1. Wireless Sensor Networks: Coverage and Energy Conservation Issues
國立交通大學 資訊工程系
曾煜棋教授
Prof. Yu-Chee Tseng

2. Research Issues in Sensor Networks
Hardware (2000)
CPU, memory, sensors, etc.
Protocols (2002)
MAC layers
Routing and transport protocols
Applications (2004)
Localization and positioning applications
Management (2005)
Coverage and connectivity problems
Power management
etc.

3. Coverage Problems In general Possible Approaches
Determine how well the sensing field is monitored or tracked by sensors.
Possible Approaches
Geometric Problems
Level of Exposure
Area Coverage
Coverage
Coverage and Connectivity
Coverage-Preserving and Energy-Conserving Problem

4. Review: Art Gallery Problem
Place the minimum number of cameras such that every point in the art gallery is monitored by at least one camera.

5. Review: Circle Covering Problem
Given a fixed number of identical circles, the goal is to minimize the radius of circles.

6. Level of Exposure Breach and support paths Exposure paths
paths on which the distance from any point to the closest sensor is maximized and minimized
Voronoi diagram and Delaunay triangulation
Exposure paths
Consider the duration that an object is monitored by sensors

7. Coverage and Connectivity
A region is k-covered, then the sensor network is k-connected if RC  2RS
Extending the coverage such that connectivity is maintained.

8. Coverage-Preserving and Energy-Conserving Protocols
Sensors' on-duty time should be properly scheduled to conserve energy.
thus extending the lifetime of the network.
This can be done if some nodes share the common sensing region.

9. The Coverage Problems in 2D Spaces
(ACM MONET, 2005)

10. Coverage Problems In general
To determine how well the sensing field is monitored or tracked by sensors
Sensors may be randomly deployed
Combining the literatures in review, we conclude that in general, given a randomly deployed sensor network, the goal of the coverage problems is to determine how well the sensing field is monitored or tracked by sensors. For example, this is a target area. The black dots are sensor nodes, and the circles represent their sensing areas. As a coverage problem is defined, we may want to know if every point in this field is covered by some sensor. If not, we need to find out the uncovered region and maybe to deploy more sensors such that any object moving in this target area will be monitored by one or more sensors.

11. Coverage Problems We formulate this problem as
Determine whether every point in the service area of the sensor network is covered by at least a sensors
Why a sensors?
Localization, positioning, and tracking applications
Fault-tolerance
More specifically, in our work, we formulate the coverage problem as a decision problem. It’s goal is to determine whether every point in the service area of the sensor network is covered by at least alpha sensors. In most cases, alpha equaling one will be sufficient for applications such as monitoring. However, for localization, positioning, and tracking applications, alpha should be equal to or larger than three. Moreover, an alpha larger than one may be chosen for fault tolerance.

12. The 2D Coverage Problem Is this area 1-covered?
This area is not only 1-covered, but also 2-covered!
What is the coverage level of this area?
This region is not covered by any sensor!
So this area is not 1-covered!
The 2D Coverage Problem
1-covered means that every point in this area is covered by at least 1 sensor
2-covered means that every point in this area is covered by at least 2 sensors
Coverage level = a means that this area is a-covered
Before we focus on the coverage problem in a 3-dimensional field, we first give an example of the coverage problem in a 2-dimensional field. In a 2-dimensional sensing field, we want to know whether this field is 1-covered. 1-covered means that every point in this area is covered by at least 1 sensor. From this figure, we can see that because this region is not covered by any sensor, so this area is not sufficiently 1-covered. Now, the deployment has been changed. Still, we want to know whether this field is 1-covered. In fact, this sensing field is not only 1-covered, but also 2-covered, since every point in the area is covered by at least 2 sensors. So, when given a deployment in a 2-dimensional sensing field, the coverage problem is to determine what is the coverage level of this area? If the coverage level is alpha, it means that this area is alpha-covered, in other words, every point in this area is covered by at least alpha sensors.

13. Sensing and Communication Ranges
1Honghai Zhang and Jennifer C. Hou, ``On deriving the upper bound of a-lifetime for large sensor networks,'' Proc. ACM Mobihoc 2004, June 2004

14. Assumptions
Each sensor is aware of its geographic location and sensing radius.
Each sensor can communicate with its neighbors.
Difficulties:
O(n2) regions divided by n circles
How to determine boundaries of these regions?

