1. Coverage and Connectivity Issues in Sensor Networks
Ten-Hwang Lai
Ohio State University

2. A Sensor Node Memory (Application) Transmission range Processor
Sensing
range
Sensor
Actuator
Network
Interface

3. Sensor Deployment How to deploy sensors over a field?
Deterministic, planned deployment
Random deployment
Desired properties of deployments?
Depends on applications
Connectivity
Coverage

4. Coverage, Connectivity
Every point is covered by 1 or K sensors
1-covered, K-covered
The sensor network is connected
K-connected
Others 1 8 R 2 7 6 3 4 5

5. Coverage & Connectivity: not independent, not identical
If region is continuous & Rt > 2Rs
Region is covered sensors are connected Rt Rs

6. Problem Tree coverage connectivity probabilistic algorithmic
per-node homo
homo heterogeneous
barrier
coverage
k-connected
blanket coverage

7. Connectivity Issues

8. Power Control for Connectivity
Adjust transmission range (power)
Resulting network is connected
Power consumption is minimum
Transmission range
Homogeneous
Node-based

9. Power control for k-connectivity
For fault tolerance, k-connectivity is desirable.
k-connected graph:
K paths between every two nodes
with k-1 nodes removed, graph is still connected
1-connected connected connected

10. Two Approaches Probabilistic Algorithmic
How many neighbors are needed?
Algorithmic
Gmax connected
Construct a connected subgraph
with desired properties

11. Growing the Tree
coverage
connectivity
probabilistic algorithmic

12. Probabilistic Approach
How many neighbors are necessary and/or sufficient to ensure connectivity?

13. How many neighbors are needed?
Regular deployment of nodes – easy
Random deployment (Poisson distribution)
N: number of nodes in a unit square
Each node connects to its k nearest neighbors.
For what values of k, is network almost sure connected?
P( network connected ) → 1, as N → ∞

14. An Alternative View N ∞ P( network connected ) → 1, as N →
A square of area N.
Poisson distribution of a fixed density λ.
Each node connects to its k nearest neighbors.
For what values of k, is the network almost sure connected?
P( network connected ) → 1, as N → N ∞

15. A Related Old Problem Packet radio networks (1970s/80s)
Larger transmission radius
Good: more progress toward destination
Bad: more interference
Optimum transmission radius?

16. Magic Number Kleinrock and Silvester (1978) 6 is the magic number.
Model: slotted Aloha & homogeneous radius R & Poisson distribution & maximize one hop progress toward destination.
Set R so that every station has 6 neighbors on average.
6 is the magic number.

17. More Magic Numbers Tobagi and Kleinrock (1984)
Eight is the magic number.
Other magic numbers for various protocols and models:
5, 6, 7, 8

18. Are Magic Numbers Magic?
Xue & Kumar (2002)
For the network to be almost sure connected, Θ(log n) neighbors are necessary and sufficient.
Heterogeneous radius
8, 7, 6, 5
(Magic numbers)

19. Θ(log n) neighbors needed for connectivity
N: number of nodes (or area). K: number of neighbors.
Xue & Kumar (2002):
If K < log N, almost sure disconnected.
If K > log N, almost sure connected.
2004, improved to log N and log N K
0.074 log n log n

20. Penrose (1999): “On k-connectivity for a geometric random graph”
As n → infinity
Minimum transmission range required
R(n): for graph to be k-connected
R’(n): for graph to have degree k
Homogeneous radius
R(n) and R’(n) are almost sure equal
P( R(n) = R’(n) ) → 1, as n → infinity.
If every node has at least k neighbors then network is almost sure k-connected.

21. Any contradiction? Xue & Kumar (improved by others): Penrose:
If every node connects to its
Log n nearest neighbors, almost sure connected.
0.3 Log n nearest neighbors, almost sure disconnected.
Node-based radius
Penrose:
If every node has at least 1 neighbor, then almost sure 1-connected.
Homogeneous radius

22. Applying Asymptotic Results
Applying Xue & Kumar’s result
“The K-Neigh Protocol for Symmetric Topology Control in Ad Hoc Networks”
Blough et al, MobiHoc’03.
Applying Penrose’s result
“On the Minimum Node Degree and Connectivity of a Wireless Multihop Network”
Bettstetter, MobiHoc’02.

