Post on 19-Dec-2015
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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…