15. The Proposed Solution
We try to look at how the perimeter of each sensor’s sensing range is covered.
How a perimeter is covered implies how an area is covered
… by the continuity of coverage of a region
By collecting perimeter coverage of each sensor, the level of coverage of an area can be determined.
a distributed solution

16. How to calculate the perimeter cover of a sensor?
Si is 2-perimeter-covered

17. Relationship between k-covered and k-perimeter-covered
THEOREM. Suppose that no two sensors are located in the same location. The whole network area A is k-covered iff each sensor in the network is k-perimeter-covered.

18. Detailed Algorithm
Each sensor independently calculates its perimeter-covered.
k = min{each sensor’s perimeter coverage}
Time complexity: nd log(d)
n: number of sensors
d: number of neighbors of a sensor

19. Simulation Results

20. A Toolkit

21. Summary New coverage problems!
We have proposed efficient polynomial-time solutions.
Simulation results and a toolkit based on proposed solutions are presented.
Applications of the solutions are discussed.

22. The Coverage Problem in 3D Spaces
(IEEE Globecom 2004)

23. What is the coverage level of this 3D area?
The 3D Coverage Problem
Now, since our goal is to define the coverage problem in a 3-dimensional space, we consider a 3-dimensional sensing field such as a cuboid in this figure. As the definition in the previous example, this time we want to determine the coverage level of this 3-dimensional sensing field.

24. The 3D Coverage Problem Problem Definition Assumptions:
Given a set of sensors in a 3D sensing field, is every point in this field covered by at least a sensors?
Assumptions:
Each sensor is aware of its own location as well as its neighbors’ locations.
The sensing range of each sensor is modeled by a 3D ball.
The sensing ranges of sensors can be non-uniform.
So, we define the 3-dimensional coverage problem as follows: Given a set of sensors in a 3-dimensional sensing field, is every point in this field covered by at least alpha sensors? Here, we assume that each sensor is aware of its own location as well as its neighbors’ locations. Furthermore, the sensing range of each sensor is modeled by a 3-dimensional ball, and is known by all of its neighbors. The sensing ranges of sensors can be non-uniform. In other words, the sensing ranges can be different.

25. Overview of Our Solution
The Proposed Solution
We reduce the geometric problem from a 3D space to one in a 2D space, and further to one in a 1D space.
To solve the 3-dimensional coverage problem, we try to reduce this problem from a 3-dimensional space to one in a 2-dimensional space, and further to one in a 1-dimensional space.

26. Reduction Technique (I)
3D => 2D
To determine whether the whole sensing field is sufficiently covered, we look at the spheres of all sensors
Theorem 1: If each sphere is a-sphere-covered, then the sensing field is a-covered.
Sensor si is a-sphere-covered if all points on its sphere are sphere-covered by at least a sensors, i.e., on or within the spheres of at least a sensors.
Here are the reduction steps. Our goal is to determine whether the whole sensing field is sufficiently covered. To do this, we simply look at the spheres of all sensors. Theorem 1 states that if each sphere is alpha-sphere-covered, then the entire sensing field will be alpha-covered. A sensor is alpha-sphere-covered if all points on its sphere are on or within the spheres of at least alpha sensors. This theorem successfully reduces the problem in a 3-dimensional space to one in a 2-dimensional space. Once the sphere of each sensor is alpha-sphere-covered, the entire sensing field will be alpha-covered.

27. Reduction Technique (II)
2D => 1D
To determine whether each sensor’s sphere is sufficiently covered, we look at how each spherical cap and how each circle of the intersection of two spheres is covered.
(refer to the next page)
Corollary 1: Consider any sensor si. If each point on Si is a-cap-covered, then sphere Si is a-sphere-covered.
A point p is a-cap-covered if it is on at least a caps.
So, now we want to know how the sphere of each sensor is covered. To determine whether each sensor’s sphere is sufficiently covered, we look at how each spherical cap and how each circle of the intersection of two spheres is covered. What is a spherical cap and what is a circle? We see when the sensing ranges of two neighboring sensors intersect with each other, the spherical cap Cap(i, j) is defined as the intersection of the sphere of sensor si and the sensing ball of sensor sj, and the circle Cir(i, j) is defined as the intersection of the spheres of these two sensors. To validate this reduction, Corollary 1 claims that if each point on a sphere is alpha-cap-covered, then this sphere will be alpha-sphere-covered. Here, a point is alpha-cap-covered if it is on at least alpha caps.

28. Cap and Circle
When the sensing ranges of two sensors intersect with each other, the spherical cap Cap(i, j) is defined as the intersection of the sphere of sensor si and the sensing ball of sensor sj. The circle Cir(i, j) is the intersection of the spheres of these two sensors.

