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The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,

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Presentation on theme: "The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,"— Presentation transcript:

1 The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor, Zvi Lotker and David Peleg WRAWN Reykjavik, IcelandJuly 2011

2 Goal Study Topological Properties of Reception Maps and their applications to Algorithmic Design

3 Stations with radio device Synchronous operation Wireless channel No centralized control Stations with radio device Synchronous operation Wireless channel No centralized control S1S1 S2S2 S3S3 S4S4 S5S5 Wireless Radio Networks d

4 Physical Models Attempting to model attenuation and interference explicitly Most commonly used: Signal to Interference plus Noise Ratio (SINR) Most commonly used: Signal to Interference plus Noise Ratio (SINR)

5 transmission power of station s i Path loss parameter (usually 2α6) Distance between s i and point p Receiver point p R d Station s i R d Station s i R d Physical Model: Received Signal Strength (RSS) Received Signal Strength Receiver point p R d

6 RSS of station s j Receiver point Interfering stations in R d Physical Model: interference Interference

7 RSS of station S j Noise Interference Physical Models: Signal to interference & noise ratio Receiver point station s i

8 Station s i is heard at point p d - S iff Fundamental Rule of the SINR model Reception Threshold (>1)

9 S1S1 S1S1 S2S2 S2S2 S4S4 S4S4 S5S5 S5S5 S3S3 S3S3 The SINR Map A map characterizing the reception zones of the network stations

10 Reception Point Sets: Zones and Cells Reception Zone of Station s i Cell := Maximal connected component within a zone. Cell := Maximal connected component within a zone. Zone H 1 Cell of H 3

11 Reception Point Sets: Zones and Cells Reception Zone of Station s i Cell := Maximal connected component within a zone. Cell := Maximal connected component within a zone. Zone H 1 Cell of H 3 1 st Cell of H 1

12 Null Cell The Null Zone Null Zone := The zone where no station is heard

13 Wireless Computational Geometry Voronoi Diagram SINR Diagram What is it Good For?

14 A: Compute SINR(s i,p) for every s i in time O(n) Consider point p in the plane. By definition, p hears at most one station of S. Q: Does p hear any of the stations? s2s2 s4s4 s3s3 s1s1 Suppose all stations in S = {s 1, s 2,…,s n } transmit simultaneously. p ? Motivation: Point Location Problems

15 15 Algorithmic Question s2s2 s4s4 s3s3 s1s1 Can we answer point location queries FASTER?

16 Given a query point p: Relay answer by nearby grid vertices. Given a query point p: Relay answer by nearby grid vertices. In pre-processing stage: (1) Form a grid (2) Calculate answers on its vertices In pre-processing stage: (1) Form a grid (2) Calculate answers on its vertices s4s4 s3s3 s1s1 s2s2 p Idea:

17 Picture formed by sampling in pre- processing Picture formed by sampling in pre- processing s4s4 s3s3 s2s2 s1s1 Problem: What if reception regions are skinny /wiggly?

18 s4s4 s3s3 s2s2 s1s1 p Problem: Querying Point P: Might lead to a false answer Querying Point P: Might lead to a false answer

19 Requires studying Topology / geometry of reception zones Requires studying Topology / geometry of reception zones s4s4 s3s3 s2s2 s1s1 Problem: Can such odd shapes occur in practice?

20 All stations transmit with power 1 (Ψ i =1 for every i) H1H1 H2H2 H3H3 H4H4 Uniform Power Networks

21 Theorem (Convexity) The reception zone H i is convex for every 1 i n not convex Uniform Power: Whats Known? [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC09]

22 Theorem (Convexity) The reception zone H i is convex for every 1 i n Theorem (Fatness) The reception zone H i is fat for every 1 i n not fat Uniform Power: Whats Known? [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC09]

23 Set H is fat if there is a point p such that the ratio Δ Δ/δ = O(1) H δ p = Δ radius(smallest circumscribed ball of H centered at p) δ radius(largest inscribed ball of H centered at p) is bounded by a constant Fatness

24 Application (Point Location) A data structure constructed in polynomial time and supporting approximate point location queries of logarithmic cost Theorem (Convexity) The reception zone H i is convex for every 1 i n Theorem (Fatness) The reception zone H i is fat for every 1 i n [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC09] Uniform Power: Whats Known?

25 What are the fundamental properties of SINR maps for such networks? Non-Uniform SINR Diagrams Stations may transmit with varying transmitting powers (different Ψ i values) Stations may transmit with varying transmitting powers (different Ψ i values)

26 ψ1ψ1 ψ2ψ2 With non-uniform power: no problem 1111 With uniform power: impossible Why Using Non-Uniform Powers? r1r1 r2r2 s2s2 s1s1

27 Non-convex Disconnected (5 stations) Possibly many singular points (4 stations) Non-uniform Diagrams are Complicated... How Does it Look Like?

