Deployment of Mobile Routers Ensuring Coverage and Connectivity
Coverage and Connectivity Issues in Sensor Networks
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Coverage and Connectivity Issues in Sensor Networks
Ten-Hwang Lai
Ohio State University
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A Sensor Node
Processor
Sensor Actuator NetworkInterface
Memory(Application)
Transmission range
Sensing range
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Sensor Deployment
How to deploy sensors over a field?– Deterministic, planned deployment– Random deployment
Desired properties of deployments? – Depends on applications– Connectivity– Coverage
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Coverage, Connectivity
Every point is covered by 1 or K sensors– 1-covered, K-covered
The sensor network is connected– K-connected
Others
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Coverage & Connectivity: not independent, not identical
If region is continuous & Rt > 2Rs
Region is covered sensors are connected
Rs
Rt
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coverage connectivity
Problem Tree
homo heterogeneous
probabilistic algorithmic
per-node homo
k-connected
blanket coverage
barrier coverage
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Connectivity Issues
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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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Probabilistic– How many neighbors are needed?
Algorithmic– Gmax connected– Construct a connected subgraph
with desired properties
Two Approaches
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coverage connectivity
Growing the Tree
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 →∞
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An Alternative View
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
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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, Θ(log n) neighbors are necessary and sufficient.
Heterogeneous radius
8, 7, 6, 5(Magic numbers)
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Θ(log n) neighbors needed for connectivity
N: number of nodes (or area). K: number of neighbors.
Xue & Kumar (2002):
– If K < 0.074 log N, almost sure disconnected.
– If K > 5.1774 log N, almost sure connected.
2004, improved to 0.3043 log N and 0.5139 log N
K0.074 log n 5.1774 log n
0.3043 0.5139
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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 & 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.
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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.
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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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coverage connectivity
Growing the Tree
probabilistic algorithmic
per-node homo radiusradius
Xue&Kumar 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 ∩ 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.
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coverage connectivity
Growing the Tree
probabilistic algorithmic
per-node homo
various connected subgraphs
Homogeneous max trans. range
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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
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coverage connectivity
Growing the Tree
homo heterogeneous
probabilistic algorithmic
per-node homo
k-connected
max range
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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?
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Is the perimeter covered?
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K-covered
1-covered2-covered3-covered
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K-Coverage Problem
Given: region, sensor deployment, integer k Question: Is the entire region k-covered?
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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 network’s 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?
Sleep Wake up Go to Work?
workyes
no
Wake-up rate?
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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) = λ exp(- λt)
• exponential distribution• λ = average # of wake-ups per unit time
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Wake-up rates
f(t) = λ exp(- λt)
f(t) = λ’ exp(- λ’t)
A
B
A + B: f(t) = (λ + λ’) exp(- (λ + λ’) t)
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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 byλ := λ (λd / λm)
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Go to work or return to sleep?
Depends on whether there is a working node nearby.
Go back to sleep go to work
Rp
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Is the resulting network covered or connected?
If Rt ≥ (1 + √5) 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– MobiCom’03
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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 = √3 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.1. Select a starting node. Turn it on and broadcast
a power-on message.
2. 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 carryingSender’s positionDesired 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
PEAS OGDC
K-covered?
How many sensorsare needed?
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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:
N nodes = expected # of active sensors in a sensing disk.
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On k Coverage 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
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Critical Value
M: average # of active sensors in each sensing disk.
M > log(np): almost sure covered. M < log(np): almost sure not covered.
N nodes
log(np)
not covered
Infocom’03: log n 4 log n
covered
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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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coverage connectivity
Growing the Tree
homo heterogeneous
probabilistic algorithmic
per-node homo
k-connected
blanket coverage
barrier coverage
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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.
L R
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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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coverage connectivity
Growing and Growing
homo heterogeneous
probabilistic algorithmic
per-node homo
k-connected
blanket coverage
barrier coverage
Thank You