23. Applying Penrose’s result to power control (Bettstetter, MobiHoc’02)
Nodes deployed randomly.
Given: number of nodes n, node density λ, transmission range R.
P = Probability(every node has at least k neighbors) can be calculated.
Adjust R so that P ≈ 1.
With this transmission range, network is k-connected with high probability.

24. Application 1 N = 500 nodes A = 1000m x 1000m 3-connected required
With R = 100 m, G has degree 3 with probability 0.99.
Thus, G is 3-connected with high probability.
500 nodes

25. Application 2: How many sensors to deploy?
A = 1000m x 1000m
R = 50 m
3-connected required
N = ?
Choose N such that P( G has degree 3) is sufficiently high.

26. Growing the Tree coverage connectivity probabilistic algorithmic
per-node homo radius
radius
Xue&Kumar Penrose

27. Algorithmic Approach

28. Gmax: network with maximum transmission range
Gmax: assumed to be connected
Construct a connected subgraph of Gmax
With certain desired properties
Distributed & localized algorithms
Use the subgraph for routing
Adjust power to reach just the desired neighbor
What subgraphs?

29. What Subgraphs? Gmax(V): Network with max trans range
RNG(V): Relative neighborhood graph
GG(V): Gabriel graph
YG(V): Yao graph
DG(V): Delaunay graph
LMST(V): Local minimum spanning tree graph
GG(V):

30. Desired Properties of Proximity Graphs
PG ∩ Gmax is connected (if Gmax is)
PG is sparse, having Θ(n) edges
Bounded degree
Degree RNG, GG, YG ≤ n – 1 (not bounded)
Degree of LMST ≤ 6
Small stretch factor
Others
See “A Unified Energy-Efficient Topology for Unicast and Broadcast,” Mobicom 2005.

31. Growing the Tree coverage connectivity probabilistic algorithmic
per-node homo
Homogeneous max trans. range
various connected subgraphs

32. Maximum transmission range
Homogeneous
Same max range for all nodes
PG ∩ Gmax is connected (if Gmax is)
Heterogeneous
Different max ranges
PG ∩ Gmax is not necessarily connected
(even if Gmax is)
PG: existing PGs

33. Growing the Tree coverage connectivity probabilistic algorithmic
per-node homo
max range
homo heterogeneous
k-connected

34. Some references
N. Li and J. Hou, L Sha, “Design and analysis of an MST-based topology control algorithms,” INFOCOM 2003.
N. Li and J. Hou, “Topology control in heterogeneous wireless control networks,” INFOCOM 2004.
N. Li and J. Hou, “FLSS: a fault-tolerant topology control algorithm for wireless networks,” Mobicom 2004.

35. Coverage Issues

36. Simple Coverage Problem
Given an area and a sensor deployment
Question: Is the entire area covered? 1 8 R 2 7 6 3 4 5

37. Is the perimeter covered?

38. K-covered
1-covered
2-covered
3-covered

39. K-Coverage Problem Given: region, sensor deployment, integer k
Question: Is the entire region k-covered? 1 8 R 2 7 6 3 4 5

40. Is the perimeter k-covered?

41. Reference
C. Huang and Y. Tseng, “The coverage problem in a wireless sensor network,”
In WSNA, 2003.
Also MONET 2005.

42. Density (or topology) Control
Given: an area and a sensor deployment
Problem: turn on/off sensors to maximize the sensor network’s life time

43. PEAS and OGDC
PEAS: A robust energy conserving protocol for long-lived sensor networks
Fan Ye, et al (UCLA), ICNP 2002
“Maintaining Sensing Coverage and Connectivity in Large Sensor Networks”
H. Zhang and J. Hou (UIUC), MobiCom 2003

44. PEAS: basic ideas How often to wake up?
How to determine whether to work or not?
Wake-up rate?
yes
Sleep
Wake up
Go to
Work?
work no

