## Part B: Network Design for Energy Optimization  
### Tracking Moving Entities  
#### Problem Description  
We wish to solve the problem of _localizing and continuously tracking mobile objects_ moving in a domain described by a set of set of three-dimensional curves, \(S\), over a period of time \(T\) i.e. we want the wireless sensor network to be able to detect the position of any moving object, at any time \(t\) in \(T\). We allow multiple targets that arise in the network area at random locations and at random times. The movement of each target can follow an arbitrary but continuous path i.e. we disallow the target to instantaneously \"jump\" to another location; still, we can handle such discontinuities as multiple targets.  
In our setting, the set \(S\) of 3D curves is the set of possible trajectories of objects moving for some time within the period \(T\). Such a moving object might follow a part of a curve in \(S\), and possibly arrive to an intersection of curves and then follow another curve.  
We can assume that each such curve in \(S\) is specified (for \(t=0,\ldots,T\)) by an analytic equation (e.g. in \(x,y,z\) coordinates). Thus, we can compute a curve (route) _piece_\(\tau_{ij}\) for the curve (route) \(R_{i}\), via some criterion (e.g. to split \(R_{i}\) into pieces of equal length). We, then, wish to deploy some sensors (eitherstanding or even moving for some time) in some (initial) places within \(S\). A basic demand for such a deployment consists of the following rule:  
(Rule R) For every point \(\stackrel{{\rightarrow}}{{p}}=(x,y,z)\) in \(S\) and for every \(t\in[0,T]\) there are at least three sensors, active at time \(t\), whose sensing range includes the point \(\stackrel{{\rightarrow}}{{p}}\). Here, for a sensor \(\sigma\) operating at time \(t\), its _sensing range_ is a sphere of some radius \(R_{\sigma}(t)\) and with center the position of \(\sigma\) at time \(t\).  
Note that rule R guarantees localization of any moving target in \(S\) at any time \(t\) in \(T\), via triangulation.  
We follow a _network design approach_ to this problem. Let us assume, hypothetically, that we could have an abundance of sensors, each characterized by an initial position, an operation period, and an initial available energy (battery) that allows a particular implementation of sensing ranges during the operation of the sensor. Then, the decision to actually select one sensor (with the initial position, operating period and available energy) has a certain _cost_. Our goal is to be able to select a subset of sensors that implement rule (R) and is of nearly minimal cost.  
The problem we study is related (but different) to the problems of network coverage and tracking. In fact, we extend the well-studied coverage problems by being able to track the moving path, and by also taking time into account. On the other hand, to reduce the energy dissipation and overhead of our tracking solution, we avoid some of the collaborative information processing components (like which nodes should sense, which have useful information and should communicate, which should receive information and how often). Thus, we are not dealing directly with queries of the type \"how many targets are in a certain region during a certain time interval\". Still, our solution performs the collaborative processing tasks (triangulation by at least three sensors) that allow localization of the targets as they move in the network. Also, we discuss how alternative collaborative processing methods can be combined with our approach to provide full tracking.