# The Topology of Wireless Communication

## Presentation on theme: "The Topology of Wireless Communication"— Presentation transcript:

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

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

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

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)

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)

Physical Model: interference

Physical Models: Signal to interference & noise ratio

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

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

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

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

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

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

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)

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

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.

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

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

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

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

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]

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]

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

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]

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?

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

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)

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

SINR Map & Voronoi Diagram

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]

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

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

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
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→∞

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→∞

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).

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.

Weaker Forms of Convexity

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

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

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

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

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

“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

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

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

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

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

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

“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

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)

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

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

Connectivity & Convexity in Higher Dimensions

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

Example: Linear Network
In 2-Dim: Connected ψ1 > ψ2

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.

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

[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

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

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

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.

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}

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

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

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

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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