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Placement Optimization of UAV-Mounted MobileBase Stations

Jiangbin Lyu,Member, IEEE, Yong Zeng,Member, IEEE, Rui Zhang,Fellow, IEEEand Teng Joon Lim,Fellow, IEEE

Abstract—In terrestrial communication networks without fixedinfrastructure, unmanned aerial vehicle (UAV)-mounted mobilebase stations (MBSs) provide an efficient solution to achievewireless connectivity. This letter aims to minimize the numberof MBSs needed to provide wireless coverage for a group ofdistributed ground terminals (GTs), ensuring that each GT iswithin the communication range of at least one MBS. We propose apolynomial-time algorithm with successive MBS placement,wherethe MBSs are placed sequentially starting on the area perimeter ofthe uncovered GTs along a spiral path towards the center, until allGTs are covered. Each MBS is placed to cover as many uncoveredGTs as possible, with higher priority given to the GTs on theboundary to reduce the occurrence of outlier GTs that each mayrequire one dedicated MBS for its coverage. Numerical resultsshow that the proposed algorithm performs favorably comparedto other schemes in terms of the total number of required MBSsand/or time complexity.

Index Terms—Unmanned aerial vehicles, mobile base stationplacement, user coverage, geometric disk cover problem

I. I NTRODUCTION

With their maneuverability and increasing affordability,un-manned aerial vehicles (UAVs) have many potential applicationsin wireless communication systems [1]. In particular, UAV-mounted mobile base stations (MBSs) can be deployed toprovide wireless connectivity in areas without infrastructurecoverage such as battlefields or disaster scenes. Unlike terrestrialbase stations (BSs), even those mounted on ground vehicles,UAV-mounted MBSs can be deployed in any location andmove along any trajectory constrained only by their aeronauticalcharacteristics, in order to cover the ground terminals (GTs) ina given area based on their known locations. When the UAV-GTchannels are dominated by line-of-sight (LOS) links, the authorsin [2] use a K-means clustering algorithm to partition the GTs tobe served byp UAVs, while each UAV has a capacity constraintand the unsupported GTs are served by the fixed ground BSs.The authors in [3] adopt a probabilistic LOS channel model andstudy the 3-dimensional (3D) placement of a single aerial BSto offload as many GTs as possible from the ground BS.

In this letter, we assume that the GT locations are knownand the UAVs are flying at a fixed altitudeH , while theUAV-GT channels are dominated by LOS links whose channelquality mainly depends on the UAV-GT distance. We considerthe scenario where no ground BSs are available and the UAV-mounted MBSs are backhaul-connected via satellite links, whileeach MBS has an equivalent coverage radius ofr projected onthe ground, as shown in Fig. 1. We thereby focus on the MBSplacement problem to provide wireless coverage for all GTs ina given area. This can be formulated as the Geometric DiskCover (GDC) problem [4], whose objective is to cover a setof K nodes (GTs) in a region with the minimum number of

The authors are with the Department of Electrical and Computer Engi-neering, National University of Singapore (email:{elelujb, elezeng, elezhang,eleltj}@nus.edu.sg).

r

H

Fig. 1: A wireless communication system with UAV-mounted MBSs

disks of given radiusr. The GDC problem can be optimallysolved by the core-sets method [5] whose theoretical boundsonthe running time are exponential inK. Since the GDC problemis NP-hard in general, a strip-cover-with-disks algorithmwasproposed in [4], which divides the plane into equal-width stripsand solves the problem locally over the GTs within each strip.The computational complexity is reduced thanks to this strip-based partitioning which, however, may lead to significant per-formance loss since the GTs in different strips are independentlyconsidered though certain GTs in adjacent strips could in factbe covered by the same MBS.

This letter proposes a new MBS placement algorithm byplacing the MBSs sequentially, starting from the perimeterofthe area boundary in an inward spiral manner until all GTs arecovered. In the proposed spiral placement algorithm, each MBSis first positioned to cover at least one uncovered GT near thearea boundary, and then its position is adjusted inwards towardthe area center to cover as many additional uncovered GTs aspossible. This localized strategy has low complexity and doesnot partition the coverage area into independent regions, henceovercoming the limitations of the strip-based algorithm. Ourproposed algorithm has a polynomial-time complexityO(K3) inthe worst case, which is comparable to the strip-based algorithmbut much lower than the core-sets method. Numerical resultsshow that for small networks requiring only a few MBSs, wherethe theoretical minimum can be found by the core-sets method,the proposed algorithm provides the near-optimal performancein terms of the number of required MBSs. Moreover, theproposed algorithm also outperforms other heuristic schemes interms of the number of required MBSs and time complexity onaverage for networks of different sizes, including the strip-basedalgorithm, the K-means clustering algorithm, and the randomplacement algorithm. Note that the proposed algorithm can beconsidered as a new approach to solve the GDC problem ingeneral and thus can be used in other pertinent applications.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a wireless system withK GTs, which aredenoted by the setK = {1, 2, · · · ,K} and at known locationsgiven by {wk}k∈K, where wk ∈ R

