Optimal deployment of sustainable UAV networks for providing wireless coverage

Optimal deployment of sustainable UAV networks for providing wireless coverage

Xiao Zhang and Lingjie Duan X. Zhang and L. Duan are with Engineering Systems and Design Pillar, Singapore University of Technology and Design, Singapore. E-mail: {zhang_xiao, lingjie_duan}@sutd.edu.sg.
Abstract

Recent years have witnessed increasingly more uses of Unmanned Aerial Vehicle (UAV) networks for rapidly providing wireless coverage to ground users. Each UAV is constrained in its energy storage and wireless coverage, and it consumes most energy when flying to the top of the target area, leaving limited leftover energy for hovering at its deployed position and providing wireless coverage. The literature largely overlooks this sustainability issue of UAV network deployment to prolong the UAV network’s residual lifetime for providing wireless coverage, and we aim to maximize the minimum leftover energy storage among all UAVs after their deployment. We also practically consider the No-Fly-Zones (NFZs) constraint to tell that, UAVs cannot be deployed to anywhere even if their energy storages allow. When all UAVs are deployed from a common UAV station, we propose an optimal deployment algorithm, by jointly optimizing UAVs’ flying distances on the ground and final service altitudes in the sky. We show that a UAV with larger initial energy storage should be deployed further away from the UAV station for balancing multi-UAVs’ energy consumption in the flight. We also show that, due to NFZs, the optimization problem becomes more difficulty and the whole UAV network consumes more energy. We solve it optimally in time for a number n of UAVs. Moreover, when UAVs are dispatched from different initial locations, we first prove that any two UAVs will not fly across each other in the flight as long as they have the same initial energy storage, and then design a fully polynomial time approximation scheme (FPTAS) of time complexity to arbitrarily approach the optimum with relative error . Further, we consider that UAVs may have different initial energy storages under the constraint of NFZs, and we prove this problem is NP-hard. Despite of this, we successfully propose a heuristic algorithm to solve it by balancing the efficiency and computation complexity well. Finally, we extend the FPTAS to a 3D scenario and validate theoretical results by extensive simulations.

Unmanned Aerial Vehicle Networks, Sustainable Wireless Coverage, No-Fly-Zone, Approximation Algorithm.

I Introduction

Recently, there are increasingly more exercises and commercial uses of Unmanned Aerial Vehicle (UAV) networks for rapidly providing wireless coverage to ground users (e.g., [1] [2] [3]). In these applications, UAVs serve as flying base stations to serve a geographical area (e.g., cell edge or disaster zone) out of the capacity or reach of territorial base stations. The continuing development of UAV applications for providing wireless coverage still faces two key challenges. First, since each UAV’s coverage radius for providing wireless coverage (though adjustable by its deployed altitude) is small, it consumes most energy when flying over a long distance to the top of the target area. This leaves limited leftover energy for the UAV network’s hovering and providing wireless coverage afterwards, results in a severe sustainability issue. The endurance of each UAV’s on-board energy storage is fundamentally limited by its weight and aircraft size. A large number of UAVs need to cooperate to balance their energy consumption during deployment before providing full coverage to wireless users in the distant and large area. Second, more countries have set up sizable No-Fly-Zones (NFZs) which prohibit UAVs to be deployed inside [4]. Usually, NFZs include restricted areas, prohibited areas, and danger areas (military ranges). Take Singapore as a typical urban city example, Figure 1 shows that NFZs in orange widely cover airports, air bases and military context 111https://garuda.io/what-you-must-know-about-drone-no-fly-zones-nfz/. UAVs can only fly at very low altitude to cross these NFZs before reaching their final deployment positions. In the future, there will be more airspace restrictions for UAVs to protect public security and reserve civil privacy, and the optimal UAV network deployment should adapt to these NFZ constraints. To the best of our knowledge, none of the existing work study the UAV network deployment problem for providing wireless coverage over a target area with NFZ constraint.

