Guy Even, Zvi Lotker, Dana Ron, Shakhar Smorodinsky

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Conflict-free colorings of simple geometric
regions with applications to frequency
assignment in cellular networks
Guy Even, Zvi Lotker, Dana Ron, Shakhar
Smorodinsky Tel Aviv University
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Now thats a pretty LONG title!!!
Guy, are you sure you you didnt forget to add
something to the title?
Conflict-free colorings of simple geometric
regions with applications to frequency
assignment in cellular networks
Guy Even, Zvi Lotker, Dana Ron, Shakhar
Smorodinsky Tel-Aviv University
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cellular networks a base-station
every client within range can communicate with
base station
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cellular networks multiple base-stations
more antennas ? increase covered region
backbone network between base-stations
radio link client ? base-station
mobile clients dynamically create links with
base-stations
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interfering base-stations
base-stations using same frequency ? interference
in intersection of regions
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non-interfering base-stations
base-stations use different frequencies ? no
interference!
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base-station frequency assignment
Coloring intersecting base-stations must use
different frequencies too restrictive every
base can serve region of intersection. but,
one is enough!
Most models deal with interference between pairs
of base-stations, 3rd base-station cant resolve
an interference.
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Def Conflict-free coloring

• Coloring
• regions that cover a point P
• N(P) regions d P ? d
• point P is served by region d, if
• CF-coloring all covered points are served.

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What is the min colors needed in a CF-coloring ?
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What is the minimum number of colors we need ?
every 2 adjacent disks must have different
colors
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What is the minimum number of colors we need ?
Answer 3 colors
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What is the min colors needed in a CF-coloring?
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Answer 4 colors
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Hardness Min CF-coloring of unit disks

• NPC reduction similar to CCJ90
• vertex coloring of planar graph
• ?
• Vertex coloring of intersection graphs of unit
  disks
• Reduction implies also that
• (4/3-?)-approximation is NPC.

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arrangements of unit disks
a cell
Topological arrangement sub-division of plane
into cells.
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examples of arrangements
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set-system representation
coalesce cells with identical neighbors
disk-cell edge if cell in disk
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primal/dual set-systems
primal sets elements
dual elements sets
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arrangements of unit disks
arrangement corresponding to dual set system
skip
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self-duality

• A collection of set-systems A is self-dual if
  (X,R)? A implies that (R,X) ? A.

points arbitrary disks NOT self-dual
Consider set systems of points unit disks
X set of points in the plane R set of
ranges induced by intersection with unit
disks. Claim set systems of points unit
disks are self-dual.
More general points regions Claim set
system of points regions is self-dual
if regions are translations of a
centrally-symmetric body (e.g. square, hexagon,
rectangle).
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CF-coloring of points wrt ranges

• Coloring
• Require for every range d, there exists a color
  i, such that P?d ?(P)i contains a single
  point.
• Compare with coloring regions so that every
  point is served
• Simply means CF-coloring of the dual set system.

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CF-coloring of disks

• THM 1 poly-time algorithm for CF-coloring.
• Input arrangement of n disks in the plane
• Output CF-coloring of disks using O(log n)
  colors.

Tight ?arrangements of unit disks that require
?(log n) colors

• THM 2 poly-time algorithm for CF-coloring.
• Input X? R2 centers of n disks in the plane
• Output coloring ? of X using O(log n) colors,
  such that for every radius r, ? is a CF-coloring
  of D(X,r).

D(X,r) set of disks of radius r centered at
points of X
Uniform coloring ALG not given the radius. Same
coloring good for all radiuses.
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uniform CF-coloring of congruent disks
Reduction to CF-coloring of points wrt disks
(dual-of-dual)

• Notation
• X? R2 centers of n disks
• r gt 0 common radius
• D(X,r) set of n disks of radius r centered at
  points of X
• Y set of representatives from cells in
  arrangement D(X,r)
• Primal set-system (Y, D(X,r))
• Goal CF-color D(X,r) using O(log n) colors.

Uniform coloring radius r is not known
Dual set-system (X , D(Y,r)) Equivalent goal
CF-color points X wrt disks D(Y,r) using O(log n)
colors. Extended goal CF-color X wrt all disks
using O(log n) colors.
implies THM 2 (uniform coloring of disks)
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CF-color X wrt all disks using O(log n) colors

• Trivial empty range ranges with single point
• Remaining ranges with ? 2 points.
• Observation
• minimal ranges are the edges of the Delaunay
  graph of X.

Planarity of Delaunay graph ? independent set
X/4.
ALG (X,i) find an independent set IND?X in
DG(X), color every point x?IND with color
i recurse ALG(X-IND, i1)
IND?X/4 implies O(log n) colors!
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Correctness CF-color X wrt all disks
ALG (X,i) find an independent set IND?X in
DG(X), color every point x?IND with color
i recurse ALG(X-IND, i1)
Claim ALG(X,0) finds a CF-coloring of X wrt to
all disks

• Proof Fix disk D, and apply induction on size
  of range SD ?X.
• If S1, trivial.
• If S?2, then S?IND, because S contains an edge
  of DG(X).
• Eventually, IND stabs S, and then
• 0 lt S-IND lt S
• colors(S-IND) gt color(IND)
• Induction hyp. (S-IND) contains point with
  distinct color gt i
• ? S contains a point with distinct color. QED.

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Generalize CF-coloring of X wrt other regions

• THM 3 if regions are congruent homothetic copies
  of a
• centrally-symmetric convex body, then exists a
  CF-coloring of X
• wrt regions using O(log n) colors.

