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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, Zvi Lotker and David Peleg WRAWN Reykjavik, IcelandJuly 2011

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

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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

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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)

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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

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

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

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

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

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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

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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

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

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

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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

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

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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:

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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?

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

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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?

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

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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]

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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]

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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

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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?

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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)

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

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

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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:

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SINR Map & Voronoi Diagram

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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.

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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

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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

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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

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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

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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

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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)?

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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

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Weaker Forms of Convexity

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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:

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

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

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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)

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

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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

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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

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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

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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

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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

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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 })

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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 ?

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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

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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)

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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

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Connectivity & Convexity in Higher Dimensions

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

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

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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

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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

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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

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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

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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

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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

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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

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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

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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:

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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

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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

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