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The Topology of Wireless Communication

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, Iceland July 2011. Goal. Study Topological Properties of Reception Maps.

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The Topology of Wireless Communication

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

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

  3. Wireless Radio Networks d S1 • Stations with radio device • Synchronous operation • Wireless channel • No centralized control S2 S3 S4 S5

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

  5. Physical Model: Received Signal Strength (RSS) • Receiver point p∈ Rd • Receiver point p∈ Rd • transmission power of stationsi • Station • si ∈ Rd Received Signal Strength • Distance between siand point p Path loss parameter (usually 2≤α≤6)

  6. Physical Model: interference • Interfering stations in Rd • RSS of station sj • Receiver point Interference

  7. Physical Models: Signal to interference & noise ratio • stationsi • Receiver point • RSS of station Sj • Interference • Noise

  8. Fundamental Rule of the SINR model Station si is heard at point p ∈d- S iff • Reception Threshold (>1)

  9. The SINR Map A map characterizing the reception zones of the network stations S1 S2 S5 S3 S4

  10. Reception Point Sets: Zones and Cells Reception Zone of Station si Cell := Maximal connected component within a zone. 1st Cell of H1 Cell of H3 Zone H1

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

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

  13. Motivation: Point Location Problems s2 s1 s3 s4 Suppose all stations in S = {s1, s2,…,sn} transmit simultaneously. Consider point p in the plane. By definition, p hears at most one station of S. p ? Q:Does phear any of the stations? A: Compute SINR(si,p) for every si in time O(n)

  14. Algorithmic Question s2 s1 s3 s4 15 Can we answer point location queries FASTER?

  15. Idea: s2 s1 s3 s4 In pre-processing stage: (1) Form a grid (2) Calculate answers on its vertices p Given a query point p: Relay answer by nearby grid vertices.

  16. Problem: What if reception regions are skinny /wiggly? s2 Picture formed by sampling in pre-processing s1 s3 s4

  17. Problem: Querying Point P: Might lead to a false answer p s2 s1 s3 s4

  18. Problem: Can such odd shapes occur in practice? s2 Requires studying Topology / geometry of reception zones s1 s3 s4

  19. All stations transmit with power 1 (Ψi=1for every i) Uniform Power Networks H3 H4 H1 H2

  20. Uniform Power: What’s Known? Theorem (Convexity) The reception zone Hi is convexfor every 1 ≤ i ≤ n convex not [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

  21. Uniform Power: What’s Known? Theorem (Convexity) The reception zone Hi is convexfor every 1 ≤ i ≤ n Theorem (Fatness) The reception zone Hi is fatfor every 1 ≤ i ≤ n not fat [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

  22. Set H is fat if there is a point p such that the ratio Fatness δ Δ 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 H p Δ/δ= O(1)

  23. Uniform Power: What’s Known? Theorem (Convexity) The reception zone Hi is convexfor every 1 ≤ i ≤ n Theorem (Fatness) The reception zone Hi is fatfor every 1 ≤ i ≤ n Application (Point Location) A data structure constructed in polynomial time and supporting approximate point location queries of logarithmic cost [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

  24. Non-Uniform SINR Diagrams Stations may transmit with varying transmitting powers (different Ψivalues) What are the fundamental properties of SINR maps for such networks?

  25. Why Using Non-Uniform Powers? ψ2 ψ1 1 1 r2 1 s2 1 r1 s1 With uniform power: impossible With non-uniform power: no problem

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

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

  28. SINR Map & Voronoi Diagram

  29. Uniform SINR Map & Voronoi Diagram H1 H2 H5 H3 Vor1 H4 H1 Vor5 H2 H5 H3 Vor3 H4 Vor4 Vori := Vornoi Cell of station si∈S. Lemma [Uniform Map and Voronoi Diagram] Hi⊆ Vori For every uniform reception zone Hi [Avin, Emek, Kantor, Lotker, Peleg and Roddity, PODC 09]

  30. Weighted Voronoi Diagram Planar subdivision with circular edges WVor(V): Weighted system V=〈S,W〉where: • S = {s1, s2 ,…, sn} =set of points in d • wi R+= weight of point si

  31. Weighted Voronoi Diagram V=〈S,W〉 • S = {s1, s2 ,…, sn} • wi= weights The weighted Voronoi diagram WVor(V) partitions the plane into n zones, where

  32. Properties • Facts: • The Weighted Voronoi Diagram WVor(V) is not necessarily connected • [Aurenhammer, Edelsbrunner; 84] • The number of cells in WVor(V) is at most O(n2)

  33. Non-Uniform SINR map & Weighted Voronoi Diagram Given a wireless network A: VA=〈S,W〉= weighted Voronoi diagram with weights wi = ψi1/α Transmission Energy Lemma: Hi(A) ⊆ WVori(VA) for every station si, β≥1 Note: Since weights decay with α, Hi(A) ⊆ Vori(VA) when α→∞

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

  35. Proof Sketch s1 s1 s1 s3 s3 S5 S3 s5 s5 WVor1 WVor1 WVor1 S4 s4 s4 1. Consider a network where H1 is not connected. 2. Replace each other station by a set of m weak stations at the same position and transmission energy=ψi/m. H1 remains the same but WVor1 becomes much larger.

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

  37. Classification of Non-Convex Cells “vanilla” non-convexity free hole occupied hole

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

  39. 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: The “No-Free-Hole” Property s2 s4 s6 s5 s3 if Φ(C) ⊆ Hi Φ(C) C C then C ⊆Hi S1

  40. The Big Question Do SINR zones satisfy the “no-free-hole” property ? 43

  41. “No-Free-Hole” in 1-Dim Networks Consider a 1-Dim n-station wireless network A s2 s3 s1 s3 s2 s4 Theorem (No-Free-Hole Property in 1-D) The reception zones of A enjoy the “no-free-hole” property Theorem (Number of Cells in 1-D) The number of cells in A is bounded by 2n-1 (tight)

  42. Number of Cells in 1-Dim Maps s2 s3 s1 s3 s2 s4 • Order S = {s1,…, sn} 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

  43. Claim: The Zone of the Weakest Station is Connected Assume otherwise… si st (WEAKEST) s1 s2 x1 x2 xt xi • Due to NFH there exists some station si in between 46

  44. Claim: The Zone of the Weakest Station is Connected si s1 s2 st (WEAKEST) x1 x2 xt xi a b Closer to stronger Station, si Contradiction to the fact it is a reception cell of st. 47

  45. Claim: Due to step t, at most 2 cells are added st si s4 s1 si x1 x4 xt xi a b xi Cannot be divided Can be divided into at most two cells . Overall, due to stage t at most two cells are added 48

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

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

  48. Lower Bound on Number of Cells (in 2-Dim) Theorem: There exist 2-Dim n-station wireless networks where s1 has Ω(n) cells

  49. Lower Bound on Number of Cells (in 2) Idea: • Strong Station s1 located at center of radius R circle • 4n weak stations organized in n O(1) x O(1) squares • The 4 weak stations block s1 reception on square boundary; • s1 is still heard in square center R>2n Ψ1=O(n2) Square: 4 interfering weak stations

  50. Connectivity & Convexity in Higher Dimensions

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