Fig. 1: The distribution of No-Fly-Zones (NFZs) in orange in Singapore.
Five deployment scenarios Performance Running time Place
Same UAV station & same initial energy storages for UAVs optimal Section III-A
Same UAV station & different initial energy storages for UAVs optimal Algorithm 1 in Section III-B
Different UAV stations & same initial energy storages (1+)-approximation Algorithm 3 in Section IV-A
Different UAV stations & different initial energy storages near optimal Algorithm 4 in Section IV-B
UAV stations on two ends in 3D & different initial energy storages (1+)-approximation Algorithm 5 in Section V
TABLE I: Summary of our algorithms for deploying sustainable UAV networks in different scenarios.

The deployment of UAVs as flying base stations are attracting growing research interests and the literature focuses on UAV-enabled wireless communications in service phase after deployment (e.g., [5, 6, 7, 8, 9]).

Recent work on UAV-enabled communications have studied multiple issues such as air-to-ground transmission modeling [6] [7], interference management [10], and UAV trajectory planning [11]. For example, [6] and [7] investigate the optimal service altitude for a single UAV, where a larger service altitude of the UAV increases the line-of-sight opportunity of air-to-ground transmission but incurs a larger path loss. [10] studies the mutual interference of UAV downlink links and analyzes the link coverage probability between UAV and ground users. [11] uses UAV-enabled base station to serve multiple users on the ground and jointly optimize the transmit power and UAV trajectory to maximize the average throughput per user. [8] and [12] study the energy-efficient UAV movement and UAV-user link scheduling when serving users, and [13] studies how a UAV should dynamically adapt its location to user movements.

Due to a UAV’s small wireless service coverage, it consumes most energy when flying over a long distance to the top of the target area, leaving limited leftover energy for the UAV network’s hovering and wireless coverage in service phase. It is important to optimize the UAV network deployment before the actual service phase, yet this sustainable deployment issue is largely overlooked in the literature. There are very few works studying the network deployment phase ([2] [9]). For example, [2] studies the UAV-user interaction for learning users’ truthful locations from strategic users before UAV deployment. [14] studies the economics issues (e.g., pricing and energy allocation) for deploying UAV-provided services. [9] aims to minimize the delay of deployment a UAV network till fully covering the target area. We note that in the literature of sensor networks, there are similar deployment problems (e.g., [15] [16]). Yet such results cannot apply to our sustainable UAV deployment problem. Unlike sensors on the ground, UAVs should be deployed to the air and the optimal deployment should take into account the correlation between each UAV’s service altitude and wireless coverage radius.

In this paper, we study this new energy sustainability issue of UAV network deployment to prolong the UAV network’s residual lifetime for providing wireless coverage. We also consider the practical NFZ distribution which complicates the deployment algorithm design. We aim to maximize the minimum leftover energy storage among all UAVs to prolong the UAV network lifetime after their deployment.

Our key novelty and main contributions are summarized as follows. We also present Table I to summarize our proposed algorithms for five different deployment scenarios.

  • Novel sustainable UAV deployment under energy and NFZs constraints (Section II): To our best knowledge, this is the first paper to study the energy sustainability issue for deploying a UAV network and we aim to provide long enough UAV-provided services to a distant target area. We jointly optimize multi-UAVs’ flying distances on the ground and service altitudes in the sky for energy saving purpose. We practically consider the correlation between each UAV’s service altitude and its coverage radius as well as the NFZ distribution for maximizing the whole UAV network’s lifetime after deployment.

  • Optimal deployment by balancing multi-UAVs’ energy consumptions in their flights (Section III): When UAVs are initially located in the same UAV station, we first propose an optimal deployment algorithm in constant running time without considering NFZ, by jointly optimizing UAVs’ flying distances on the ground and service altitudes in the sky. We show that a UAV with larger initial energy storage should be deployed further away on the ground for balancing multi-UAVs’ energy consumptions in the flights. Moreover, due to NFZs constraint, the problem become more complicated, and we present an optimal algorithm in time, in which some UAVs’ final locations are selected and moved to the edges of NFZs, resulting in more energy consumption for the whole UAV network.