45. How often to wake up?
Desired: the total wake-up rate around a node equals some given value

46. Inter Wake-up Time f(t) = λ exp(- λt) exponential distribution
λ = average # of wake-ups per unit time

47. Wake-up rates A B A + B: f(t) = (λ + λ’) exp(- (λ + λ’) t)

48. Adjust wake-up rates Working node knows
Desired total wake-up rate λd
Measured total wake-up rate λm
When a node wakes up, adjusts its λ by
λ := λ (λd / λm)

49. Go to work or return to sleep?
Depends on whether there is a working node nearby. Rp
Go back to sleep go to work

50. Is the resulting network covered or connected?
If Rt ≥ (1 + √5) Rp and … then
P(connected) → 1
Simulation results show good coverage

51. OGDC: Optimal Geographical Density Control
“Maintaining Sensing Coverage and Connectivity in Large sensor networks”
Honghai Zhang and Jennifer Hou
MobiCom’03

52. Basic Idea of OGDC Minimize the number of working nodes
Minimize the total amount of overlap

53. Minimum overlap
Optimal distance = √3 R

54. Minimum overlap

55. Near-optimal

56. OGDC: the Protocol Time is divided into rounds.
In each round, each node runs this protocol to decide whether to be active or not.
Select a starting node. Turn it on and broadcast a power-on message.
Select a node closest to the optimal position. Turn it on and broadcast a power-on message. Repeat this.

57. Selecting starting nodes
Each node volunteers with a probability p.
Backs off for a random amount of time.
If hears nothing during the back-off time, then sends a message carrying
Sender’s position
Desired direction

58. Select the next working node
On receiving a message from a starting node
Each node computes its deviation D from the optimal position.
Sets a back-off timer proportional to D.
When timer expires, sends a power-on message.
On receiving a power-on message from a non-starting node

60. PEAS vs. OGDC

61. Coverage Issues How many sensors are needed? density control
K-covered?
How many sensors
are needed?
PEAS OGDC

62. How many sensors to deploy?
A similar question for k-connectivity
Depends on:
Deployment method
Sensing range
Desired properties
Sensor failure rate
Others

63. Unreliable Sensor Grid: Coverage and Connectivity, INFOCOM 2003
Active
Dead
p: probability( active )
r: sensing range
Necessary and sufficient condition for area to be covered?
N nodes

64. Conditions for Asymptotic Coverage
Necessary:
Sufficient:
= expected # of
active sensors in a sensing disk.
N nodes

65. On k­Coverage in a Mostly Sleeping Sensor Network, Mobicom’04
Almost sure k-covered:
Almost sure not k-covered:
Covered or not covered depending on how it approaches 1

66. Critical Value
M: average # of active sensors in each sensing disk.
M > log(np): almost sure covered.
M < log(np): almost sure not covered.
log(np)
not covered
covered
N nodes
Infocom’03: log n log n

67. Poisson or Uniform Distribution
Similar critical conditions hold.

68. Application of Critical Condition
P: probability of being active
R: sensing range
N: number of sensors?

69. Growing the Tree coverage connectivity probabilistic algorithmic
per-node homo
homo heterogeneous
barrier
coverage
k-connected
blanket coverage

70. Blanket vs. Barrier Coverage
Blanket coverage
Every point in the area is covered (or k-covered)
Barrier coverage
Every crossing path is k-covered

71. Recent Results
Algorithms to determine if a region is k-barrier covered.
How many sensors are needed to provide k-barrier coverage with high probability?

72. Is a belt region k-barrier covered?
Construct a graph G(V, E)
V: sensor nodes, plus two dummy nodes L, R
E: edge (u,v) if their sensing disks overlap
Region is k-barrier covered iff L and R are k-connected in G. R L

73. Donut-shaped region
K-barrier covered iff G has k essential cycles.

74. Critical condition for k-barrier coverage
Almost sure k-covered:
Almost sure not k-covered: s
1/s

75. Thank You Growing and Growing coverage connectivity
probabilistic algorithmic
per-node homo
homo heterogeneous
barrier
coverage
Thank You
k-connected
blanket coverage