**5. The lifetime dependency graph model**

In this section, we introduce the Lifetime Dependency (LD) Graph as a model for the maximum lifetime coverage problem defined in Section 3. This model is a key contribution of this dissertation since the heuristics and algorithms that follow in subsequent sections rely heavily on the LD Graph.

Recall from Section 3 that given a sensor network and a set of static targets, the maximum lifetime sensor scheduling problem is to select a subset of sensors that covers all targets and then periodically shuffle the members of this subset so as to maximize the total time for which the network can cover all targets.

Since these sensors are powered by batteries, energy is a key constraint for these networks. Once the battery has been exhausted, the sensor is considered to be dead. The lifetime of the network is defined as the amount of time that the network can satisfy its coverage objective, i.e., the amount of time that the network can cover its *area* or *targets* of interest. Having all the sensors remain "on" would ensure coverage but this would also significantly reduce the lifetime of the network as the nodes would discharge quickly. A standard approach taken to maximizing the lifetime is to make use of the overlap in the sensing regions of individual sensors caused by the high densit y of deployment. Hence, only a subset of all sensors need to be in the "on" or "sense" state, while the other sensors can enter a low power "sleep" or "off" state. The members of this active set, also known as a *cover* set, are then periodically updated so as to keep the network alive for longer duration. In using such a scheduling scheme, there are two problems that need to be addressed. First, we need to determine how long to use a given cover set and then we need to decide which set to use next. This problem has been shown to be NP-complete [1, 6].

A key problem here is that since a sensor can be a part of multiple covers, these covers have an impact on each other, as using one cover set reduces the lifetime of another set that has sensors common with it. By making greedy choices, the impact of this dependency is not being considered, since none of the heuristics in the literature study this reduction in the lifetime of other cover sets caused by using a sensor that is a member of several such sets. The earlier disjoint formulations mentioned in the previous section, entirely avoided this problem by preventing it.

We capture this dependency between covers by introducing the concept of a local Lifetime Dependency (LD) Graph. This consists of the cover sets as nodes with any two nodes connected if the corresponding covers intersect. The graph is an example of an intersection graph since it represents the sensors common to different cover sets. By looking at the graph locally (fixed 1-2 hop neighbors), we are able to construct all the local covers for the local targets and then model their dependencies. Based on these dependencies, a sensor can then prioritize its covers and negotiate these with its neighbors. We also present some simple heuristics based on the graph. The material presented in this section was published in [35].

**Figure 1.** A sensor network

8 Will-be-set-by-IN-TECH

lives. A target is defined as a *sink* if it is the shortest-life target for at least one sensor covering that target. Otherwise, it is a *hill*. To guard against leaving a target uncovered during a shuffle, each target is assigned an in-charge sensor. For each sink, its in-charge sensor is the one with the largest battery for which this is the shortest-life target. For a hill target, its in-charge is that neighboring sensor whose shortest-life target has the longest life. An in-charge sensor does not switch off unless its targets are covered by someone. Apart from this, the rules are identical as those in LBP protocol. DEEPS relies on two-hop information to make these decisions.

In this section, we introduce the Lifetime Dependency (LD) Graph as a model for the maximum lifetime coverage problem defined in Section 3. This model is a key contribution of this dissertation since the heuristics and algorithms that follow in subsequent sections rely

Recall from Section 3 that given a sensor network and a set of static targets, the maximum lifetime sensor scheduling problem is to select a subset of sensors that covers all targets and then periodically shuffle the members of this subset so as to maximize the total time for which

Since these sensors are powered by batteries, energy is a key constraint for these networks. Once the battery has been exhausted, the sensor is considered to be dead. The lifetime of the network is defined as the amount of time that the network can satisfy its coverage objective, i.e., the amount of time that the network can cover its *area* or *targets* of interest. Having all the sensors remain "on" would ensure coverage but this would also significantly reduce the lifetime of the network as the nodes would discharge quickly. A standard approach taken to maximizing the lifetime is to make use of the overlap in the sensing regions of individual sensors caused by the high densit y of deployment. Hence, only a subset of all sensors need to be in the "on" or "sense" state, while the other sensors can enter a low power "sleep" or "off" state. The members of this active set, also known as a *cover* set, are then periodically updated so as to keep the network alive for longer duration. In using such a scheduling scheme, there are two problems that need to be addressed. First, we need to determine how long to use a given cover set and then we need to decide which set to use next. This problem has been

A key problem here is that since a sensor can be a part of multiple covers, these covers have an impact on each other, as using one cover set reduces the lifetime of another set that has sensors common with it. By making greedy choices, the impact of this dependency is not being considered, since none of the heuristics in the literature study this reduction in the lifetime of other cover sets caused by using a sensor that is a member of several such sets. The earlier disjoint formulations mentioned in the previous section, entirely avoided this problem

We capture this dependency between covers by introducing the concept of a local Lifetime Dependency (LD) Graph. This consists of the cover sets as nodes with any two nodes connected if the corresponding covers intersect. The graph is an example of an intersection graph since it represents the sensors common to different cover sets. By looking at the graph locally (fixed 1-2 hop neighbors), we are able to construct all the local covers for the local targets and then model their dependencies. Based on these dependencies, a sensor can then

**5. The lifetime dependency graph model**

heavily on the LD Graph.

the network can cover all targets.

shown to be NP-complete [1, 6].

by preventing it.