  • Near-optimal UAV deployment from different initial locations: In Section IV-A, when dispatching UAVs from different initial locations, we first prove that any two UAVs of the same energy storage should not fly across each other from their initial locations. This helps us simplify the sustainable network deployment problem by fixing the UAVs’ final position order on the ground. Then we successfully design a fully polynomial time approximation scheme (FPTAS) of time complexity to arbitrarily approach the optimum with relative error .

  • Extension to deploying UAVs with different initial energy storages and in 3D: In Section IV-B, we further consider that UAVs may have different initial energy storages before deploying from different initial locations, where now it may be optimal for two UAVs to fly across each other. Then we prove this problem is NP-hard in general. Still, we propose a heuristic algorithm to balance the performance efficiency and computation complexity well. Finally, in Section V, we further extend the proposed FPTAS to a general 3D scenario where UAV stations are located on the two ends on the ground and we also validate the deployment schemes with simulations.

Ii System Model and Problem Formulation

This section introduces our system model and problem formulation for deploying multi-UAVs to provide full wireless coverage to a target service area. The target area includes potential users in an activity to be served (e.g., people celebrating new year in the fifth avenue in Manhattan) and we first model the target area as a line interval in 1D, as shown in Figure 2. Here, a number of UAVs in a set are initially rested in 1D ground locations with initial energy storages before their deployment. Later in Section V, we will extend to model the target area in 2D (see Figure 9) and generalize our design of UAV deployment algorithms to 3D by considering altitude. We denote any UAV ’s final position after deployment as at hovering altitude . According to the air-to-ground transmission model estimated in [6] and [7], the wireless coverage radius of UAV concavely increases with the service altitude due to increasing LoS benefit to negate the path loss till a reasonably large altitude. After deployment, UAV at position covers in . We require a full coverage over the target interval by deploying cooperative UAVs, i.e., .

Fig. 2: An illustrative example of deploying two UAVs from their initial locations to provide wireless coverage to the target area , where UAV with energy storage is deployed from initially to at service altitude with coverage radius . Neither or can fall into NFZ within .

Due to the NFZ policy, any UAV cannot be finally deployed within NFZ. Suppose there are in total NFZs in this area, and we model particular NFZ as a sub-interval in . For UAV , we only allow . Yet it is still allowed for UAVs to just bypass NFZs at low altitude according to [4]. To bypass NFZs, Figure 2 shows that UAV first flies horizontally from to , then flies vertically up to . It travels a normalized distance , where tells the different energy consumptions per unit horizontally and vertically flying distances. In practice, it is more energy consuming to fly vertically to the sky than horizontally over the ground. One may wonder the relationship between flying distance and the energy consumption for a UAV. According to [17], the energy consumption of UAV is proportional to the deployment distance at rate , which is estimated as Watt hour per km (Wh/km) for UAV prototype MD4-3000 and Wh/km for UAV prototype DJI S1000.

After deployment, UAV only has leftover energy to hover in the service phase and keep providing wireless coverage. To prolong the whole UAV network lifetime, we aim to maximize the minimum leftover energy among all the UAVs as the lifetime of the UAV network. Once one UAV uses up its energy, we can no longer guarantee full wireless coverage over and the UAV network’s lifetime is ended up. Our sustainable UAV deployment problem is

(1)

This max-min problem solving requires UAVs to cooperate with each other in deployment distance and altitude to evenly use up their energy. The problem (1) belongs to the domain of combinatorial optimization and the deployment solution is a specific combination of ordered UAVs above the ground, which is generally exponential in the number of UAVs and difficult to solve.

Iii Sustainable UAV deployment from a co-located UAV Station

In this section, we study problem (1) when dispatching UAVs all from the nearest UAV ground station (i.e., for ). This is the case for covering a not huge service area , and we do not need more UAVs from other distant UAV stations. Without loss of generality, we assume that , UAV , which is symmetric to the case of . Note that for the case of , we can divide the line interval into two subintervals, i.e., and , and apply our deployment algorithm (as presented later) similarly over both subintervals. Since all UAVs’ initial locations are identical in the section, we normalize the initial location to , UAV for ease of exposition. For ease of presentation, we first skip the NFZ consideration in the first subsection and will add back later to tell its effect.

