The definition of perimeter coverage has been proved useful to determine the cov-erage level of a sensor network in [9]. However, the network connectivity issue has not been studied. For a sensor network to operate successfully, sensors must maintain both sensing coverage and network connectivity. Below we develop some fundamentals to achieve this goal
Definition 5 Consider any sensor si. The neighboring set of si, denoted as N(i), is the set of sensors each of whose sensing region intersects with si’s sensing region.
Definition 6 Consider any sensor si. We say that siis k-direct-neighbor-perimeter-covered, or k-DPC, if siis k-perimeter-covered and sihas a link to each node in N(i).
Similarly, we say that si is k-multihop-neighbor-perimeter-covered, or k-MPC, if si
is k-perimeter-covered and si has a (single- or multi-hop) path to each node in N(i).
si
Figure 3.1: Proof of Lemma 1: (a) the path construction, and (b) possible cases of sx.
Note that the above definitions, though slightly different from what is defined in [9], would make possible deriving the following joint coverage-and-connectivity requirements on a network.
Lemma 1 Consider any two sensors si and sj. If each sensor in S is 1-MPC, there must exist a communication path between si and sj.
Proof. This proof is by construction. If si’s sensing region intersects with sj, by Definition 6, there must exist a path between si and sj, which proves this lemma.
Otherwise, draw a line segment L connecting si and sj, as illustrated in Fig. 3.1(a).
Let L intersect si’s perimeter at point p. Since si is 1-MPC, by Definition 6, there must exist a sensor sx in N(i) which covers p and has a path to si. In addition, either sx must cover sj, or sx’s perimeter must intersect L at a point, namely q, which is closer to sj than p is. Fig. 3.1(b) shows several possible combinations of sx
and rx. In the former case, by Definition 6, there must exist a path between sx and sj, and thus si and sj, which proves this lemma. In the latter case, there must exist another sensor sy in N(x) which covers q. We can repeat the above argument until a sensor sz is found which either covers sj or intersect L at a point, say r, inside sj’s sensing range. In either case, there must exist a path between sz and sj, which
proves this lemma. 2
a
(a) (b)
Figure 3.2: Observations of Theorem 2 and Theorem 3: (a) The network is 2-covered and 1-connected. The removal of sensor a will disconnect the network, and (b) The network is 2-covered and 2-connected but no sensor is 2-DPC. Note that the sensing and communication ranges of each sensor are the same and are represented by circles.
Theorem 2 A sensor network is covered and 1-connected iff each sensor is k-MPC.
Proof. For the “if” part, we have to guarantee both the coverage and connectivity.
The fact that the network is k-covered has been proved by Theorem 1 because each sensor which is k-MPC is also k-perimeter-covered. In addition, Lemma 1 can guarantee that the network is 1-connected, hence proving the “if” part.
For the “only if” part, we have to show that each sensor is k-perimeter-covered and has a path to each sensor whose sensing region intersects with its region. The first concern can be ensured by Theorem 1, while the second concern can be ensured
by the fact that the network is 1-connected. 2
Theorem 3 A sensor network is covered and connected if each sensor is k-DPC.
Proof. Coverage has been guaranteed by Theorem 1 since a sensor which is k-DPC is k-perimeter-covered by definition. For the connectivity part, if we remove any k − 1 nodes from the network, it is not hard to see that each of the rest of sensors must remain 1-DPC. This implies that these sensors are also 1-MPC, and by Lemma 1 there must exist a path between any pair of these sensors. As a result, the network is still connected after the removal of any k − 1 nodes, which proves
this theorem. 2
a
Figure 3.3: An example to compare Theorem 3 with results in [24, 30]. Solid circles and dotted circles are sensors’ sensing ranges and communications ranges, respectively.
Below we make some observations about Theorem 2 and Theorem 3. First, a major difference is that Theorem 2 can guarantee only 1 connectivity, while Theo-rem 3 can guarantee k connectivity. This is because, in a network where each sensor is k-MPC, the removal of any sensor may disconnect the network. For example, in the network in Fig. 3.2(a), each sensor is 2-MPC. By Theorem 2, the network is 2-covered and 1-connected. However, if we remove sensor a, the network will be partitioned into two components. Interestingly, although the network remains 2-covered, it is no longer connected.
Second, the reverse direction of Theorem 3 may not be true. That is, if a network is k-covered and k-connected, sensors in this network may not be k-DPC. Fig. 3.2(b) shows an example in which the network is 2-covered and 2-connected. However, each node has a neighbor (with overlapping sensing range) to which there is no direct communication link.
Third, Theorem 3 is stronger than the results in [24, 30]. It is clear that when two sensors have overlapping sensing range, there is a direct communication link between these two sensors if the communication distance is at least twice the sensing distance.
So what can be determined by [24, 30] can also be determined by Theorem 3.
Furthermore, when the above assumption does not exist, Theorem 3 may still work while [24, 30] do not. For example, Theorem 3 can determine that the network in Fig. 3.3 is 1-covered and 1-connected, when some sensors’ communication ranges are less than twice their sensing ranges.
si
Figure 3.4: Proof of the Lemma 2.