## 2 Network Environment and Problem Formulation
### Notations  
This paper aims to construct a data collection trajectory along which the mobile sink can collect all readings generated by all sensors while the data-relaying workloads of all sensors can be balanced. To achieve this purpose, the mobile sink traverses different tracks for different number of sweep repetitions to collect data. For the ease of presentation, we first introduce some notations.  
Let \(J_{i}=(x^{\prime}_{n},x^{\prime}_{n-1},\ldots,x^{\prime}_{1})\) denote the _regular trajectory_\(i\) that a mobile sink initially traverses, track \(k_{n}\) for \(x^{\prime}_{n}\) repetitions of sweeps and then traverses track \(k_{n-1}\) for \(x^{\prime}_{n-1}\) repetitions of sweeps and so on. After traversing track \(k_{1}\) for \(x^{\prime}_{i}\) repetitions of sweeps, the mobile sink is said to move along the trajectory \(J_{i}\) in one round. Let \(e^{\mathit{con}}_{i,k}\) denote the energy consumption of sensor \(s_{i}\) when the mobile sink completes the movement of trajectory \(J_{k}\) in one round. The trajectory \(J_{k}\) is called an _energy-balanced trajectory_, noted as \(J^{\mathit{E-T}}_{i}\), if \(e^{\mathit{con}}_{a,k}=e^{\mathit{con}}_{k,k}\) holds, where \(\forall_{s_{i},s_{i}\in S}\). Let \(l_{j}=2\pi(2 jr-r)\) denote the movement length of mobile sink moving along track \(k_{j}\) for one repetition of sweep. Let \(|J_{i}|\sum_{j=i}^{n}x^{\prime}_{j}\) denote the total movement length of mobile sink moving along the trajectory \(J_{i}=(x^{\prime}_{n},x^{\prime}_{n+1},\ldots,x^{\prime}_{1})\) in one round. Let \(J^{\mathit{EB-T}}\) denote the set of all energy-balanced trajectories. An energy-balanced trajectory \(J^{\mathit{EB-T}}_{i}\) is referred to as the _energy-balanced trajectory with minimal length_, noted as \(J^{\mathit{EB-T}}_{\mathit{min}}\), if it satisfies the Equation (1):  
\[\left|J^{\mathit{EB-T}}_{i}\right|=\min\left(J^{\mathit{EB-T}}_{k}\right| \right)\ \ \forall J^{\mathit{EB-T}}_{k}\in J^{\mathit{EB-T}} \tag{1}\]  
For instance, consider a circular monitoring region which is geographically partitioned into three circular tracks \(k_{3}\), \(k_{2}\), and \(k_{1}\). The \(l_{3}\), \(l_{2}\), and \(l_{1}\) are \(10\pi r\), \(6\pi r\), and \(2\pi r\), respectively. A regular trajectory \(J_{a}=(5,\,2,\,1)\) represents that mobile sink initially traverses track \(k_{3}\) for five sweep repetitions and then traverses track \(k_{2}\) for two sweep repetitions and finally traverses track \(k_{1}\) for one sweep repetition. If trajectory \(J_{a}\) is an energy-balanced trajectory, this indicates that all sensors will be energy-balanced when the mobile sink completes the movement of trajectory \(J_{a}\) in each round. Furthermore, since \(J_{a}=(5,\,2,\,1)\) is an energy-balanced trajectory, trajectory \(J_{b}=(5i,\,2i,\,i)\) must be the other energy-balanced trajectories, where \(\forall i\in N\). Obviously, as shown in Equations (2) and (3), the value of \(|J_{a}|\) must be less  
Figure 2: Monitoring region \(M\) is geographically partitioned into \(n=R/2r\) circular tracks which are traversed by mobile sink.  
than or equal to those of \(|J_{b}|\) and therefore trajectory \(J_{a}\) is denoted as \(J_{\min}^{\frac{\alpha-T}{2}}\). Some additional notations are summarized in Table 1.  
\[|\,J_{a}\,|=5\times 10\pi r+2\times 6\pi r+1\times 2\pi r=64\pi r \tag{2}\]  
\[|\,J_{b}\,|\equiv 5i\times 10\pi r+2i\times 6\pi r+i\times 2\pi r=64i\pi r,\,\, \forall i\in N \tag{3}\]