Guy Even, Zvi Lotker, Dana Ron, Shakhar Smorodinsky

Tel Aviv University

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

Conflict-free colorings of simple geometric regions

with applications to frequency

assignment in cellular networks

Now that’s a pretty LONG title!!!Guy, are you sure you you didn’t forget to add something to the title?

r=range

every client within range cancommunicate with base station

cellular networks – a base-station

more antennas

increase covered region

cellular networks – multiple base-stations

backbone network:between base-stations

radio link:client base-station

mobile clients: dynamicallycreate links with base-stations

interfering base-stations

base-stations using same frequency

interference in intersection of regions

non-interfering base-stations

base-stations use different frequencies

no interference!

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 can´t resolve an interference.

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.

)()'(:)(' dddPNd

dP

Nregions :

1

2

What is the min #colors needed in a CF-coloring ?

What is the minimum number of colors we need ?

every 2 “adjacent” disks must have different colors

Answer: 3 colors

What is the minimum number of colors we need ?

What is the min #colors needed in a CF-coloring?

Answer: 4 colors

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.

arrangements of unit disks

Topological arrangement: sub-division of plane into cells.

a cell

examples of arrangements

7 cells : all non-empty subsets

6 cells : missing red-blue cell

7 cells: missing red-blue cell but brown cell appears twice.

(view it as a single cell combinatorially equiv. to previous arrangement)

set-system representation 1

23

4

5

6

7

1

2

3

4

5

6

disks

1

2

3

4

5

7

6

cells

coalesce cells with identical neighbors

1

2

3

4

5

7

6

disks cells

1 2 3 4 51 2 3 4 5

6

7

disk-cell edge if cell in disk

primal/dual set-systems

primal: sets elements

dual: elements sets

arrangements of unit disks

arrangement corresponding to dual set system:

)()( PDXXDP

skip

self-dualityA collection of set-systems A is self-dual if

(X,R) A implies that (R,X*) A.

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).

“points & arbitrary disks”NOT self-dual

CF-coloring of points wrt ranges

• Coloring:

• Require: for every range d, there exists a color i, such that {Pd: (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.

Npoints:

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.

D(X,r) = set of disks of radius r centered at points of XUniform coloring: ALG not given the radius.

Same coloring good for all radiuses.

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).

uniform CF-coloring of congruent disks

Notation:X R2: centers of n disks

r > 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)

Reduction: to CF-coloring of points wrt disks (“dual-of-dual”)

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.

ALG (X,i) :

find an independent set INDX in DG(X),

color every point xIND with color i

recurse: ALG(X-IND, i+1)

Planarity of Delaunay graph independent set |X|/4.

IND|X|/4 implies O(log n) colors!

Correctness: CF-color X wrt all disksALG (X,i) :

find an independent set INDX in DG(X),

color every point xIND with color i

recurse: ALG(X-IND, i+1)

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 S=D X.If |S|=1, trivial.If |S|2, then SIND, because S contains an edge of DG(X).Eventually, IND stabs S, and then:

1. 0 < |S-IND| < |S|2. colors(S-IND) > color(IND)3. Induction hyp.: (S-IND) contains point with distinct color > i

S contains a point with distinct color. QED.

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.

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.

constant ratio approximation algorithms

THM 6: O(1)-apx algorithms for CF-coloring:

- arrangements of axis-parallel squares

- arrangements of axis-parallel rectangles ifconstant

)min(

)max(,

)min(

)max(

length

length

width

width

- arrangements of axis-parallel “unit” hexagons

- arrangements of axis-parallel hexagons ifratios of side lengths are constant.

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 sectors…progress by Har-Peled & Somorodinsky.

• Capacitated versions: center may serve a limited #clients

indexed arrangements

• assign indexes to disks (not arbitrary!).

• represent set system by diagram

(i.e. is cell covered by disk?)cells

disks

2 4 5

7 8 9

N(cell) is an interval

N(cell) is not an interval

Interval property of arrangements

• Full interval property: interval property and,

for every interval [i,j], there exists a cell such that

N(v) = [i,j].

• Indexed arrangement: every disk has an index.

• Interval property: if, for every cell v,

there exist i j such that: N(v) = [i,j].

• Chain: an indexed arrangement that satisfies

the full interval property

Equivalent DEF:dual set system representationisomorphic to the set system

({1,…,n}, {[i,j]} )

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.

CF-colorings of chainsClaim: 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.

).(log)()1(),(max 1

:equation recurrence

i nnfinfif f(n)

coloring chain with O(log n) colors

Back to thms

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

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 44 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)

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.

boundary arc

non-boundary arc

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.

distance(centers) 1

angle of intersection at least 2/3

decomposition of boundary disks:disks on one side of a line

- 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

r

H

This is where proof fails for non-identical disks

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.

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

• previous method gives 2

colors per disk.

• 4 chains & each disk in

2 chains.

• partition disks into

parts.

• 2 chains in each part.

2

4

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.

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.

2

4

Observation: if all disks belong to same tile,

then ALG uses at most 10OPT + 4 colors

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!

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

Outline

• cellular networks – Frequency Assignment Problem

• conflict-free coloring – Model of FAP

• primal/dual range spaces

• results

• more results

• open problems

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…