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![Page 1: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/1.jpg)
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
![Page 2: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/2.jpg)
Goal
Study Topological Properties of Reception Maps
and their applications to Algorithmic Design
![Page 3: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/3.jpg)
Stations with radio
device
Synchronous operation
Wireless channel
No centralized control
S1
S2
S3
S4
S5
Wireless Radio Networks
d
![Page 4: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/4.jpg)
Physical Models
Attempting to model attenuation and interference explicitly
Most commonly used:Signal to Interference plus Noise Ratio
(SINR)
![Page 5: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/5.jpg)
),(
,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
![Page 6: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/6.jpg)
}\{
),(},{isSj
ji psEpsSI
RSS of station sjReceiver pointInterfering
stations in Rd
Physical Model: interference In
terf
ere
nc
e
![Page 7: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/7.jpg)
NpsSI
psEpsSINR
i
ii
},{
,),(
RSS of station Sj
NoiseInterference
Physical Models: Signal to interference & noise ratio
Receiver point
station si
![Page 8: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/8.jpg)
Station si is heard at point p ∈d - S iff
),( psS I N R i
Fundamental Rule of the SINR model
Reception Threshold (>1)
![Page 9: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/9.jpg)
S1
S2
S4
S5 S3
The SINR Map
A map characterizing the
reception zones of the network
stations
![Page 10: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/10.jpg)
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
![Page 11: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/11.jpg)
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
![Page 12: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/12.jpg)
Wireless Computational Geometry
VoronoiDiagram
SINRDiagram
What is it Good For?
![Page 13: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/13.jpg)
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
![Page 14: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/14.jpg)
15
Algorithmic Question
s2
s4
s3
s1
Can we answer point location queries
FASTER?
![Page 15: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/15.jpg)
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:
![Page 16: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/16.jpg)
Picture formed
by
sampling in pre-
processing s4
s3
s2
s1
Problem:
What if reception regions are skinny /wiggly?
![Page 17: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/17.jpg)
s4
s3
s2
s1
p
Problem:
Querying Point P:
Might lead to a
false answer
![Page 18: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/18.jpg)
Requires studying
Topology /
geometry
of reception zones s4
s3
s2
s1
Problem:
Can such odd shapes occur in practice?
![Page 19: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/19.jpg)
All stations transmit with power 1
(Ψi=1 for every i)
H1 H2
H3
H4
Uniform Power Networks
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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]
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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]
![Page 22: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/22.jpg)
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
![Page 23: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/23.jpg)
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?
![Page 24: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/24.jpg)
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)
![Page 25: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/25.jpg)
ψ1
ψ2
With non-uniform power: no problem
11 1 1
With uniform power: impossible
Why Using Non-Uniform Powers?
r1 r2 s2 s1
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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:
![Page 28: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/28.jpg)
SINR Map & Voronoi Diagram
![Page 29: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/29.jpg)
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.
![Page 30: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/30.jpg)
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
![Page 31: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/31.jpg)
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
![Page 32: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/32.jpg)
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
![Page 33: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/33.jpg)
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
![Page 34: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/34.jpg)
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)?
![Page 35: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/35.jpg)
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
![Page 36: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/36.jpg)
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:
![Page 37: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/37.jpg)
occupiedhole
freehole
“vanilla” non-convexity
Classification of Non-Convex Cells
![Page 38: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/38.jpg)
The “No-Free-Hole” Conjecture
A free hole cannot occur in an SINR map
Classification of Non-Convex Cells
![Page 39: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/39.jpg)
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)
![Page 40: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/40.jpg)
43
The Big Question
Do SINR zones satisfy the “no-free-
hole” property ?
![Page 41: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/41.jpg)
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
![Page 42: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/42.jpg)
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
![Page 43: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/43.jpg)
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
![Page 44: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/44.jpg)
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
![Page 45: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/45.jpg)
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
![Page 46: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/46.jpg)
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?
![Page 47: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/47.jpg)
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
![Page 48: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/48.jpg)
Theorem:There exist 2-Dim n-station wireless networks where s1 has Ω(n) cells
Lower Bound on Number of Cells (in 2-Dim)
![Page 49: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/49.jpg)
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
![Page 50: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/50.jpg)
Connectivity & Convexity in Higher Dimensions
![Page 51: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/51.jpg)
s1 s2
H1H1 H2
ψ1 > ψ2
In 1-Dim: Disconnected map
Example: Linear Network
![Page 52: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/52.jpg)
ψ1 > ψ2
In 2-Dim:Connected
Example: Linear Network
![Page 53: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/53.jpg)
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
![Page 54: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/54.jpg)
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
![Page 55: The Topology of Wireless Communication Merav Parter Department of Computer Science and Applied Mathematics Weizmann Institute Joint work with Erez Kantor,](https://reader038.fdocuments.in/reader038/viewer/2022103111/5518c1b9550346b31f8b55da/html5/thumbnails/55.jpg)
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
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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 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
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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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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
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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, 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
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Thank You for Listening!