2×1 represents the two-dimensional (2D) coordinates of thek-th GT on the horizon-tal plane (ground). Assume that the UAV-GT communication

2

channels are dominated by LOS links. Though simplified, theLOS model offers a good approximation for practical UAV-GTchannels, and enables us to investigate the coverage problem inthis letter. Other practical issues such as multi-access and UAV-GT association can be considered separately [6]. We assume thatthe transmit power is fixed and the minimum required signal-to-noise ratio (SNR) at the receiver for reliable communications isgiven. Under the LOS model, the UAV-GT channel power gainfollows the free-space path-loss model, which is determined bythe UAV-GT link distance. Assume that the UAVs are flying at agiven altitudeH and their maximum coverage radius projectedon the ground plane corresponding to the SNR threshold isspecified byr, as shown in Fig. 1.

For cost minimization, we aim to deploy the minimumnumber of MBSs (UAVs) so that each GT is served by atleast one MBS within its communication radiusr. Note thatthis does not preclude the possibility that some GTs may becovered by more than one MBSs. In such scenarios, the inter-cellinterference issue needs to be addressed by, e.g., proper channelassignment and power control after deploying the MBSs, whichis out of the scope of this letter. Denoting byM = {1, ...,M}the set of MBSs to be deployed, the problem can be formulatedas follows.

(P1) :

min{um}m∈M

|M|

s.t. minm∈M

‖wk − um‖ ≤ r, ∀k ∈ K,

where |M| = M denotes the cardinality of the setM, um ∈R

2×1 denotes the horizontal coordinates of MBSm, and theEuclidean norm‖wk − um‖ is the distance between GTk andMBS m projected on the ground plane.

(P1) is also known as the GDC problem [4], which is NPhard in general. The GDC problem is also related to thep-centerproblem [7], which aims to locatep centers (MBS locations) ofthe smallest disks to cover allK nodes (GTs), given by

(P2) :

min{um}p

m=1

ρ

s.t. minm=1,··· ,p

‖wk − um‖ ≤ ρ, ∀k ∈ K,

whose optimal valueρ∗ is the smallest radius of thep disksrequired to cover allK GTs. If ρ∗ ≤ r, then all GTs canbe covered by thep MBSs in (P1) andMmin ≤ p, whereMmin denotes the optimal value of (P1). The GDC problem (P1)can thus be converted into a series ofp-center problems withincreasingp values, until the smallest number of MBSs requiredto cover all GTs is found. Unfortunately, (P2) is in generaldifficult to solve optimally due to the non-convex constraintmin

m=1,··· ,p‖wk − um‖ ≤ ρ, ∀k ∈ K, whose left-hand side is the

minimum of convex functions and hence is non-convex. In fact,the p-center problem is also NP-hard, whose optimal solutionrequires computational complexity ofO(pK) using brute forcesearch [8], which is infeasible even for moderate values ofp and K. Recent progress is based on the exploitation of asmall subset of GTs called core-sets [5]. A branch-and-boundalgorithm to traverse the partitions of possible core-setsusingdepth-first strategy is given in [5], which can find the optimalsolution to thep-center problem for small values ofp (p ≤ 8),although the worst-case complexity is stillO(pK).

III. SPIRAL MBS PLACEMENT ALGORITHM

In this section, we propose an efficient heuristic algorithmtosolve (P1) approximately based on successive MBS placement.The main idea is to place the MBSs sequentially along the areaperimeter, which is defined as the path connecting the extremepoints (referred to as the boundary GTs) of the convex hullof all uncovered GTs. Each MBSm is guaranteed to cover atleast one boundary GTk0, and those GTs at a distance of morethan 2r away fromk0 are removed from consideration, sincethey cannot be jointly covered withk0 by the same MBSm.Sincek0 is at the boundary, MBSm will be placed inwardstoward the area center to cover as many uncovered GTs aspossible, with higher priority given to the GTs on the boundaryto reduce the occurrence of outlier GTs that each may requireone dedicated MBS for its coverage. After MBSm is placed,the area perimeter of the remaining uncovered GTs shrinks atthe local region neark0. The above process repeats to place thenext MBSm+ 1 counterclockwisely next to MBSm, and thearea perimeter gradually shrinks until all GTs are covered.As aresult, the connecting line of the placed MBSs looks like a spiralwhich starts from the area boundary and counterclockwiselyrevolves inwards toward the area center. We therefore nameour proposed algorithm as thespiral MBS placement algorithm,which is summarized in Algorithm 1.

