wireless ad hoc networking

Coverage and Connectivity of Wireless Sensor Networks 9
Theorem 1, if all other sensors are at least 2-perimeter-covered, the sensing
field is 2-covered.
This algorithm can be easily translated to a distributed protocol, where
each sensor only needs to collect local information to make its decision.
The computational cost for each sensor is dominated by sorting the intersection
angles. This incurs a time complexity of O(d log d), where d is the
maximum number of sensors that are neighboring to another sensor. Compared
to the algorithm in Ref. [2], which requires O(d 2 ) for each sensor
to observe the intersection points within its coverage, this algorithm incurs
lower computational complexity.
Note that when a sensor s i is k-perimeter-covered, it is not necessary that
the sensing range of s i is k-covered. In Figure 1.4, sensor s 1 is 2-perimetercovered
but its sensing range is not 2-covered since the shadow region
contained in s 1 is only 1-covered. Since this algorithm looks at perimeter
coverage, this can be explained by observing that s 2 is not 2-perimetercovered
(the dashed segment of s 2 ’s perimeter is only 1-perimeter-covered).
Although this property differs from the algorithm in Ref. [2], when gathering
all sensors’ local decisions, the coverage level of the whole field can be
correctly determined.
Ref. [3] further proposes a new algorithm for determining the coverage
level of a three-dimensional (3D) space. Given a set of sensors in a 3D sensing
field, the goal is to determine if this field is sufficiently α-covered, where
α is a given integer, in the sense that every point in the field is covered by
at least α sensors. The sensing range of each sensor is modeled by a 3D
ball. At the first glance, the 3D coverage problem seems very difficult since
even determining the subspaces divided by the spheres of sensors’ sensing
ranges is very complicated. However, the authors show that tackling
1
s 1
s 2
Figure 1.4 An example of the difference between 2-perimeter-coverage and
2-coverage. The sensor s 1 is 2-perimeter-covered but not 2-covered.