Iii-a Deployment of UAVs without NFZ consideration

During the deployment, UAVs should cooperate to cover the whole target area and balance their energy consumptions. We have the following result for multi-UAV cooperation.

Fig. 3: Two neighboring UAVs’ coverages overlap.
Proposition 1.

At the optimal solution to problem (1), all the UAVs have the same amount of leftover energy storage after deployment, i.e., , in which . Their coverage radii do not overlap with each other and seamlessly cover the target interval. That is, , as illustrated in Figure 4.

Proof.

First, we show that any two neighboring UAVs will not overlap in the optimal solution by contradiction. Suppose in the optimal solution, the coverages of two UAVs and overlap as shown in Figure 3. In this case, we can simply lower the service altitude of and move left relatively to remove the overlap, while decrease its energy consumption for moving. We can see that we can obtain better solution and the leftover energy storages of both UAVs do not increase by removing the overlap. This is a contradiction. Next, suppose two UAVs and seamlessly cover a subinterval of in the optimal solution, but they have different leftover energy (i.e., , ). If and is the bottleneck, we can lower the service altitude of and move left relatively to decrease its energy consumption while increase the service altitude of for keeping exactly the same total coverage. In this way, the bottleneck has more leftover energy and the objective of problem (1) is further improved. Otherwise, and is the bottleneck, we can lower the service altitude of and move right relatively to decrease its energy consumption. In the meantime, we increase the service altitude of and move right to cover more for keeping exactly the same total coverage. In the end, the bottleneck is increased and our proof is completed. ∎

Corollary 1.

In the special case when UAVs further have the same initial energy storage (i.e.,  for ), in the optimal solution, a UAV deployed further away (i.e., with large on the ground) should be placed to a lower altitude for balancing multi-UAV energy consumptions during the deployment.

By Proposition 1, we know that UAVs’ coverage radii do not overlap and they seamlessly cover the interval. When the initial energy storage is identical, we expect the same flying distance to keep the same leftover energy among UAVs. The more energy consumes in flying horizontal distance, less energy is left for flying vertically up to service altitude. Therefore, a UAV deployed further away on the ground should be placed to a lower altitude, while a closer UAV should be placed to a higher altitude for balancing multi-UAV energy consumption during deployment. It should be noted that if UAVs have different initial storage, the result above may not hold.

Fig. 4: In the optimal solution, all the UAVs have the same leftover energy and their non-overlapping coverages seamlessly reach full coverage of target area .

Note that when UAV has larger initial energy storage , after travelling the same distance in the flight, it will have more energy left in battery. We have following proposition to show that the UAV with larger energy storage should be dispatched further away in the optimal solution. Then, we can compute the leftover energy storage objective in problem (1) by dispatching the UAVs according to the ordering of their initial energy storages.

Proposition 2.

Without the loss of generality, suppose the initial energy storages of UAVs satisfy . Then the ground destinations of UAVs satisfy in the optimal solution to problem (1).

Proof.

At the optimum, we denote each UAV’s leftover energy storage after deployment as . Consider two neighboring UAVs and with initial energy storages and satisfying . They seamlessly cover a continuous interval with touching point , as shown in the upper subfigure of Figure 5. We have , and for leaving the same residual energy after deploying the two UAVs. We prove by contradiction by supposing at optimality given , then we have . As illustrated in the lower subfigure of Figure 5, we then swap and to show a better solution is actually achieved. Specifically, we move to at altitude such that to cover the same starting point in the target area. We can see that covers . Then we divide our discussion, depending on the relationship between and .

If , then due to larger coverage and . In this case, we can simply move to and to cover prior , as UAV has larger energy storage at than UAV does at . Since now, these two UAVs unnecessarily overlap in their coverage and we can further improve this solution beyond the optimal solution (before UAVs’ swapping). This completes our proof by contradiction for this case.