Algorithm 1 Spiral MBS Placement AlgorithmInput: GT setK, with known locations{wk}k∈K.Output: MBS setM, with optimized locations{um}m∈M.Initialization: Uncovered GT setKU ← K; M = ∅;m = 1.

1: while KU 6= ∅ do2: Find boundary GT setKU,bo ⊆ KU and list them in counter-

clockwise order. Update inner GT setKU,in ← KU \ KU,bo. Ifm = 1, randomly pick a GTk0 ∈ KU,bo.

3: Refine MBS locationu to coverk0 and as many boundary GTsas possible, by calling[u,Pprio] = LocalCover(wk0

, {k0},KU,bo\{k0}). Let Knew,bo ← Pprio.

4: Refine MBS locationu to coverKnew,bo and as many innerGTs as possible, by calling[u,Pprio] = LocalCover(u, Knew,bo ,KU,in). Let um = u, Knew ← Pprio.

5: M←M∪ {m}, KU ← KU \ Knew , m← m+ 1.6: FromKU,bo \Knew,bo , pick the first uncovered boundary GTk′

0

counterclockwisely next tok0. Let k0 ← k′0.

7: end while

We use the example in Fig. 2 to illustrate the notations andthe main steps of our spiral algorithm. Denote byKU ⊆ Kthe subset of uncovered GTs, which is initialized toK at thebeginning of Algorithm 1.KU is partitioned into the innerGT subsetKU,in and the boundary GT subsetKU,bo, wherethe boundary GTs can be listed in counterclockwise order asKU,bo = {1, 2, 3, 4, 5, 6, · · · } initially (dark blue triangles), andKU,in = KU \ KU,bo (light blue triangles). The path connectingthese boundary GTs is referred to as the area perimeter of theuncovered GTs, as shown in Fig. 2. We use the convex hull todefine the boundary GTs, whereas other boundary definitions[9] can also be used which produce similar results.

We give higher priority to the boundary GTs in the way that acertain subset of boundary GTs are guaranteed to be covered byeach newly placed MBS. To place the first MBS, we randomlychoose a boundary GTk0 which is guaranteed to be covered,e.g., GT 3 at the lower left corner denoted by a red triangle (step2 in Algorithm 1). Then we refine the MBS locationu to cover

3

r

2r

0k

'

0k

Fig. 2: Illustration of the spiral algorithm

k0 and as many boundary GTs as possible (step 3). In this case,the boundary GTs 2 and 4 can be covered, and hence are addedinto the prioritized setPprio = {2, 3, 4} which is guaranteed tobe covered first. Then we proceed to cover GTs fromPprio andas many inner GTs as possible (step 4). In this case, the innerGTs 7 and 8 can be covered. The final location of the first MBSis denoted by a green square, which is the center of the coveringdisk of radiusr, denoted by a dashed green circle. After placingthe first MBS, the area perimeter shrinks at the local region nearGT k0, with GT 1 directly connected to GT 5 in this case. Toplace the next MBS, we pick the first uncovered boundary GTk′0 counterclockwisely next tok0, which in this case is GT 5,and updatek0 ← k′

0(step 6). Then the above steps are repeated

to place the second MBS which covers GTs 5, 6 and 11. Theabove process repeats until all GTs are covered.

Note that we have used aLocalCover procedure in steps 3and 4 of Algorithm 1, which refines the new MBS locationu toguarantee to cover GTs from the given prioritized setPprio (e.g.,the initial boundary GTk0), and then to cover as many GTs aspossible from the secondary set of GTs (e.g., uncovered innerGTs), denoted asPsec. Mathematically, this can be formulatedas the following optimization problem.

