallel rectangles ≤8 colors For general case, apply divide and conquer

24. Coloring axis-parallel rectangles Obtain Coloring with 8log n colors For general case, apply divide and conquer

25. Summary CF-coloring General: Works for any hypergraph • i =0 • While (R ) do { • ii+1 • Find a Coloring ofH(R) with ``few’’ colors • R’ largest color class of  • RR\R’ • } Applied to regions with union complexity u(n)

26. Brief History • [Even, Lotker, Ron, Smorodinsky 03] • Anyndiscs can be CF-colored withO(log n)colors. Tight! • Finding optimal coloring is NP-HARD even for congruent discs. (approximation algorithms are provided) • Forpts w.r.t discs (or homothetics), O(log n)colors suffice. • [Har-Peled, Smorodinsky 03] • Randomized framework for ``nice’’ regions, relaxed colorings, higher dimensions, VC-dimension …

27. Brief History (cont) • [Alon, Smorodinsky 05]O(log3 k) colors for n discs s.t. each intersects at most k others. • (Algorithmic) Online version: • [Fiat et al., 05] pts arrive online on a line. CF-color w.r.t intervals. O(log2n) colors. • [Chen 05] [Bar-Noy, Hillaris, Smorodinsky 05]O(log n) colors w.h.p • [Kaplan, Sharir, 05] pts arrive online in the plane • CF-color w.r.t congruent discs. O(log3n) colors w.h.p • [Chen 05]CF-color w.r.t congruent discs. • O(log n) colors w.h.p

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