Shakhar Smorodinsky

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On the Chromatic Number of Some Geometric
Hypergraphs
Shakhar Smorodinsky Courant Institute, New-York
University (NYU)
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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
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Example
R1,2,3,4, 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
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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
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Motivation for CF-colorings Frequency Assignment
in cellular networks
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1
2
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Goal Minimize the total number of frequencies
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Another model radius of coverage is determined
by clients power
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A CF-Coloring Framework for R 1. Find a proper
coloring of R
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2. Color regions in largest color class with 1
and remove them
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3. Recurse on remaining regions
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3
4
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3
2
2
4
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Problem Conflict-Free Coloring of Points w.r.t
Discs
Any (non-empty) disc contains a unique color
A Coloring of pts
is Conflict Free if
4
1
3
2
4
3
3
2
1
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What is Conflict-Free Coloring of pts w.r.t
Discs?
A Coloring of pts
1
is Conflict Free if
1
3
2
4
3
3
2
1
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Lower bounds for CF-coloring
Lower Bound log n
Easy n points on a line
3
1
2
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Points on a line Upper Bound (cont)
log n colors suffice (when pts colinear) Divide
Conquer
1
3
1
3
2
3
2
3
Color median with 1
Recurse on right and left Reusing colors!
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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
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Proof of Upper Bound f(n) O(log n)
Consider the Delauney Graph i.e., the empty
pairs graph
n pts
It is planar. Hence, by the four color Thm ?
large independent set
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent

Pn
1. Color IS with 1 2. Remove IS
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent!

Pn
1.Color IS with 1 2. remove IS
3. Construct the new Delauney graph and iterate
(O(log n) times) on remaining pts
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Proof of f(n) O(log n) (cont)

• IS? P s.t. IS ? n/4 and
• IS is independent!

Pn
1.Color IS with 1 2. remove IS
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4
3. Iterate (O(log n) times) on remaining pts
3
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Proof of f(n) O(log n) (cont)
Algorithm is correct
Consider a non-empty disc
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maximal color is unique
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1
1
4
3
1
2
maximal color 3
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Proof of f(n) O(log n) (cont)
maximal color i is unique
Proof Assume i is not unique and ignore colors lt
i
i
maximal color i
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Proof of f(n) O(log n) (cont)
maximal color i is unique
Assume i is not unique and ignore colors lt i
i
i
maximal color i
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Proof maximal color i is unique
Consider the ith iteration
independent
i
i
i
A third point exists
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Proof maximal color i is unique
Consider the ith iteration
Contradiction!
i
i
i
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• General Framework for CF-coloring a Hypergraph
  H (V,E)
• Find a large Independent Set V in
• Delauney graph
• 2. Color V with i ii1
• 3. Recurse on V\V

Works as long as the hypergraph is shrinkable
(i..e, every hyperedge contains an edge)
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Framework doesnt work for general hypergraphs
such as
R Discs Not shrinkable
Three small discs form an independent set in the
Delaunay graph
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New Framework for CF-coloring Summary

• CF-coloring a finite family of regions R
• i 0
• While (R ? ?) do
• i ?i1
• Find a Proper Coloring ? of H(R) with few
  colors
• R? largest color class of ? R ? i
• R ? R \R

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Framework for CF-coloring (cont)

• i0
• While (R ? ?) do
• i ?i1
• Find a Coloring ? of H(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
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Framework for CF-coloring (cont)

• i0
• While (R ? ?) do
• i ?i1
• Find a Coloring ? of H(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 color i
i th iteration Not monochromatic
Not discard at ith iteration
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New Framework (cont)

• CF-coloring a finite family of regions R
• i 0
• While (R ? ?) do
• i ?i1
• Find a Coloring ? of H(R) with few
  colors
• R? largest color class of ? R ? i
• R ? R \R

Key question Can we make ? use only few
colors?
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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
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Chromatic number of H(R) Definition Union
Complexity
1
4
2
Union complexity vertices on boundary
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Thm R Regions s.t. any n have union complexity
bounded by u(n) then ?(H(R)) o(u(n)/n)
Example pseudo-discs
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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)
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How about axis-parallel rectangles?
Union complexity could be quadratic !!!
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Coloring axis-parallel rectangles
8 colors
For general case, apply divide and conquer
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Coloring axis-parallel rectangles
Obtain Coloring with 8log n colors
For general case, apply divide and conquer
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R Discs
R is four colorable iff the Four-Color THM
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Summary CF-coloring
General Works for any hypergraph

• i 0
• While (R ? ?) do
• i ?i1
• Find a Coloring ? of H(R) with few
  colors
• R? largest color class of ?
• R ? R \R

Applied to regions with union complexity u(n)
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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

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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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