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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, Iceland July 2011 1

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**and their applications to Algorithmic Design**

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

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**Wireless Radio Networks**

Stations with radio device Synchronous operation Wireless channel No centralized control Insert in click

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**Signal to Interference plus Noise Ratio**

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

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**Physical Model: Received Signal Strength (RSS)**

Station si ∈ Rd Receiver point p∈ Rd Receiver point p∈ Rd transmission power of station si Received Signal Strength Distance between si and point p Path loss parameter (usually 2≤α≤6)

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**Physical Model: interference**

Interfering stations in Rd RSS of station sj Receiver point Interference

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**Physical Models: Signal to interference & noise ratio**

station si Receiver point RSS of station Sj Interference Noise

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**Fundamental Rule of the SINR model**

Station si is heard at point p ∈d - S iff Increase font of SINR Reception Threshold (>1) 8 8

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**A map characterizing the reception zones of the network stations**

The SINR Map A map characterizing the reception zones of the network stations S1 S2 S4 S5 S3 Replace photo, explain

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**Reception Point Sets: Zones and Cells**

Reception Zone of Station si Cell := Maximal connected component within a zone. Cell of H3 Zone H1

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**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

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

Cell

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**Wireless Computational Geometry**

What is it Good For? Wireless Computational Geometry Fundamental to understanding the behavior of wireless networks Development of networks algorithms Vision: Voronoi diagrams in studying proximity queries and related issues in computational geometry Voronoi Diagram SINR Diagram

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**Motivation: Point Location Problems**

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. s2 s4 s3 s1 p ? Q: Does p hear any of the stations? A: Compute SINR(si,p) for every si in time O(n)

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**Algorithmic Question Can we answer point location queries FASTER? s2**

15 Can we answer point location queries FASTER?

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**Idea: In pre-processing stage: (1) Form a grid**

(2) Calculate answers on its vertices s4 s3 s1 s2 p Given a query point p: Relay answer by nearby grid vertices.

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**What if reception regions are skinny /wiggly?**

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

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

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**Can such odd shapes occur in practice?**

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

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**Uniform Power Networks**

All stations transmit with power 1 (Ψi=1 for every i) H3 H4 H1 H2 20

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**Uniform Power: What’s Known?**

Theorem (Convexity) The reception zone Hi is convex for every 1 ≤ i ≤ n not convex [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

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**Uniform Power: What’s Known?**

Theorem (Convexity) The reception zone Hi is convex for every 1 ≤ i ≤ n Theorem (Fatness) The reception zone Hi is fat for every 1 ≤ i ≤ n not fat [Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

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**Fatness Set H is fat if there is a point p such that the ratio =**

Δ radius(smallest circumscribed ball of H centered at p) δ radius(largest inscribed ball of H centered at p) Δ is bounded by a constant H δ p Δ/δ = O(1) 23

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**Uniform Power: What’s Known?**

Theorem (Convexity) The reception zone Hi is convex for every 1 ≤ i ≤ n Theorem (Fatness) The reception zone Hi is fat for 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]

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**Non-Uniform SINR Diagrams**

Stations may transmit with varying transmitting powers (different Ψi values) What are the fundamental properties of SINR maps for such networks?

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**Why Using Non-Uniform Powers?**

ψ1 ψ2 r1 1 1 r2 1 s2 1 s1 With uniform power: impossible Click 1: “How hard is this problem?” Our basic question is: “How hard is this problem?” With non-uniform power: no problem 26 26

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**Disconnected (5 stations)**

How Does it Look Like? Non-uniform Diagrams are Complicated... a singularity is in general a point at which a given mathematical object is not defined Possibly many singular points (4 stations) Non-convex Disconnected (5 stations)

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**Types Of Questions: “Counting” Questions: “Visual” Questions:**

Maximal number of connected cells in n-station SINR map “Niceness” properties: Weaker Convexity? Algorithmic Tools: Point Location

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**SINR Map & Voronoi Diagram**

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**Hi ⊆ Vori Uniform SINR Map & Voronoi Diagram**

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

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**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

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**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

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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

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**Non-Uniform SINR Map & Weighted Voronoi Diagram**

Station si transmitting with Ψi WVori :=Weighted Voronoi zone of si∈S with weight wi=Ψi 1/α Lemma [Non-Uniform Map and Weighted Voronoi Diagram] Hi ⊆ WVori For every non-uniform reception zone Hi Note: Since weights decay with α, Hi(A) ⊆ Vori(VA) when α→∞

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**Weighted Voronoi Diagram**

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→∞

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**There exists a wireless network A such that **

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).