28 Maximal number of connected cells in n-station SINR map Counting Questions: Niceness properties: Weaker Convexity? Visual Questions: Point Location Algorithmic Tools: Types Of Questions:

29 SINR Map & Voronoi Diagram

30 Lemma [Uniform Map and Voronoi Diagram] H i Vor i For every uniform reception zone H i H1H1 H2H2 H4H4 H3H3 H5H5 [Avin, Emek, Kantor, Lotker, Peleg and Roddity, PODC 09] H1H1 H2H2 H4H4 H3H3 H5H5 Vor 1 Vor 4 Vor 5 Vor 3 Uniform SINR Map & Voronoi Diagram Vor i := Vornoi Cell of station s i S.

31 WVor(V): Weighted system V= S,W where: S = {s 1, s 2,…, s n } = set of points in d w i R + = weight of point s i Planar subdivision with circular edges Weighted Voronoi Diagram

32 V= S,W S = {s 1, s 2,…, s n } w i = weights The weighted Voronoi diagram WVor(V) partitions the plane into n zones, where Weighted Voronoi Diagram

33 Facts: 1.The Weighted Voronoi Diagram WVor(V) is not necessarily connected 2. [Aurenhammer, Edelsbrunner; 84] The number of cells in WVor(V) is at most O(n 2 ) Properties

34 Lemma [Non-Uniform Map and Weighted Voronoi Diagram] H i WVor i For every non-uniform reception zone H i Non-Uniform SINR Map & Weighted Voronoi Diagram Station s i transmitting with Ψ i WVor i :=Weighted Voronoi zone of s i S with weight w i = Ψ i 1/α Note: Since weights decay with α, H i (A) Vor i (V A ) when α

35 Lemma: H i (A) WVor i (V A ) for every station s i, β1 Given a wireless network A: V A = S,W = weighted Voronoi diagram with weights w i = ψ i 1/α Note: Since weights decay with α, H i (A) Vor i (V A ) when α Non-Uniform SINR map & Weighted Voronoi Diagram Transmission Energy

36 Fact: There exists a wireless network A such that a given cell of WVor(V A ) contains more than one cell of H(A). Can Number of Cells in H(A) be Bounded by Number of Cells in WVor(VA)?

37 WVor 1 S5S5 S4S4 S3S3 s1s1 s1s1 s3s3 s4s4 s5s5 2. Replace each other station by a set of m weak stations at the same position and transmission energy=ψ i /m. H 1 remains the same but WVor 1 becomes much larger. s1s1 s3s3 s4s4 s5s5 WVor 1 1. Consider a network where H 1 is not connected. Proof Sketch

38 Weaker Forms of Convexity

39 Maximal number of connected cells in n-station SINR map Counting Questions: Niceness properties: Weaker Convexity? Visual Questions: Point Location Algorithmic Tools: Types Of Questions:

40 occupied hole free hole vanilla non-convexity Classification of Non-Convex Cells

41 The No-Free-Hole Conjecture A free hole cannot occur in an SINR map Classification of Non-Convex Cells

42 S1S1 A collection of convex shapes C in d enjoys the no- free-hole property if for every shape C C that is free of interfering stations: C s2s2 s3s3 s4s4 C s6s6 s5s5 if Φ (C) H i then C H i The No-Free-Hole Property Φ(C)

43 43 The Big Question Do SINR zones satisfy the no-free-hole property ?

44 Theorem (Number of Cells in 1-D) The number of cells in A is bounded by 2n-1 (tight) s2s2 s3s3 s1s1 s3s3 s4s4 s2s2 Consider a 1-Dim n-station wireless network A Theorem (No-Free-Hole Property in 1-D) The reception zones of A enjoy the no-free-hole property No-Free-Hole in 1-Dim Networks

45 Order S = {s 1,…, s n } in non-increasing order of energy Add stations one by one Should show that: 1. The zone of the weakest station is connected 2. Each step t adds at most 2 cells Order S = {s 1,…, s n } in non-increasing order of energy Add stations one by one Should show that: 1. The zone of the weakest station is connected 2. Each step t adds at most 2 cells s2s2 s3s3 s1s1 s3s3 s4s4 s2s2 Number of Cells in 1-Dim Maps

46 46 s t ( WEAKEST) xtxt s1s1 x1x1 s2s2 x2x2 sisi xixi Assume otherwise… Due to NFH there exists some station s i in between Claim: The Zone of the Weakest Station is Connected

47 47 s t (WEAKEST) xtxt ab s1s1 x1x1 s2s2 x2x2 sisi xixi Contradiction to the fact it is a reception cell of s t. Closer to stronger Station, s i Claim: The Zone of the Weakest Station is Connected

48 48 s1s1 x1x1 sisi xixi ab s4s4 x4x4 sisi xixi xtxt stst Cannot be divided Can be divided into at most two cells. Overall, due to stage t at most two cells are added Claim: Due to step t, at most 2 cells are added

49 Proof Outline: Show that the zone of the weakest station is connected. Show that each step t adds at most 2 cells each step t adds at most 2 cells Number of Cells in 1-Dim Maps Order S = {s 1,…, s n } in non-increasing order of energy (i>j if Ψ i Ψ j ). Add stations one by one: station s i is added at step i. Denote: H(A t ) the SINR diagram at step t (on S t = {s 1,…, s t })

50 Conjecture: For a d-dimensional n-station network A, the reception zones of H(A) enjoy the no-free-hole property in d No-Free-Hole Property in d ?

