1. 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

2. 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?

3. r=range every client within range can communicate with base station cellular networks – a base-station

4. 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

5. interfering base-stations base-stations using same frequency  interference in intersection of regions

6. non-interfering base-stations base-stations use different frequencies  no interference!

7. 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.

8. 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

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

10. What is the minimum number of colors we need ? every 2 “adjacent” disks must have different colors

11. Answer: 3 colors What is the minimum number of colors we need ?

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

13. Answer: 4 colors

14. 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.

15. arrangements of unit disks Topological arrangement: sub-division of plane into cells. a cell

16. 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)

17. set-system representation 1 2 3 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 diskscells 12345 12345 6 7 disk-cell edge if cell in disk

18. primal/dual set-systems primal: sets elements dual: elements sets

19. arrangements of unit disks arrangement corresponding to dual set system: skip

20. 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

21. 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.

22. 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).

23. 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”)

24. 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!

25. 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.

26. 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.

27. 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.

28. 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.

29. 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

31. indexed arrangements assign indexes to disks (not arbitrary!). represent set system by diagram (i.e. is cell covered by disk?) cells disks 245 7 89 N(cell) is an interval N(cell) is not an interval

32. 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]} )

33. 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.

34. 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.

35. coloring chain with O(log n) colors Back to thms

36. 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

37. 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)

38. 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

39. 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

40. 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

41. 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.

42. 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.

43. 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.

44. 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

45. 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!

48. 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

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

50. 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…