**6. Symbols and definitions**

Let us begin with a few basic conventions and definitions that will be used in the rest of this dissertation. Individual chapters will introduce additional notation as and when necessary. The notation presented here applies to the basic LD grapg model and will be utilized for all the chapters that follow.

We will use *s*1,*s*2, etc., to represent sensors, *t*1, *t*2, etc., to represent targets, and *C*, *C*� , etc., to denote covers.

Let us assume we have *n* sensors and *m* targets, both stationary.

Consider the sensor network in Figure 1 with *n* = 8,*s* = {*s*1,*s*2, ...,*s*8} and *m* = 3 targets, *t*1, *t*2, and *t*3.

We will employ the following definitions, illustrated using this network.


which need to be avoided. Likewise, the possible covers for the only target of sensor *s*<sup>3</sup> are {*s*1}, {*s*3}, {*s*4} and {*s*5}.

• *lt*(*C*) = *mins*∈*Cb*(*s*), the maximum lifetime of a cover. The bottleneck sensor of the cover {*s*2,*s*3} is *s*<sup>3</sup> with the weakest battery of 1. Therefore, *lt*({*s*2,*s*3}) = 1.

An optimal lifetime schedule of length 6 for this network is ({*s*1,*s*6}, 1), ({*s*1,*s*7}, 1), ({*s*1,*s*8}, 1), ({*s*2,*s*3}, 1), ({*s*2,*s*4}, 1), ({*s*2,*s*5}, 1)) where each tuple is a cover for the entire network followed by its duration.