29. k-cap-covered p is 2-cap-covered (by Cap(i, j) and Cap(i, k)).
For example, point p is 2-cap-covered because it is on Cap(i, j) and Cap(i, k).

30. Reduction Technique (III)
2D => 1D
Theorem 2: Consider any sensor si and each of its neighboring sensor sj. If each circle Cir(i, j) is a-circle-covered, then the sphere Si is a-cap-covered.
A circle is a-circle-covered if every point on this circle is covered by at least a caps.
So, according the Corollary 1, once we can determine that each point on a sphere is on at least alpha caps, we can prove this sphere will be alpha-sphere-covered. So, now we want to know whether every point on each sphere is on at least alpha caps, in the sense that each sphere is alpha-cap-covered. To do this, we turn to look at the circles on this sphere. Theorem 2 states that if each circle on some sphere is alpha-circle-covered, then this sphere will be alpha-cap-covered. Here, a circle is alpha-circle-covered if every point on this circle is covered by at least alpha caps.

31. k-circle-covered Cir(i, j) is 1-circle-covered (by Cap(i, k)).
For example, Cir(i, j) is 1-circle-covered since every point on Cir(i, j) is covered by Cap(i, k).
Cir(i, j)

32. Reduction Technique (IV)
2D => 1D
By stretching each circle on a 1D line, the level of circle coverage can be easily determined.
This can be done by our 2-D coverage solution.
Finally, by stretching each circle on a 1-dimensional line, we can determine the coverage level of the entire sensing field.

33. Reduction Example =>
We have proven the above theorems and corollary in our paper. Now here is the illustration of the above reduction. It also gives a picture of how we solve the 3-dimensional coverage problem. To determine the coverage level of the a 3-dimensional sensing field, we look at the sphere of each sensor. Because once the sphere of each sensor is covered, the entire field will be sufficiently covered.

34. Reduction Example =>
And then, to determine the sphere coverage of each sensor, we consider the circles on this sphere. This is because that once every point on each circle on this sphere is covered, this sphere will be covered, which in turn proves that the entire sensing field will be covered. In this example, sensor si has neighbors sj, sk, and sl, so there will be three circles on the sphere of si. Now consider only one circle at a time, for example, Cir(i, j). Here comes the question. How to determine the coverage level of such a circle?

35. Calculating the Circle Coverage
Suppose that Cir(i, j) is covered by these caps. For simplification, we now only look at the caps in this way.

36. Calculating the Circle Coverage
=>
Now, if we consider only the intersection of Cir(i, j) and Cap(i, k), we can calculate the angle theta where Cir(i, j) is covered by Cap(i, k).

37. Calculating the Circle Coverage
=>
In this way, we will obtain the angle where Cir(i, j) is covered by each cap. Then, we draw a circle representing Cir(i, j) on the plane, and mark the angles on Cir(i, j). For example, Cir(i, j) is covered by Cap(i, k) from angle n to angle b. It is covered by Cap(i, l) from angle f to angle k. And it is covered by Cap(i, m) from angle j to angle a. So, we can mark these angles on this graph.

38. Calculating the Circle Coverage
=>
Afterwards, we stretch the circle on a 1-dimensional line. The coverage level of this circle will be the minimum number of caps covering a part of this circle. For example, we can see from this figure that some part of this circle is covered by only two caps, and no other part is covered by less caps. Therefore, the coverage level of this circle will be 2.

39. The Complete Algorithm
Each sensor si independently calculates the circle coverage of each of the circle on its sphere.
sphere coverage of si =
min{ circle coverage of all circles on Si }
overall coverage = min{ sphere coverage of all sensors }
So, in the complete algorithm, each sensor independently calculates the circle coverage of each of the circle on its sphere, and the sphere coverage will be the minimum circle coverage among all circles on its sphere.
Finally, the overall coverage of the entire sensing field will be the minimum sphere coverage of all sensors.
So, once we compute the circle coverage of each circle on the sphere of a sensor, we can determine the sphere coverage of this sensor, and the overall coverage level will be obtained. This is the way we compute the coverage level of a 3-dimensional sensing field.

40. Complexity To calculate the sphere coverage of one sensor: O(d2logd)
d is the maximum number of neighbors of a sensor
Overall: O(nd2logd)
n is the number of sensors in this field
Now we look at the complexity of such an algorithm. To calculate the sphere coverage of one sensor, it takes time O(d squared log d), where d is the maximum number of neighbors of a sensor. And the overall complexity in the field will be O(n d squared log d), where n is the number of sensors in this field.

