1. 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

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

3. Wireless Radio Networks
Stations with radio device
Synchronous operation
Wireless channel
No centralized control
Insert in click

4. 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)

5. 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)

6. Physical Model: interference
Interfering stations in Rd
RSS of station sj
Receiver point
Interference

7. Physical Models: Signal to interference & noise ratio
station si
Receiver point
RSS of station Sj
Interference
Noise

8. 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

9. 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

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

11. 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

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

13. 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

14. 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)

15. Algorithmic Question Can we answer point location queries FASTER? s215
Can we answer point location queries FASTER?

16. 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.

17. 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

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

19. 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

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

21. 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]

22. 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]

23. 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

24. 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]

25. 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?

26. 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

27. 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)

28. 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

29. SINR Map & Voronoi Diagram

30. 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]

31. 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

32. 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

33. 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

34. 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→∞

35. 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→∞

36. 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).

37. 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.

38. Weaker Forms of Convexity

39. 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

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

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

42. 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

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

44. “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

45. 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

46. 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

47. 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

48. 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

49. 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

50. “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

51. 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)

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

53. 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

54. Connectivity & Convexity in Higher Dimensions

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

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

57. 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.

58. 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

59. [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

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

61. 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

62. 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.

63. 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}

64. 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

65. 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

66. 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

67. 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

68. Thank You for Listening!