Examples of centrally-symmetric convex
bodies Disks, squares, rectangles, regular
polygons with even vertices
uniform coloring construction only needs
centers common scaling factor not given.
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bi-criteria algorithms for unit-disks
THM 4 Inflate radius by ?. Poly-time algorithm
for coloring inflated disks using O(log (1/ ?))
colors so that all points in unit disks are
served.
? 1/2O(opt) ? opt colors!
THM 5 Poly-time algorithm for coloring unit
disks using O(log (1/ ?)) colors so that all but
? -fraction of points in unit disks are served.
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constant ratio approximation algorithms

• THM 6 O(1)-apx algorithms for CF-coloring
• arrangements of axis-parallel squares
• arrangements of axis-parallel rectangles if

- arrangements of axis-parallel unit hexagons -
arrangements of axis-parallel hexagons if ratios
of side lengths are constant.
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Open questions

• O(1)-approximation algorithm for disks (have one
  for case of intersecting unit disks).
• CF-coloring of arrangements of regions similar to
  coverage areas of antennas 60º sectorsprogress
  by Har-Peled Somorodinsky.
• Capacitated versions center may serve a limited
  clients

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indexed arrangements

• assign indexes to disks (not arbitrary!).
• represent set system by diagram
• (i.e. is cell covered by disk?)

cells
N(cell) is an interval
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N(cell) is not an interval
disks
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Interval property of arrangements

• Indexed arrangement every disk has an index.
• Interval property if, for every cell v,
• there exist i ? j such that N(v) i,j.

• Full interval property interval property and,
• for every interval i,j, there exists a cell
  such that
• N(v) i,j.

Equivalent DEF dual set system
representation isomorphic to the set
system (1,,n, i,j )

• Chain an indexed arrangement that satisfies
  
• the full interval property

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chains
Claim for every n, there exists a chain C(n)
of n unit circles.
Proof index circles from left to right
same proof works with axis-parallel squares,
hexagons, etc.
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CF-colorings of chains

• Claim every CF-coloring of C(n) requires ?(log
  n) colors.
• proof query which disk serves cell v
  N(v)1,n?
• color of this disk appears once (unique
  color).

-red disk partitions chain into 2 disjoint
chains. -pick larger part, and continue
queries recursively.
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coloring chain with O(log n) colors
Back to thms
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theorem for unit disks

• a tile a square of unit diameter.
• local density ?(A(C)) of arrangement A(C)
  max disk centers in tile.
• Theorem There exists a poly-time algorithm
• Input a collection C of unit disks
• Output a CF-coloring of C
• Number of colors O(log ?(A(C)))
• Tightness see chains

BY every set-system can be CF-colored using
O(log2 C) colors
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reduction to case all disks centers in the same
tile
-Tile the plane diameter(tile) 1.
center(unit disk) ? tile ? tile ? unit
disk -Assign a palette to each tile (periodically
to blocks of 4?4 tiles), so disks from different
tiles with same palette do not intersect.
suffices now to CF-color disks with centers in
the same tile. (in particular, intersection of
all disks contains the tile)
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reduction to case all disks in the same tile
have a boundary arc
boundary disk disk with a boundary
arc. Reduction based on lemma ?boundary
disks ?disks. ? need to consider only boundary
disks in tile.
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boundary arcs
set of disks C - all centers in same tile - all
disks have a boundary arc Lemma every disk in C
has at most two boundary arcs.
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decomposition of boundary disksdisks on one
side of a line
This is where proof fails for non-identical disks
- all the disks cut r twice - ?two disks
intersect once - ?boundary disk WRT H has one
boundary arc in H - no nesting of boundary
disks - boundary disks WRT H are a chain
H
r
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decomposition of boundary disks (assume that
all the disks have precisely one boundary arc)

• pick 4 disks (that intersect
• extensions of vert sides)
• color 4 circles with
• 4 new distinct colors

• remaining disks
• 4 disjoint chains.
• color each chain.

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decompositions of boundary disks(disks that have
2 boundary arcs)
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decompositions of boundary disks(disks that have
2 boundary arcs)

• Lemma pairs of chains have the same orders.
• use 1 indexing for both chains.
• colors of disk in 2 chains agree.

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summary of CF-coloring algorithm

• Tiling 16 palettes
• Decomposing boundary disks 4 disks
• 4 chains of disks with 1 boundary arc
• 4 ? log (boundary disks in tile)
• chains of disks with 2 boundary arcs
• 6 ? log (boundary disks in tile)
• ? O(log(max (boundary disks in tile))) colors.

Observation if all disks belong to same
tile, then ALG uses at most 10?OPT 4 colors
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applications a bi-criteria algorithm

• C set of unit disks with ?C non-empty
• CF(C) min colors in CF-coloring of C
• C? Disk(x,1 ?) x center of unit disk in C
• Serve ?C with a coloring of C? .
• CORO exists coloring of C? that serves ?(C)
  using O(log 1/ ?) colors.
• Proof dilute centers so that dmin ? ?.
• CORO ? 1/2O(CF(C)) ? CF(C) colors!

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far from optimal

• ALG uses log n colors
• but, OPT uses only 4 colors
• reason ALG ignores help from disks centered
  in other tiles.
• local OPT ? global OPT

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Outline

• cellular networks Frequency Assignment Problem
• conflict-free coloring Model of FAP
• primal/dual range spaces
• results
• more results
• open problems

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More results

• Arrangements of squares constant approximation
  algorithm.
• Arrangements of regular polygons constant
  approximation algorithm. (also for case of
  constant angle types.
• Open problems constant approximation for unit
  disks, non-identical disks
• OPEN NP-completeness

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