Coverage and Connectivity Issues in Sensor Networks

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
Network Interface
Actuator
Sensor
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
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Coverage Connectivity not independent, not
identical

• If region is continuous Rt gt 2Rs
• Region is covered sensors are connected

Rt
Rs
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Problem Tree
coverage
connectivity
probabilistic algorithmic
per-node homo
homo heterogeneous
barrier coverage
k-connected
blanket coverage
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Connectivity Issues
8
Power Control for Connectivity

• Adjust transmission range (power)
• Resulting network is connected
• Power consumption is minimum
• Transmission range
• Homogeneous
• Node-based

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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 2-connected
3-connected
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Two Approaches

• Probabilistic
• How many neighbors are needed?
• Algorithmic
• Gmax connected
• Construct a connected subgraph
• with desired properties

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Growing the Tree
coverage
connectivity
probabilistic algorithmic
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Probabilistic Approach

• How many neighbors are necessary and/or
  sufficient to ensure connectivity?

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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 ?

8
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An Alternative View
N

• A square of area N.
• Poisson distribution of a fixed density ?.
• 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 ?

8
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A Related Old Problem

• Packet radio networks (1970s/80s)
• Larger transmission radius
• Good more progress toward destination
• Bad more interference
• Optimum transmission radius?

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Magic Number

• Kleinrock and Silvester (1978)
• 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.

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More Magic Numbers

• Tobagi and Kleinrock (1984)
• Eight is the magic number.
• Other magic numbers for various protocols and
  models
• 5, 6, 7, 8

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Are Magic Numbers Magic?

• Xue Kumar (2002)
• For the network to be almost sure connected,
  T(log n) neighbors are necessary and sufficient.
• Heterogeneous radius

8, 7, 6, 5 (Magic numbers)
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T(log n) neighbors needed for connectivity

• N number of nodes (or area). K number of
  neighbors.
• Xue Kumar (2002)
• If K lt 0.074 log N, almost sure disconnected.
• If K gt 5.1774 log N, almost sure connected.
• 2004, improved to 0.3043 log N and 0.5139 log N

0.3043 0.5139
K
0.074 log n 5.1774
log n
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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.

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Any contradiction?

• Xue Kumar (improved by others)
• 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

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Applying Asymptotic Results

• Applying Xue Kumars result
• The K-Neigh Protocol for Symmetric Topology
  Control in Ad Hoc Networks
• Blough et al, MobiHoc03.
• Applying Penroses result
• On the Minimum Node Degree and Connectivity of a
  Wireless Multihop Network
• Bettstetter, MobiHoc02.

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Applying Penroses result to power control
(Bettstetter, MobiHoc02)

• Nodes deployed randomly.
• Given number of nodes n, node density ?,
  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.

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Application 1

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

500 nodes
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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.

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Growing the Tree
coverage
connectivity
probabilistic algorithmic
per-node homo radius radius
XueKumar Penrose
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Algorithmic Approach
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• 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?

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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)
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Desired Properties of Proximity Graphs

• PG n Gmax is connected (if Gmax is)
• PG is sparse, having T(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.

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

• Homogeneous
• Same max range for all nodes
• PG n Gmax is connected (if Gmax is)
• Heterogeneous
• Different max ranges
• PG n Gmax is not necessarily connected
• (even if Gmax is)
• PG existing PGs

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

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Coverage Issues
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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
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Is the perimeter covered?
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K-covered

• 1-covered
• 2-covered
• 3-covered

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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
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Is the perimeter k-covered?
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Reference

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

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Density (or topology) Control

• Given an area and a sensor deployment
• Problem turn on/off sensors to maximize the
  sensor networks life time

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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

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PEAS basic ideas

• How often to wake up?
• How to determine whether to work or not?

Wake-up rate?
yes
Wake up
Sleep
Go to Work?
work
no
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How often to wake up?

• Desired the total wake-up rate around a node
  equals some given value

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Inter Wake-up Time

• f(t) ? exp(- ?t)
• exponential distribution
• ? average of wake-ups per unit time

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Wake-up rates
A
f(t) ? exp(- ?t)
B
f(t) ? exp(- ?t)
A B f(t) (? ?) exp(- (? ?)
t)
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Adjust wake-up rates

• Working node knows
• Desired total wake-up rate ?d
• Measured total wake-up rate ?m
• When a node wakes up, adjusts its ? by
• ? ? (?d / ?m)

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Go to work or return to sleep?

• Depends on whether there is a working node nearby.

Rp
Go back to sleep go to work
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Is the resulting network covered or connected?

• If Rt (1 v5) Rp and then
• P(connected) ? 1
• Simulation results show good coverage

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OGDC Optimal Geographical Density Control

• Maintaining Sensing Coverage and Connectivity in
  Large sensor networks
• Honghai Zhang and Jennifer Hou
• MobiCom03

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Basic Idea of OGDC

• Minimize the number of working nodes
• Minimize the total amount of overlap

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Minimum overlap
Optimal distance v3 R
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Minimum overlap
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Near-optimal
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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.

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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
• Senders position
• Desired direction

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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

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PEAS vs. OGDC
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Coverage Issues

density control
K-covered?
How many sensors are needed?
PEAS OGDC
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How many sensors to deploy?

• A similar question for k-connectivity
• Depends on
• Deployment method
• Sensing range
• Desired properties
• Sensor failure rate
• Others

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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
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Conditions for Asymptotic Coverage
Necessary Sufficient
expected of active sensors
in a sensing disk.
N nodes
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On kCoverage in a Mostly Sleeping Sensor
Network, Mobicom04

• Almost sure k-covered
• Almost sure not k-covered
• Covered or not covered depending on how it
  approaches 1

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Critical Value

• M average of active sensors in each sensing
  disk.
• M gt log(np) almost sure covered.
• M lt log(np) almost sure not covered.

log(np)
not covered
covered
N nodes
Infocom03 log n 4 log n
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Poisson or Uniform Distribution

• Similar critical conditions hold.

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Application of Critical Condition

• P probability of being active
• R sensing range
• N number of sensors?

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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

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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?

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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
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Donut-shaped region

• K-barrier covered iff G has k essential cycles.

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Critical condition for k-barrier coverage

• Almost sure k-covered
• Almost sure not k-covered

s
1/s
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Growing and Growing
coverage
connectivity
probabilistic algorithmic
per-node homo
homo heterogeneous
barrier coverage
Thank You
k-connected
blanket coverage