Next, we consider , then and we move UAV rightwards to such that for seamless coverage from UAV . As shown in Figure 5, , . If given , we have , implying . Therefore, we have . By using up the same amount of energy for both UAVs, we cover a larger total coverage than the optimal solution, telling that UAV swapping provides a better solution than the assumed optimal solution. This completes our proof by contradiction for this subcase.

Now, we only need to consider the other subcase of . As and , then we have . Since and , we have . Due to and , and for leaving the same residual energy . Moreover, due to , we have .

After swapping the two UAVs, we have . Note that implies . Since is an increasing function in our interested scope, we have and further . By combining this with the first equation in this paragraph, we have and for leaving the same residual energy . Overall, we have , and . To show the contradiction, we just need to prove for enlarging the total coverage with the same . Simply, we only need to prove .

We let , then . We define function and the derivative is . implies . As is an increasing concave function, and function is decreasing. Thus, for . It follows that is an increasing function for . Since , either or is less than . Specifically, if , then we have due to ; if , then we have due to . In both cases, we thus have due to . This completes our proof by contradiction for this final subcase.

Fig. 5: Proof illustration of two UAVs and with , for showing UAV should be deployed closer than .

By Proposition 2, we can determine the ground deployment order of UAVs according to . Based on Propositions 1 and 2, we are ready to determine the optimal destination of each UAV for keeping the leftover energy storage identical. Specifically, due to non-overlapping full coverage of area , we first have

(2)

Recall that, the leftover energy storage of is , as shown on the left hand side of

(3)

which indicates any two UAVs’s leftover energy storages are equal. Here, any UAV ’s final destination is . As we now express by , we only have ’s left and we have unknowns in equations in Equations (2) and (3). Solving these equations only needs a constant time . Then we obtain the final locations for each UAV .

(a) Case 1: UAV relocates to the left-edge of NFZ .
(b) Case 2: UAV relocates to the right-edge of NFZ .
(c) Case 3: UAVs and relocates to and , respectively.
Fig. 6: Procedures to bypass NFZ.

Iii-B Incorporation of NFZs for multi-UAV deployment

As presented in Section III-A, we can compute the maximum minimum leftover energy storage directly by solving Equations (2) and (3) without considering NFZs. However, if the destinations of the UAVs fall into some NFZ, it is not a feasible solution. Without loss of generality, we assume there is one NFZ, i.e., (, ).222If there is more than one NFZ, we can similarly discuss each UAV’s possibility to fly to any NFZ and there are just more combinations of Cases 1-3 as in this subsection. In this case, we need to consider three cases for redesigning the deployment algorithm, and at most two UAVs are finally deployed to the edges of the NFZ.

Case 1: UAV is chosen among all to dispatch to the left-edge of NFZ (), as shown in Figure 6(a). This UAV is not necessarily the one within NFZ according to (3)-(2). Similar to Proposition 1, we can show that at the bottleneck UAVs have the same leftover energy storage as and they seamlessly cover including NFZ without any overlap in their coverages. That is,

(4)
(5)

where UAV is located to (, ) and UAV is located to (). By solving (4)-(5), we can determining the deployment of UAVs as well as . Still we need to check and make sure that the other UAVs are able to cover by keeping at least energy after deployment. Note that they may overlap with ’s wireless coverage without reaching the bottleneck.

Case 2: UAV is chosen among all to dispatch to the right-edge of NFZ (), as shown in Figure 6(b). Similar to Proposition 1, we can show that at the bottleneck, UAVs should have the same leftover energy storage as and they are seamlessly cover without any coverage overlap. That is,

(6)
(7)

where UAV is located to (, ) and UAV is located to (). Still we need to check and make sure that the other UAVs are able to cover by keeping at least energy after deployment. Note that they may overlap with ’s wireless coverage without reaching the bottleneck.