(P3) :

maxu,Knew

|Knew|

s.t. ‖u−wk‖ ≤ r, ∀k ∈ Knew ∪ Pprio,

Knew ⊆ Psec,

whereu denotes the location of the new MBS to be placed,Knew ⊆ Psec denotes the set of GTs newly covered by thisnew MBS. Note that the first constraint in (P3) ensures thatall GTs in Knew and Pprio are covered by this new MBS.(P3) is a combinatorial optimization problem, which in generalrequires exhaustive search over all2|Psec| subsets ofPsec inorder to obtain the optimal solution, which is prohibitive even formoderately large systems. Therefore, we propose aLocalCoverprocedure with possibly sub-optimal solutions to (P3) for low-complexity implementation, as summarized in Algorithm 2.

Algorithm 2 LocalCover ProcedureProcedure [u,Pprio] = LocalCover(u, Pprio, Psec)

1: while Psec 6= ∅ do2: UpdatePsec by excluding GTs more than2r away from any

GT in Pprio. UpdatePprio (Psec) by including (excluding) GTswithin distancer to u.

3: Find GT k1 ∈ Psec with shortest distance tou. Add (remove)k1 to (from) Pprio (Psec) if it can be covered by refiningu viasolving the 1-center problem. Stop otherwise.

4: end while

We continue to use the example in Fig. 2 to illustrateAlgorithm 2. Firstly, for any givenPprio, Psec can be reducedby excluding those GTs more than2r away from any GT inPprio, since the same MBS cannot cover two GTs that are morethan 2r away from each other. This confines the search spaceto a local region nearPprio. For example, since the first MBSis guaranteed to cover GT 3, we can draw a dashed red circlecentered at GT 3 with radius2r as shown in Fig. 2, and excludethose GTs that are outside of this circle from consideration,after which only GTs 2, 3, 4, 7, 8, and 9 are left. This greatlyreduces the problem size in (P3). Secondly, the remaining GTsin Psec are sorted in ascending order of the distance to thecurrent MBS locationu, and are then successively includedbased on this order until they cannot be covered by the sameMBS. Intuitively, the number of newly covered GTs inPsec isapproximately maximized. Moreover, in step 2 of Algorithm 2,we updatePprio (Psec) by including (excluding) GTs withindistancer to u. This simple check reduces the times that the 1-center subroutine in step 3 of Algorithm 2 needs to be executed.For example, after MBS 1 covers the boundary GTs 2, 3 and 4,the algorithm finds that GT 7 is already covered and hence doesnot need to call the 1-center subroutine for GT 7 subsequently.

In step 3 of Algorithm 2, to check whether a setP of K pointscan be covered by a single disk of radiusr, we need to solve the1-center problem, which finds the locationu of the center fromwhich the maximum distance to any point inP is minimized.Several algorithms exist to solve the 1-center problem, such asthat in [10] withO(K) complexity, and a more straightforwardone in [11] withO(K2) complexity.

For our spiral algorithm, each of the MBSs to be placedneeds to run the convex hull algorithm to find the boundary GTsand list them in counterclockwise order, which has complexityO(K log b) with b ≤ K being the number of extreme points ofthe convex hull. Moreover, each MBS may also need to executethe 1-center subroutine for up toO(K) times. Since the numberof placed MBSs is at mostO(K), the overall computationalcomplexity is upper-bounded byO(K[K logK +K · C(K)]),whereC(K) is the running time of the 1-center subroutine.Note that the actual running time could be much less than thisworst-case complexity, since the size of each 1-center subroutineand the times to be executed are greatly reduced, thanks to thestrategy of excluding far-away GTs and including nearby GTsin step 2 of Algorithm 2.

To illustrate the final MBS placement results, we apply ourspiral algorithm to a numerical example withK = 80 GTs(denoted as triangles) randomly and independently scattered in asquare region of area 10 km2, where each MBS has a coverageradius r = 0.5 km, as shown in Fig. 3. We use dash-dottedred arrows to connect the MBSs which are successively placedalong the area perimeter. In this case, a total of 11 MBSs(denoted as green squares) are required and their connectingline looks like a spiral which starts from the area boundary andcounterclockwisely revolves inwards toward the area center.

To check the optimality of our spiral algorithm, we applythe core-sets method of exponential complexity in [5] withstacked-depth-first branch-and-bound search to the 80 GTs’topology in Fig. 3, which yields a minimum coverage radius of0.5231 km and 0.4829 km for 10-center and 11-center problems,respectively. Therefore, it requires a minimum of 11 MBSs tocover all 80 GTs with a coverage radius of 0.5 km, which isthe same as that achieved by our spiral algorithm. The placed

4

Fig. 3: Solutions of the spiral, strip-based and core-sets methods to theGDC problem with 80 GTs and MBS coverage radiusr = 0.5 km.

