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

3. 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 coloring H

4. 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} } Example: 1 2 3 4

5. 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-Color H

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

7. Goal: Minimize the total number of frequencies

8. Another model: radius of coverage is determined by client’s power

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

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

11. 3. Recurse on remaining regions

13. 2 2

14. 4 3

15. 1 1 1 2 2 4 3

16. 1 A Coloring of pts Problem: Conflict-Free Coloring of Points w.r.t Discs 2 1 2 3 3 3 4 is Conflict Free if: 4 Any (non-empty) disc contains a unique color

17. 1 A Coloring of pts What is Conflict-Free Coloring of pts w.r.t Discs? 2 1 2 3 3 3 4 is Conflict Free if: 1

18. Lower bounds for CF-coloring Lower Bound log n Easy: n points on a line 1 3 2  log n colors n pts n/2n/2  n / 4

19. Points on a line: Upper Bound (cont) log n colors suffice (when pts colinear) Divide & Conquer 132 Color median with 1 Recurse on right and left Reusing colors! 32 3 3 1

20. CF-coloring pts w.r.t discs Upper Bound f(n) = minimum # colors needed in worst case Thm: [Even, Lotker, Ron, Smorodinsky 03] O(log n) colors suffice! n pts

21. Proof of Upper Bound: f(n) = O(log n) Consider the Delauney Graph i.e., the “empty pairs” graph It is planar. Hence, by the four color Thm  “large” independent set n pts

22. Proof of: f(n) = O(log n) (cont)  IS  P s.t. |IS|  n/4 and IS is independent |P|=n 1. Color IS with 1 2. Remove IS 1 1 1

23. Proof of: f(n) = O(log n) (cont)  IS  P s.t. |IS|  n/4 and IS is independent! |P|=n 1.Color IS with 1 2. remove IS 3. Construct the new Delauney graph … and iterate (O(log n) times ) on remaining pts 22

24. Proof of: f(n) = O(log n) (cont)  IS  P s.t. |IS|  n/4 and IS is independent! |P|=n 1.Color IS with 1 2. remove IS 3. Iterate (O(log n) times ) on remaining pts 5 3 4

25. Algorithm is correct Proof of: f(n) = O(log n) (cont) 1 1 1 5 3 4 2 2 Consider a non-empty disc “maximal” color 3 “maximal” color is unique

26. Proof of: f(n) = O(log n) (cont) “maximal” color i is unique Proof: Assume i is not unique and ignore colors < i “maximal” color i i

27. Proof of: f(n) = O(log n) (cont) “maximal” color i is unique Assume i is not unique and ignore colors < i “maximal” color i i i

28. Proof: maximal color i is unique Consider the i ’th iteration independent i i A third point exists i

29. Proof: maximal color i is unique Consider the i ’th iteration i i Contradiction! i

30. General Framework for CF-coloring a Hypergraph H= (V,E) : 1.Find a “large” Independent Set V’ in “Delauney graph” 2. Color V’ with i; i=i+1 3. Recurse on V \ V’ Works as long as the hypergraph is “shrinkable” ( i..e, every hyperedge contains an edge )

31. Framework doesn’t work for general hypergraphs such as: R : Discs: Not “shrinkable” Three small discs form an independent set in the “Delaunay graph”

32. New Framework for CF-coloring Summary CF-coloring a finite family of regions R: 1. i =0 2.While (R   ) do { 3. i  i+1 4.Find a Proper Coloring  of H(R) with ``few’’ colors 5. R’  largest color class of  ; R’  i 6. R  R \R’ }

33. Framework for CF-coloring (cont) 1.i=0 2.While (R   ) do { 3. i  i+1 4. Find a Coloring  of H(R) with ``few’’ colors 5. R’  largest color class of  ; R’  i 6. R  R \R’ } Framework is correct! In fact, maximal color of any hyperedge is unique “ maximal ” color i Another i

34. Framework for CF-coloring (cont) 1.i=0 2.While (R   ) do { 3. i  i+1 4. Find a Coloring  of H(R) with ``few’’ colors 5. R’  largest color class of  ; R’  i 6. R  R \R’ } Framework is correct! In fact, maximal color of any hyperedge is unique “ maximal ” color i Another i i th iteration Not monochromatic Not discard at i ’th iteration

35. New Framework (cont) CF-coloring a finite family of regions R: i =0 1.While (R   ) do { 2. i  i+1 3. Find a Coloring  of H(R) with ``few’’ colors 4. R’  largest color class of  ; R’  i 5. R  R \R’ } Key question: Can we make  use only ``few” colors?

36. 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 Our Results on Proper Colorings

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

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

39. Coloring pseudo-discs Thm [Kedem, Livne, Pach, Sharir 86]: The complexity of the union of any n pseudo-discs is ≤ 6n-12 Hence, u(n)/n is a constant. By above Thm, its chromatic number is O(1)

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

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

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

43. R is four colorable iff the Four-Color THM R: Discs

44. Summary CF-coloring i =0 1.While (R   ) do { 2. i  i+1 3. Find a Coloring  of H(R) with ``few’’ colors 4. R’  largest color class of  5. R  R \R’ } u(n)u(n)  (H(R))  CF (H(R)) O(n) (pseudo discs, etc) O(1)O(log n) O(n 1+  ) Convex ``fat’’ regions, etc O(n)O(n)O(n)O(n) Applied to regions with union complexity u(n) General: Works for any hypergraph

45. Brief History [Even, Lotker, Ron, Smorodinsky 03 ] Any n discs can be CF-colored with O(log n) colors. Tight! Finding optimal coloring is NP-HARD even for congruent discs. (approximation algorithms are provided) For pts 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 …

46. Brief History (cont) [Alon, Smorodinsky 05] O(log 3 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(log 2 n) 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(log 3 n) colors w.h.p [Chen 05] CF-color w.r.t congruent discs. O(log n) colors w.h.p

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