The Topology ofWireless Communication

Merav ParterDepartment 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

and their applications to Algorithmic Design

Stations with radio

device

Synchronous operation

Wireless channel

No centralized control

S1

S2

S3

S4

S5

Wireless Radio Networks

d

Physical Models

Attempting to model attenuation and interference explicitly

Most commonly used:Signal to Interference plus Noise Ratio

(SINR)

),(

,psd

psEi

i

i

transmission power of station si

Path loss parameter (usually 2≤α≤6)

Distance between si and point p

Receiver point p∈ Rd

Station

si ∈ Rd

Physical Model: Received Signal Strength (RSS)

Received Signal Strength

Receiver point p∈ Rd

}\{

),(},{isSj

ji psEpsSI

RSS of station sjReceiver pointInterfering

stations in Rd

Physical Model: interference In

terf

ere

nc

e

NpsSI

psEpsSINR

i

ii

},{

,),(

RSS of station Sj

NoiseInterference

Physical Models: Signal to interference & noise ratio

Receiver point

station si

Station si is heard at point p ∈d - S iff

),( psS I N R i

Fundamental Rule of the SINR model

Reception Threshold (>1)

S1

S2

S4

S5 S3

The SINR Map

A map characterizing the

reception zones of the network

stations

psS I N RSRpsH id

ii ,|

Reception Point Sets: Zones and Cells

Reception Zone of Station

si

Cell :=

Maximal connected

component within a

zone.Zone H1

Cell of H3

1st Cell of H1

ise v e r y fo r ,),(| psS I N RSRpH id

NullCell

The Null Zone

Null Zone := The zone where no station is

heard

Wireless Computational Geometry

VoronoiDiagram

SINRDiagram

What is it Good For?

A: Compute SINR(si,p) for every si 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? s2

s4

s3

s1

Suppose all stations in S = {s1, s2 ,…,sn} transmit simultaneously.

p ?

Motivation: Point Location Problems

15

Algorithmic Question

s2

s4

s3

s1

Can we answer point location queries

FASTER?

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

s4

s3

s1

s2

p

Idea:

Picture formed

by

sampling in pre-

processing s4

s3

s2

s1

Problem:

What if reception regions are skinny /wiggly?

s4

s3

s2

s1

p

Problem:

Querying Point P:

Might lead to a

false answer

Requires studying

Topology /

geometry

of reception zones s4

s3

s2

s1

Problem:

Can such odd shapes occur in practice?

All stations transmit with power 1

(Ψi=1 for every i)

H1 H2

H3

H4

Uniform Power Networks

Theorem (Convexity)

The reception zone Hi is convex for every 1 ≤ i ≤ n

notconvex

Uniform Power: What’s Known?

[Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

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

notfat

Uniform Power: What’s Known?

[Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

Set H is fat if there is a point p such thatthe 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

Application (Point Location)

A data structure constructed in polynomial time and

supporting approximate point location queries of

logarithmic cost

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

[Avin, Emek, Kantor, Lotker, Peleg, Roditty; PODC’09]

Uniform Power: What’s Known?

What are the fundamental properties of

SINR maps for such networks?

Non-Uniform SINR Diagrams

Stations may transmit with

varying transmitting powers

(different Ψi values)

ψ1

ψ2

With non-uniform power: no problem

11 1 1

With uniform power: impossible

Why Using Non-Uniform Powers?

r1 r2 s2 s1

Non-convex

Disconnected (5 stations)

Possibly many singular points

(4 stations)

Non-uniform Diagrams are Complicated...

How Does it Look Like?

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:

SINR Map & Voronoi Diagram

Lemma [Uniform Map and Voronoi Diagram]

Hi ⊆ VoriFor every uniform reception zone

Hi

H1

H2

H4

H3

H5

[Avin, Emek, Kantor, Lotker, Peleg and Roddity, PODC 09]

H1

H2

H4

H3

H5

Vor1

Vor4

Vor5

Vor3

Uniform SINR Map & Voronoi Diagram

Vori := Vornoi Cell of station si∈S.

