Shakhar Smorodinsky Courant Institute, New-York University (NYU) On the Chromatic Number of Some...
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Transcript of Shakhar Smorodinsky Courant Institute, New-York University (NYU) On the Chromatic Number of Some...
Shakhar Smorodinsky
Courant Institute, New-York University (NYU)
On the Chromatic Number of Some Geometric Hypergraphs
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
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
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
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
11
3. Recurse on remaining regions
2
2
4
3
1
11
22
4
3
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’
}
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
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
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?
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
Chromatic number of H(R):
Definition: Union Complexity1
2
4
Union complexity:= #vertices on boundary
Example: pseudo-discs
Thm:
R : Regions s.t. any n have union complexity bounded by
u(n) then (H(R)) = o(u(n)/n)
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
For general case, apply divide and conquer
≤ 8 colors
Coloring axis-parallel rectangles
For general case, apply divide and conquer
Obtain Coloring with
8log n colors
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) (H(R)) CF(H(R))
O(n)
(pseudo discs, etc)
O(1) O(log n)
O(n1+)Convex ``fat’’ regions,
etc
O(n) O(n)
Applied to regions with union complexity u(n)
General: Works for any hypergraph
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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