On the Chromatic Number of Some Geometric Hypergraphs

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

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

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 -Color H CF(H) = min #colors needed to CF

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

Goal: Minimize the total number of frequencies

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

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

3. Recurse on remaining regions

2 2

3 4

1 3 2 2 1 4 1

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 6. R R R’ } ; R’ i

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

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

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

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)) ≤ 8 log |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

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

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

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

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

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

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

Summary CF-coloring i =0 1. General: Works for any hypergraph 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’ } Applied to regions with union complexity u(n) (H(R)) CF(H(R)) O(n) O(1) O(log n) O(n ) (pseudo discs, etc) O(n 1+ ) Convex ``fat’’ regions, etc

Brief History [Even, Lotker, Ron, Smorodinsky 03] n discs can be CF-colored with O(log n) colors. Tight! • Any • 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 …

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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