Case 3: two neighboring UAVs and are chosen to dispatch to the both edges of NFZ (i.e., ), as shown in Figure 6(c). The NFZ is covered by and seamlessly and these two UAVs’ coverages do not overlap. That is,

(8)
(9)

Moreover, we need to check if UAVs are able to cover and if UAVs are able to cover , by keeping at least energy after deployment.

In these three cases, we can see that the critical UAV index is still undetermined. We propose to run binary search on UAV set to find the optimal , providing the maximum leftover energy storage for the whole UAV network. Note that the binary search needs running time for each UAV in each case. For example, in Case 1, after solving (4)-(5) for each UAV easily, we still need to check the feasibility of the other UAVs on the left-hand side to fully cover in linear running time, resulting in running time for scanning through all UAVs. We summarize all the above in Algorithm 1 and have the following result.

1:  Input:UAV set , NFZ ,A continuous line interval as target area
2:  Output:: the maximum leftover energy storage of the network
3:  compute UAV final locations by solving Equations (2) and (3)
4:  run binary search to select any critical UAV for Cases 1, 2, 3 and compare to choose the maximum leftover storage as .
5:  return  
Algorithm 1 Dispatching the UAVs from the same initial location by considering NFZ
Theorem 1.

When dispatching UAVs from the same UAV station, Algorithm 1 optimally finds the maximum minimum leftover energy storage in time under the NFZ constraint.

Iv Deploying UAVs from different initial locations

In this section, we study the problem when UAVs may be initially located at different locations (e.g., different UAV ground stations or the places that UAVs rest after last task). This is especially the case under emergency when we need a lot more UAVs than those just from the nearest UAV station. Due to the UAV diversity in both initial energy storage and initial location, this problem becomes very difficult. It belongs to the domain of combinatorial optimization and the complexity is generally exponential in the number of UAVs. We aim to design approximation algorithms to maximize the minimum leftover energy storage.

Iv-a When UAVs have identical initial energy storage

In this subsection, we first study the special case when all UAVs have identical initial energy storage . Without loss of generality, we assume the UAVs are indexed according to the increasing order of their initial locations, i.e., . We first prove that all UAVs’ destinations after deployment should follow the same order as the initial locations.

Proposition 3.

Given that the initial locations of UAVs follow the order , the final destinations of UAVs preserve the same order after deployment in the optimal solution.

Proof.

Consider any two neighboring UAVs , with initial locations . We prove by contradiction here. After deployment, and cover a continuous portion of the line interval , and suppose in an optimal solution, as shown on the left-hand side of Figure 7. We can see that and . If we swap the locations of and without changing their altitudes as shown on the right hand side of Figure 7, the new and are smaller than before and they save more energy during the flight by keeping the same total coverage. Then the UAV network’s leftover energy increases and the previous allocation of is actually not optimal. This is the contradiction and we require given . ∎

Fig. 7: Illustration of initial order preserving of UAVs.

Proposition 3 greatly simplifies the algorithm design of deploying UAVs by fixing their location order after deployment. It also holds after incorporating NFZs and we next design the algorithm in two stages. First, we introduce the feasibility checking problem and design the corresponding algorithm to determine whether we can find a deployment scheme for any given leftover energy storage (see Algorithm 2). Then, we use binary search over all these feasible energy storages to find the optimum as summarized in Algorithm 3.

Iv-A1 Feasibility checking procedure

1:  Input: : a given amount of leftover energy storage for each UAV
2:  Output:Feasibility result of
3:  Compute in equation (10) and in equation (11)
4:  ;
5:  for  to  do
6:     if  then
7:        
8:        if  then
9:           
10:           
11:        end if
12:        
13:     end if
14:  end for
15:  if  then
16:     return   ()
17:  else
18:     return   () and update final positions ’s
19:  end if
Algorithm 2 Feasibility checking algorithm to keep leftover energy storage

Given any amount of leftover energy storage , we want to determine whether the budget is feasible to support UAVs to reach a full coverage of and avoid NFZs. Let denotes the maximum minimum leftover storage, we next design a feasibility checking algorithm to determine whether (infeasible) or (feasible). Note that is unknown yet and will be determined in next subsection.

