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
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
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-Color H
CF(H) = min #colors needed to
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 {
i i+1
3.
4.
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)) ≤ 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
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
≤ 6n-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
8log 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)
u(n)
(H(R))
CF(H(R))
O(n)
O(1)
O(log n)
O(n)
O(n)
(pseudo discs, etc)
O(n1+)
Convex ``fat’’ regions,
etc
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 …
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(log2 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(log3 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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