51 Gap: The number of cells in an SINR map for d-Dim n-station wireless network is at most O(n d+1 ) and at least Ω(n) Bounding #Cells in Higher Dimensions

52 Theorem: There exist 2-Dim n-station wireless networks where s 1 has Ω(n) cells Lower Bound on Number of Cells (in 2-Dim)

53 R>2n Idea: Strong Station s 1 located at center of radius R circle 4n weak stations organized in n O(1) x O(1) squares The 4 weak stations block s 1 reception on square boundary; s 1 is still heard in square center Ψ 1 =O(n 2 ) Lower Bound on Number of Cells (in 2 ) Square: 4 interfering weak stations Square: 4 interfering weak stations

54 Connectivity & Convexity in Higher Dimensions

55 s1s1 s2s2 H1H1 H1H1 H2H2 ψ 1 > ψ 2 In 1-Dim: Disconnected map Example: Linear Network

56 ψ 1 > ψ 2 In 2-Dim: Connected Example: Linear Network

57 The zone of station s i in d+1 is H i (d+1) = {s i } {p d+1 -S | SINR(s i,p)β} Consider a network in d and draw the reception map in d+1. Theorem: H i (d+1) is connected for every s i S. Connectivity of Reception Zones in d+1

58 Then there exists a continuous reception curve γ H i (d+1). In particular: γ is the hyperbolic geodesic. Stations are embedded in the hyperplane x d+1 =0 Consider two reception points p 1,p 2 H i (d+1) in upper halfplane x d+1 0. s1s1 s2s2 s3s3 p2p2 p1p1 Ɣ Setting

59 Lines (geodesic) of the model: (a) Semi-circle perpendicular to x-axis (b) Vertical line (arc of circle with infinite radius) Restricted to Y>0 Infinity The Hyperbolic Plane [The Upper Half Plane Model (Henri Poincaré,1882)] Hyperbolic line Type a Hyperbolic line Type a Hyperbolic line Type b Hyperbolic line Type b

60 The Hyperbolic Geodesic Given a suitably defined hyperbolic metric Fact: A hyperbolic geodesic (line) minimizes the distance between any two of its points Given a suitably defined hyperbolic metric Fact: A hyperbolic geodesic (line) minimizes the distance between any two of its points

61 Hyperbolic convex but not convex Convex but not hyperbolic convex Hyperbolic Convex Set A set S in the upper half plane of d+1 is hyperbolic convex if the hyperbolic line segment joining any pair of points lies entirely in S A set S in the upper half plane of d+1 is hyperbolic convex if the hyperbolic line segment joining any pair of points lies entirely in S

62 Theorem: The d+1 Zones are hyperbolic convex, hence connected. Cor: The zones in d+1 enjoy the no-free-hole property in d+1. Hyperbolic Convexity of d+1 Zones

63 Consider reception points p 1,p 2 on halfplane x d+1 0 Case HC1 [Vertical Segment] x i p1 = x i p2 For all i {1,…d} Case HC2 [Arc] x i p1 x i p2 for some i {1,…d} Hyperbolic Convexity of d+1 Zones: Proof Sketch

64 64 Closed shape C with boundary Φ(C) In the non-negative halfplane d+1 Free from interfering stations. Corollary [Hyperbolic Application ] (a)Φ(C) H i (d+1) C H i (d+1). (b) Φ(C) H i (d+1)= C H i (d+1)=. Application to Testing Reception Condition

65 Maximal number of connected cells in n-station SINR map Counting Questions: Niceness properties: Weaker Convexity? Visual Questions: Point Location Algorithmic Tools: Types Of Questions:

66 Problems No Voronoi diagram No convexity No fatness Solution Use Weighted Voronoi diagram Employ more delicate tagging & querying methods Point Location in Non-Uniform Case

67 Counting Questions:Visual Question: Algorithmic Questions: Number of cells: 1 : Linear, tight d : O(n d+1 ) d+1 : n Weaker convexity: 1 : No Free Hole d : Maximum principle of interference function. d+1 : Hyperbolic Convexity. Point Location d : New variant. d+1 : Efficient Summary

68 Thank You for Listening!


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