41. Short Summary We define the coverage problem in a 3D space.
Proposed solution
3D => 2D => 1D
Network Coverage => Sphere Coverage => Circle Coverage
Applications
Deploying sensors
Reducing on-duty time of sensors
In this paper, we define the coverage problem in a 3-dimensional space, which is a newly-defined problem. A polynomial-time solution to this problem is proposed, in which we reduce the problem from 3-dimensional space to 2-dimensional space, and further to 1-dimensional space. In other words, to determine the network coverage, we simply look at the sphere coverage of each sensor. And instead of directly calculating the sphere coverage, we only have to determine the circle coverage. In this way, we can easily determine the coverage problem in a 3-dimensional space. Finally, our work can be applied in various ways, such as deploying sensors in a 3-dimensional space, or reducing on-duty time of sensors.

42. A Decentralized Energy-Conserving, Coverage-Preserving Protocol
(IEEE ISCAS 2005)

43. Overview Goal: prolong the network lifetime
Schedule sensors’ on-duty time
Put as many sensors into sleeping mode as possible
Meanwhile active nodes should maintain sufficient coverage
Two protocols are proposed:
basic scheme (by Yan, He, and Stankovic, in ACM SenSys 2003)
energy-based scheme (by Tseng, IEEE ISCAS 2005)

44. Basic Scheme Two phases Reference time: Initialization phase:
Message exchange
Calculate each sensor’s working schedule in the next phase
Sensing phase:
This phase is divided into multiple rounds.
In each round, a sensor has its own working schedule.
Reference time:
Each sensor will randomly generate a number in the range [0, cycle_length] as its reference time.

45. Structure of Sensors’ Working Cycles
Theorem:
If each intersection point between any two sensors’ boundaries is always covered, then the whole sensing field is always covered.
Basic Idea:
Each sensor i and its neighbors will share the responsibility, in a time division manner, to cover each intersection point.

46. An Example (to calculate sensor a’s working schedule)
………
Round 1
Round 2
Round n
Initial phase
Sensing phase
Initial phase a b c d
Ref a
Ref b
Ref d
Ref c
Round i
a’s final on-duty time in round i

47. more details … The above will also be done by sensors b, c, and d.
This will guarantee that all intersection points of sensors’ boundaries will be covered over the time domain.

48. Energy-Based Scheme goal: based on remaining energy of sensors
Nodes with more remaining energies should work longer.
Each round is logically separated into two zones:
larger zone: 3T/4
smaller zone: T/4.
Reference time selection:
If a node’s remaining energy is larger than ½ of its neighbors‘, randomly choose a reference time in the larger zone.
Otherwise, choose a reference time in the smaller zone.
Work schedule selection:
based on energy (refer to the next page)

49. Energy-Based Scheme (cont.)
Frontp,i and Backp,i are also selected based on remaining energies.
richer
rich
poor
Ref a
Ref b
Ref d
Ref c
Round i

50. Two Enhancements active time optimization
k-Coverage-Preserving Protocol
(omitted)
active time optimization
Longest Schedule First (LSF)
Shortest Lifetime First (SLF)

51. Simulation Results

52. Simulation Results (cont.)

53. Summary A distributed node-scheduling protocol Advantage
Conserve energy
Preserve coverage
Handle k-coverage problem
Advantage
Distribute energy consumption among nodes

54. Conclusions Distributed solutions to the coverage problems
Both in 2D and 3D spaces
Coverage-preserving, energy-conserving protocols
Fairly distribute sensors’ energy expenditure
Significantly reduce the computational complexity

55. References
C.-F. Huang and Y.-C. Tseng, “The Coverage Problem in a Wireless Sensor Network”, ACM Mobile Networking and Applications (MONET), Special Issue on Wireless Sensor Networks.
C.-F. Huang and Y.-C. Tseng, “A Survey of Solutions to the Coverage Problems in Wireless Sensor Networks”, Journal of Internet Technology, Special Issue on Wireless Ad Hoc and Sensor Networks.
C.-F. Huang and Y.-C. Tseng, “The Coverage Problem in a Wireless Sensor Network”, ACM Int’l Workshop on Wireless Sensor Networks and Applications (WSNA) (in conjunction with ACM MobiCom), 2003.
C.-F. Huang, Y.-C. Tseng, and Li-Chu Lo, “The Coverage Problem in Three-Dimensional Wireless Sensor Networks”, IEEE GLOBECOM, 2004.
C.-F. Huang, L.-C. Lo, Y.-C. Tseng, and W.-T. Chen, “Decentralized Energy-Conserving and Coverage-Preserving Protocols for Wireless Sensor Networks”, Int’l Symp. on Circuits and Systems (ISCAS), 2005.