For UAV , we respectively denote as the leftmost point and as the rightmost point on that can be covered by this UAV with leftover storage and altitude . To cover , UAV travels horizontally distance, and it travels horizontally distance to cover . Then we have

(10)
(11)

both of which are functions of . Without loss of generality, we sequentially deploy UAVs to cover the target area from the left to right hand side, and we denote the currently covered interval as . If some UAV ’s destination falls into an NFZ , we can only move it to instead of by keeping energy. If deployed to , there will be some place within out of coverage.

In Algorithm 2, we first compute and in equations (10) and (11) in line 3, then deploy the UAVs one by one (in line 5) according to their initial locations’ order to cover from the left endpoint of target interval as in Proposition 3. As , we start with UAV and end up with . Specifically, given our currently covered interval , we check whether UAV can extend within its energy budget (i.e., in line 6). If so, we will deploy to in line 7 and cover from point . Note that we do not use a UAV if is not within . If the computed falls into some NFZ , we should deploy to on ground and the corresponding service altitude in the sky, as shown in line 8-12. If the outcome shows is feasible () in line 18, our algorithm will further return the UAVs’ final locations ’s. Otherwise, it returns that is infeasible in line 16. Overall, Algorithm 2 solves the feasibility checking problem given a particular leftover energy storage in linear running time.

1:  Input:
2:  Output:: is the selected index
3:   and
4:  while  do
5:     
6:     feasibility checking on by Algorithm 2
7:     if  is feasible then
8:        
9:     else
10:        
11:     end if
12:     if  then
13:        
14:        break
15:     end if
16:  end while
17:  return  
Algorithm 3 FPTAS for multi-UAV deployment from different initial locations

Iv-A2 Binary search over all feasible energy storages

With the help of the feasibility checking algorithm, we can verify whether a given leftover energy storage is feasible or not. The maximum leftover storage among all feasible ones is actually the optimum. Here, we apply binary search to find the maximum leftover storage after determining the search scope and step size of .

We first determine the search scope by computing its upper and lower bounds. By using UAVs to cover the line interval of length , we must have at least one UAV with coverage radius . For each UAV , the minimum moving distance is its service altitude without any ground movement. Suppose is the UAV with coverage radius or equivalently altitude , the upper bound of minimum leftover energy storage among all UAVs is . We simply choose in order to facilitate the following analysis.

We next determine the lower bound. We know that the longest horizontal distance of deploying UAV is . Without flying any distance horizontally but vertically, the maximum service altitude is to cover . Then we have the lower bound .333In case that this formula returns a negative lower bound, we can replace it by a minimum possible positive energy storage (e.g., a bar left in energy storage).

Now we propose the fully polynomial-time approximation scheme in Algorithm 3 by combining both binary search and feasibility checking in Algorithm 2. We denote the relative error as , and accordingly set the search accuracy as in line 1 of Algorithm 3. Algorithm 3 starts with in line 3 and stops once turning from feasible leftover energy storage to infeasible in lines 12-15 This is also illustrated in Figure 8. Finally, the left in line 17 is our searched optimum.

Fig. 8: Binary search on with accuracy level of .
Theorem 2.

Let be the optimal leftover energy storage in problem (1). Given any relative error , Algorithm 3 presents a fully polynomial time approximation scheme (FPTAS) with time complexity to arbitrarily approach the global optimum (i.e., ).

Proof.

The leftover energy storage of a given instance has an upper bounded of and a lower bound of . Obviously, . Choosing a small constant , we divide each into sub-intervals. Each interval has length , where . We divide by into sub-intervals as as in line 1 of Algorithm 3. Overall, we have intervals in .

Then, each operation in line 6 of binary search will shrink by applying Algorithm 2 on a given . It finally terminates with leftover energy storages and , as illustrated in Figure 8, where and . The outcome is our searched optimum. We can obtain that