WVor(V):Weighted system V= S,W⟨ ⟩ where:

S = {s1, s2 ,…, sn} = set of points in d

wi R+ = weight of point si

Planar subdivision with circular edges

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

ijany for ,),(),(

)(j

j

i

idi w

psdist

w

psdistRpVWVor

Weighted Voronoi Diagram

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(n2)

Properties

Lemma: Hi(A) ⊆ WVori(VA) for every station si, β≥1

Given a wireless network A:VA= S,W⟨ ⟩ = weighted Voronoi diagram with weights wi = ψi

1/α

Note: Since weights decay with α, Hi(A) ⊆ Vori(VA) when α→∞

Non-Uniform SINR map & Weighted Voronoi Diagram

TransmissionEnergy

Fact: There exists a wireless network A such

thata given cell of WVor(VA) contains more

thanone cell of H(A).

Can Number of Cells in H(A) be Bounded by Number of Cells in WVor(VA)?

WVor1

S5

S4

S3

s1

s1

s3

s4

s5WVor1

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.

s1

s3

s4

s5WVor1

1. Consider a network where H1 is not connected.

Proof Sketch

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:

occupiedhole

freehole

“vanilla” non-convexity

Classification of Non-Convex Cells

The “No-Free-Hole” Conjecture

A free hole cannot occur in an SINR map

Classification of Non-Convex Cells

S1

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:

Cs2s3

s4

Cs6

s5

if Φ(C) ⊆ Hi

then C ⊆Hi

The “No-Free-Hole” Property

Φ(C)

43

The Big Question

Do SINR zones satisfy the “no-free-

hole” property ?

Theorem (Number of Cells in 1-D) The number of cells in A is bounded by 2n-1 (tight)

s2s3s1 s3s4s2

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

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

s2s3s1 s3s4s2

Number of Cells in 1-Dim Maps

46

st (WEAKEST)

xt

s1

x1

s2

x2

si

xi

Assume otherwise…

Due to NFH there exists some station si in between

Claim: The Zone of the Weakest Station is Connected

47

st (WEAKEST)

xta b

s1

x1

s2

x2

si

xi

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

Closer to strongerStation, si

Claim: The Zone of the Weakest Station is Connected

48

s1

x1

si

xia b

s4

x4

si

xixt

st

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

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?

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)

Bounding #Cells in Higher Dimensions

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

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

R>2n

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

Ψ1=O(n2)

Lower Bound on Number of Cells (in 2)

Square:4 interfering

weak stations

Connectivity & Convexity in Higher Dimensions

s1 s2

H1H1 H2

ψ1 > ψ2

In 1-Dim: Disconnected map

Example: Linear Network

ψ1 > ψ2

In 2-Dim:Connected

Example: Linear Network

The zone of station si in d+1 is

Hi(d+1) = {si} ⋃ {p ∊ d+1 -S | SINR(si,p)≥β}

Consider a network in d and

draw the reception map in d+1 .

Theorem:Hi(d+1) is connected for every si ∈ S.

Connectivity of Reception Zones in d+1

Then there exists a continuous reception curve γ ⊆ Hi(d+1). In particular: γ is the hyperbolic geodesic.

Stations are embedded in the hyperplane xd+1=0

Consider two reception points

p1,p2 ∈ Hi(d+1) in upper halfplane xd+1≥ 0.

s1 s2s3

p2

p1

Ɣ

Setting

Lines (geodesic) of the model:

(a) Semi-circle perpendicular to x-axis

(b) Vertical line (arc of circle with infinite radius)

Restrictedto Y>0 Infinity

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

Hyperbolic line

Type a

Hyperbolic line

Type b

The Hyperbolic Geodesic

Given a suitably defined hyperbolic metric

Fact: A hyperbolic geodesic (“line”) minimizes the distance between any two of its points

Hyperbolic convexbut not convex

Convexbut 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

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

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)⊆Hi(d+1) C ⊆ Hi(d+1). (b) Φ(C)∩Hi(d+1)= ∅ C ∩ Hi(d+1)=∅.

s1s4

s3

s2

Application to Testing Reception Condition

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:

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

“Counting” Questions: “Visual” Question:

Algorithmic Questions:

Number of cells:1: Linear, tightd: O(nd+1)d+1: n

Weaker convexity:1: No Free Holed: Maximum principleof interference function.d+1: Hyperbolic Convexity.

Point Locationd: New variant.

d+1: Efficient

Summary

Thank You for Listening!