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 can communicate 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: dynamically create 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. 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 disks cells coalesce cells with identical neighbors diskscells 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: skip

self-duality A 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 {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.

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 X Uniform coloring: ALG not given the radius. Same coloring good for all radiuses. TightTight:  arrangements of unit disks that require  (log n) colors THM 2: poly-time algorithm for CF-coloring. –Input: X  R 2 – 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  R 2 : 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.Delaunay graph ALG (X,i) : find an independent set IND  X in DG(X), color every point x  IND 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 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, 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 S  IND, 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.  =1/2 O(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 if - arrangements of axis-parallel “unit” hexagons - arrangements of axis-parallel hexagons if ratios 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 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 representation isomorphic 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 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.

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(log 2 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 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)

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.

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. Observation: if all disks belong to same tile, then ALG uses at most 10  OPT + 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 d min  . CORO:  =1/2 O(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…