MBS locations are denoted as “×” in Fig. 3. As a benchmarkcomparison, we also apply the strip-based algorithm in [4] tothe 80 GTs’ topology in Fig. 3. It requires a total number of13 MBSs (denoted as “+” in Fig. 3) to cover all GTs, which ismore than that obtained by our spiral algorithm.

IV. N UMERICAL RESULTS

In this section, we test the algorithms for two cases withK = 80 and K = 400 GTs, respectively. In each case, werandomly and independently generate 5 topologies withK GTsin a square region of side lengthD, and apply the algorithmsto these topologies with different coverage radiusr. For eachalgorithm and eachD/r ratio, the total number of requiredMBSsM and the running timet in seconds are averaged overthe 5 topologies, respectively. Besides the core-sets methodand the strip-based algorithm, we also compare with two otherheuristic schemes. The first one is random placement, whichrandomly selects a location to place an MBS and removes thecovered region from consideration when placing the next MBS.The process repeats until all GTs are covered. The second oneisto run the K-means algorithm to partition the GTs to be coveredby p MBSs. Bisection search is performed to find the minimumnumberp to cover all GTs. Each of these two heuristics isexecuted for 100 trials on each topology andD/r ratio to findthe best trial with the minimum number of MBSs. Note that themore trials of these two heuristics (hence a longer running time),the higher likelihood of finding a solution with smaller numberof required MBSs. We used the 1-center sub-routine in [11] andthe default initialization of the K-means function in MATLAB2015b, which runs on Windows 10 with Intel-i5 3.5GHz PC and8GB RAM. The results are summarized in TABLE I.

As observed from TABLE I, the theoretical minimumMmin

obtained by the core-sets method can only be found for smallnetworks requiring only a few MBSs, e.g.,K = 80 andMmin ≤ 11 or K = 400 and Mmin ≤ 8, due to theprohibitive computational complexity of the core-sets method.In these cases, the spiral algorithm provides the near-optimalperformance in terms ofM , but is much more time-efficientthan the core-sets method. Moreover, the spiral algorithm out-performs the strip-based algorithm in terms ofM while havingcomparablet on average. Note that the gap inM between thestrip-based algorithm and the spiral algorithm becomes larger asthe ratioD/r increases. This is expected since a largerD/r ratio

TABLE I: Comparison between spiral algorithm and other schemes

K 80 400D/r 2 4 6 8 10 4 8 12 16 20

Core-sets

M 2.2 5.8 10.4 - - 7.8 - - - -t(s) 0.460 5.754 10193 - - 8004 - - - -

SpiralM 2.2 5.8 10.6 15.4 20.8 8.0 22.8 41.6 62.8 85.6t(s) 0.116 0.141 0.158 0.154 0.151 0.175 0.232 0.280 0.300 0.301

StripM 2.4 6.8 12.4 18.6 26.8 8.8 25.2 49.6 79.8 111.0t(s) 0.137 0.130 0.128 0.116 0.105 0.338 0.308 0.274 0.237 0.201

K-means

M 2.6 6.6 11.6 17.2 23.0 8.4 26.4 51.2 84.4 120.2t(s) 7.558 9.151 10.88 11.19 11.21 34.13 46.37 61.97 69.83 72.58

Ran-dom

M 3.0 8.8 17.2 26.0 35.2 10.6 36.8 75.2 116.6 162.2t(s) 0.083 0.329 1.018 1.891 3.507 1.246 14.23 39.00 87.03 122.8

means more strips in the strip-based algorithm, and consequentlylarger performance loss. Our spiral algorithm outperformsthestrip-based algorithm since each MBS is not restricted to coverGTs within each of the independent fixed strips, but instead canbe flexibly placed to reduce outlier GTs and hence the totalnumber of required MBSs. Finally, the spiral algorithm alsooutperforms the other two heuristic schemes in terms ofM andt on average for networks of different sizes.

V. CONCLUSIONS

This letter proposed a new polynomial-time successive MBSplacement solution for UAV-GT communications, termed as thespiral algorithm. The proposed algorithm is compared favorablyagainst well-known benchmark schemes in terms of the min-imum number of required MBSs to cover all GTs, includingthe optimal core-sets based algorithm but with exponentialcomplexity, the low-complexity strip-based algorithm, and twoother heuristic schemes. Future work could extend to the caseswith additional backhaul connectivity constraint betweenMBSsand adaptive MBS placement subject to moving GTs.

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