Definition 7 The direct neighboring set of si, denoted as DN(i), is the set of sensors each of which has a communication link to si and whose sensing region intersects with si’s sensing region. Similarly, the multi-hop neighboring set of si, denoted as MN(i), is the set of sensors each of which has a (single- or multi-hop) path to si and whose sensing region intersects with si’s.
Definition 8 Consider any sensor si. We say that si is k-loose-direct-neighbor-perimeter-covered, or k-LDPC, if si is k-perimeter-covered by and only by nodes in DN(i). Similarly, we say that si is k-loose-multihop-neighbor-perimeter-covered, or k-LMPC, if si is k-perimeter-covered by and only by nodes in MN(i).
We comment that for any sensor si, DN(i) ⊆ MN(i) ⊆ N(i). So the definition that siis k-LDPC is looser than that siis k-DPC in the sense that k-DPC guarantees that there is a link from si to each of N(i), but k-LDPC only guarantees that there is a link from si to each of DN(i). The definition of k-LMPC is looser than that of k-MPC in a similar sense.
Lemma 2 If each sensor in S is 1-LMPC, then the network can be decomposed into a number of connected components each of which 1-covers the sensing field A.
Proof. This proof is by construction. For any sensor si, we try to construct a connected component which fully covers A. (However, the proof does not guaran-tee that si has a path to every sensor.) If si’s sensing region can fully cover A, the construction is completed. Otherwise, by Definition 8, nodes in MN(i) must
Figure 3.5: An example of two connected components each of which 1-covers A.
perimeter-cover si’s perimeter and each has a path to si, as illustrated in Fig. 3.4.
In addition, nodes in MN(i) together with si form a larger coverage region which is bounded by perimeters of nodes in MN(i). If A is already fully covered by this region, the construction is completed. Otherwise, since each sensor is 1-LMPC, we can repeat similar arguments by extending the coverage region, until the whole field
A is covered. 2
Theorem 4 A sensor network can be decomposed into a number of connected com-ponents each of which k-covers A iff each sensor is k-LMPC.
Theorem 5 A sensor network can be decomposed into a number of k-connected components each of which k-covers A if each sensor is k-LDPC.
The proof of Theorem 4 (respectively, Theorem 5) is similar to Theorem 2 (re-spectively, Theorem 3) by replacing Lemma 1 with Lemma 2. We comment that although the results of Theorem 4 and Theorem 5 do not seem to be very desirable if one only knows that there are multiple 1- or k-connected components in the net-work, this is what we have to face in practice when deploying a sensor network. An example of Theorem 4 is shown in Fig. 3.5. Due to relatively small communication ranges compared to sensing ranges, the network is partitioned into two connected components. However, each component still provides sufficient 1-coverage.
To summarize, Theorem 4 and Theorem 5 only guarantee that the network can be sufficiently covered by each connected component, while Theorem 2 and Theorem 3 can guarantee both coverage and connectivity of the whole network. When DN(i) = N(i) or MN(i) = N(i) for each sensor si, these theorems converge. Also observe
that Theorem 4 and Theorem 5 are more practical because each sensor only needs to collect its reachable neighbors’ information to make its decision. Most applications can be satisfied if a subset of sensors is connected and can provide sufficient coverage.
The redundance caused by multiple components may be eliminated by a higher level coordinator, such as the base station, to properly schedule each component’s working time such that no two components of the network are active at the same time.
Chapter 4
Distributed Coverage and Connectivity Protocols
The quality of a sensor network can be reflected by the levels of coverage and con-nectivity that it offers. The above results provide us a foundation to determine, or even select, the quality of a sensor network by looking at how each sensor’s perimeter is covered by its neighbors. Section 4.1 shows how to translate the above results to fully distributed coverage-and-connectivity-determination protocols. When sensors are overly deployed, the coverage and connectivity of the network may exceed our expectation. In this case, Section 4.2 proposes a distributed quality selection pro-tocol to automatically adjust its coverage and connectivity by putting sensors into sleep mode and tuning sensors’ transmission power. In Section 4.3, we show how to integrate the above results into one energy-saving protocol to prolong the network lifetime.
4.1 Coverage and Connectivity Determination Pro-tocols
The goal of the protocol is to determine the levels of coverage and connectivity of the network. For a sensor to determine how its perimeter is covered, first it has to collect how its one-hop neighboring sensors’ sensing regions intersect with its sensing region and calculate the level of its perimeter coverage. Periodical BEACON messages can be sent to carry sensors’ location and sensing range information. On receiving such BEACON messages, a sensor can determine who its direct neighbors are and how
its perimeter is covered by them. As reviewed in Section 2, determining a sensor’s perimeter coverage can be done efficiently in polynomial time [9]. If the level of perimeter coverage is determined to be k in this step, we can say that this sensor is k-LDPC.
If the above level of coverage, k, is below our expectation, the sensor can flood a QUERY message to its neighbors to find out who else having overlapping sensing regions with itself. The flooding can be a localized flooding (with a certain hop limit) to save cost. Each sensor who receives the QUERY message has to check if its sensing region intersects with the source node’s sensing region. If so, a REPLY message is sent to the source node. By so doing, the source node can calculate its level of perimeter coverage based on the received REPLY messages. If the level of perimeter coverage is determined to be k in this step (k ≥ k), we can say that this sensor is k-LMPC. If this value is still below our expectation, we can take an incremental approach by flooding another QUERY with a larger hop limit, until the desired level of coverage is reached or the whole network is flooded.
After the above steps, each sensor can report its exploring result to the base station or a certain centralized sensor. Then the base station can determine the coverage and connectivity levels of the network. There are three possible cases. If each sensor is at least k-LDPC, the network is k-covered and k-connected. If some sensors are at least LMPC while others are at least LDPC, the network is k-covered and 1-connected. If there exists some sensors that are neither k-LDPC nor k-LMPC, then the network must be disconnected. In this case, it is possible that the network is still sufficiently covered but is partitioned. For example, if we remove sensor a in Fig. 3.2(a), the network is disconnected into two parts. Although these two parts together provide 2-level coverage, since sensors are unable to exchange information, such a situation can not be determined by the network.