Maximizing Lifetime Coverage and Connectivity

The Maximizing Lifetime Coverage and Connectivity (MLCC) problem extends the MLC problem. It aims to prolong the time in which all targets are covered by at least one sensor node. Moreover, it considers the need for sensor nodes to send

sensed data to a base station via a multi-hop path. A node may function only as a relay where it receives and forwards sensed data from other nodes. Other nodes may act as a relay and are also responsible for sensing targets. The network lifetime for the MLCC problem is defined as the period from the time when the network is set up to the time the network cannot meet its coverage or connectivity requirement. One common solution is to divide sensor nodes into a family of subsets, each individually maintaining network coverage and connectivity requirement [30,

49–51]. An approach is to formulate the MLCC problem as a LP that is similar in formulation to the MLC problem; see Section 2.1.1. However, a flow conservation constraint is included to ensure nodes in the calculated set covers are connected. Consequently, for each sensor node, the total amount of data sensed and received must equal the amount of data transmitted. Let R denote the data collection rate of each activated sensor node, fik denotes the amount of data transmitted from sensor

node si to sk and the total number of sensor node is I. The MLCC problem can be

formulated as follows, MAX J X j=1 tj (2.5) s.t. J X j=1 sijtj ≤ 1, ∀si (2.6) Z{Cj} = 1, ∀Cj (2.7) R J X j=1 sijtj + I X k=1 fki = I X k=1 fik, ∀si (2.8) sij =        1 si ∈ Cj 0 si ∈ C/ j (2.9)

The objective of this LP is the same as the one in Section 2.1.1, which is to maximize the sum of lifetime of each set cover. Constraint2.6ensures the expended energy of sensor nodes is not exceed their limit. Constraint 2.7 ensures all the

targets are monitored at all times. The key difference to the LP in Section 2.1.1

is the flow conservation constraint, i.e., 2.8, which ensures the sensed data, i.e., RPJ

j=1sijtj, can be forwarded to the sink. The fundamental problem in this LP

is how to construct the collection of set covers, i.e., Cj, that maintaining complete

target coverage and network connectivity.

Alfieri et al. [52] formulate the MLCC problem as a Mixed Integer Linear Pro- gramming (MILP) with the objective to maximize system lifetime. The key con- straints include coverage and conservation flow. They then propose a greedy heuris- tic that divides time into rounds; all sensor nodes have a uniform probability to be activated in each round. If the subset of sensor nodes activated in a round en- sures the required coverage and connectivity, they will be active continuously in the round. Otherwise, the heuristic discards the subset of nodes and randomly activates another set of sensor nodes. The heuristic will stop if there are no other subsets.

Zhao et al. [51] propose an algorithm that formulates the MLCC problem as an Integer Programming (IP). The objective of the IP is to maximize network lifetime subject to energy and flow conservation constraints. They then propose a greedy heuristic that divides sensor nodes into a set of disjoint set covers. The heuristic first identifies a critical target, which is the one covered by a minimal number of sensor nodes. It then chooses a sensor node to cover the critical target and determines its path to the base station for data transmission. Specifically, a sensor with a large residual energy and shorter distance to the base station has a higher priority to be selected into a path. The sensor nodes involved in this path also monitor the targets within their sensing range. The heuristic then repeatedly determines the next critical target and the sensor node to cover until all targets are covered.

Liu et al. [49] consider a scenario where each sensor node can only monitor one target at a time. Moreover, each target can only be watched by one sensor. They first use a LP to maximize system lifetime subject to energy and flow conservation constraints. The result of this LP gives the length of time that a sensor node monitors a target and the data flow between any two sensor nodes. They then

build a bipartite graph between sensor nodes and targets, which are connected by edges weighted by their corresponding monitoring time. After which, they construct set covers by extracting perfect matchings from the bipartite graph that yield the minimal monitoring time. Finally, they build a sensor node surveillance tree for each set cover based on the data flow calculated from the LP. The tree is rooted at a base station and all leaf sensor nodes are active.

Another work is by Liu et al. [50] where they extend the single coverage problem in [49] to a (h, k) coverage problem: a sensor node is able to watch h targets and each target must be watched by k sensor nodes at any time. Their approach is similar to that of [49]. They first solve an LP with the objective of maximizing the system lifetime subject to energy and conservation flow constraints. Then they build a bipartite graph that represents the time that sensor nodes monitor targets. They improve upon the algorithm in [49] to construct set covers by extracting the perfect (h, k) matchings. After that, they build a sensor node surveillance tree based on the data flow calculated from the LP to guarantee connectivity.

The reviewed MLCC protocols significantly improve network lifetime as com- pared to algorithms designed for the MLC problem when taking communication cost into account [49]. However, the network lifetime remains restricted by the lim- ited battery capacity of sensor nodes. To this end, the next section reviews works using energy harvesting sensor nodes where they have the ability to replenish their battery from ambient source.