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**Proof Sketch 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. s1 s3 s4 s5 WVor1 s1 s3 s4 s5 WVor1 WVor1 S5 S4 S3 s1 H1 remains the same but WVor1 becomes much larger.

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**Weaker Forms of Convexity**

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**Types Of Questions: “Counting” Questions: “Visual” Questions:**

Maximal number of connected cells in n-station SINR map “Niceness” properties: Weaker Convexity? Algorithmic Tools: Point Location

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**Classification of Non-Convex Cells**

“vanilla” non-convexity occupied hole free hole

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**Classification of Non-Convex Cells**

The “No-Free-Hole” Conjecture A free hole cannot occur in an SINR map

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**The “No-Free-Hole” Property**

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: if Φ(C) ⊆ Hi Φ(C) s2 s3 C C s4 s6 then C ⊆Hi S1 s5 42

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**Do SINR zones satisfy the “no-free-hole” property ?**

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

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**“No-Free-Hole” in 1-Dim Networks**

Consider a 1-Dim n-station wireless network A s1 s2 s3 s2 s4 s3 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) 44

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**Number of Cells in 1-Dim Maps**

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 45 45

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**Due to NFH there exists some station si in between**

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

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**Contradiction to the fact it is a reception cell of st.**

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

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**Claim: Due to step t, at most 2 cells are added**

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

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**Number of Cells in 1-Dim Maps**

Order S = {s1,…, sn} in non-increasing order of energy (i>j if Ψi≤ Ψj). Add stations one by one: station si is added at step i. Denote: H(At) the SINR diagram at step t (on St = {s1,…, st}) 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

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**“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 50

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**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)

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**Lower Bound on Number of Cells (in 2-Dim)**

Theorem: There exist 2-Dim n-station wireless networks where s1 has Ω(n) cells

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**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

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**Connectivity & Convexity in Higher Dimensions**

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**Example: Linear Network**

In 1-Dim: Disconnected map H1 H2 s1 s2 ψ1 > ψ2

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**Example: Linear Network**

In 2-Dim: Connected ψ1 > ψ2

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**Connectivity of Reception Zones in d+1**

Consider a network in d and draw the reception map in d+1 . The zone of station si in d+1 is Hi(d+1) = {si} ⋃ {p ∊ d+1 -S | SINR(si,p)≥β} Theorem: Hi(d+1) is connected for every si ∈ S.

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**Setting Stations are embedded in the hyperplane xd+1=0**

Consider two reception points p1,p2 ∈ Hi(d+1) in upper halfplane xd+1≥ 0. p2 Ɣ p1 Need to show that for any point p ∈ Hi(d+1) there exists curve Ɣ connecting p and si. That is contained in Hi(d+1) . Then there exists a continuous reception curve γ ⊆ Hi(d+1). In particular: γ is the hyperbolic geodesic. s1 s2 s3 58

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**[The Upper Half Plane Model (Henri Poincaré,1882)]**

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

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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

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**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 Hyperbolic convex but not convex Convex but not hyperbolic convex 61

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**Hyperbolic Convexity of d+1 Zones**

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.

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**Hyperbolic Convexity of d+1 Zones: Proof Sketch**

Consider reception points p1,p2 on halfplane xd+1≥0 Case HC1 [Vertical Segment] xip1= xip2 For all i ∈ {1,…d} Case HC2 [Arc] xip1≠ xip2 for some i ∈ {1,…d}

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**Application to Testing Reception Condition**

Closed shape C with boundary Φ(C) In the non-negative halfplane d+1 Free from interfering stations. Corollary [Hyperbolic Application ] Φ(C)⊆Hi(d+1) C ⊆ Hi(d+1). (b) Φ(C)∩Hi(d+1)= ∅ C ∩ Hi(d+1)=∅. 64

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**Types Of Questions: “Counting” Questions: “Visual” Questions:**

Maximal number of connected cells in n-station SINR map “Niceness” properties: Weaker Convexity? Algorithmic Tools: Point Location

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**Point Location in Non-Uniform Case**

Problems No Voronoi diagram No convexity No fatness Solution Use Weighted Voronoi diagram Employ more delicate tagging & querying methods

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**Summary “Counting” Questions: “Visual” Question:**

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

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**Thank You for Listening!**

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