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IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. X, NO. Y, MONTH 20XX 1

Practical Heterogeneous Wireless ChargerPlacement with Obstacles

Xiaoyu Wang, Student Member, IEEE, Haipeng Dai, Member, IEEE,Weijun Wang, Jiaqi Zheng, Member, IEEE, Nan Yu, Guihai Chen, Member, IEEE,

Wanchun Dou, Member, IEEE, and Xiaobing Wu, Member, IEEE

Abstract—This paper considers the problem of practical Heterogeneous wIreless charger Placement with Obstacles (HIPO), i.e.,given a number of heterogeneous rechargeable devices distributed on a 2D plane where obstacles of arbitrary shapes exist, deployingheterogeneous chargers with a given cardinality of each type, i.e., determining their positions and orientations, the combination ofwhich we name as strategies, on the plane such that the rechargeable devices achieve maximized charging utility. After presenting ourpractical directional charging model, we first propose to use a piecewise constant function to approximate the nonlinear chargingpower, and divide the whole area into multi-feasible geometric areas in which a certain type of chargers have constant approximatedcharging power. Next, we propose the Practical Dominating Coverage Set extraction algorithm to reduce the unlimited solution space toa limited one by exacting a finite set of candidate strategies for all multi-feasible geometric areas. Finally, we prove the problem falls inthe realm of maximizing a monotone submodular function subject to a partition matroid constraint, which allows a greedy algorithm tosolve with approximation ratio of 1

2− ε. We conduct experiments to evaluate the performance. Results show that our algorithm

outperforms the comparison algorithms by at least 33.49% on average.

Index Terms—Charger placement, Heterogeneity, Obstacles.

F

1 INTRODUCTION

R ECENTLY, Wireless Power Transfer (WPT) technologyhas experienced rapid development due to its conve-

nience such as no-wiring, no-contact, and reliability. Wire-less Power Consortium, which aims to promote the stan-dardization of WPT, has grown to include 275 companiesincluding Apple and Huawei in 2017. By a recent report,there are more than 300 million commercial products basedon WPT technology in use [1].

In a WPT system, devices are typically equipped with di-rectional antennas to achieve high energy transfer efficiencyby focusing the energy in narrow energy beams. Motivatedby this fact, [2], [3] propose the directional charging modelfor which wireless chargers (or rechargeable devices) can

• This work was supported in part by the National Key R&D Program ofChina under Grant No. 2018YFB1004704, in part by the National Nat-ural Science Foundation of China under Grant No. 61872178, 61502229,61832005, 61672276, 61872173, 61802172, and 61321491, in part bythe Natural Science Foundation of Jiangsu Province under Grant No.BK20181251, in part by the Fundamental Research Funds for the CentralUniversities under Grant 021014380079, in part by the Key Research andDevelopment Project of Jiangsu Province under Grant No. BE2015154and BE2016120, and the Collaborative Innovation Center of Novel Soft-ware Technology and Industrialization, Nanjing University, in part bythe Jiangsu High-level Innovation and Entrepreneurship (Shuangchuang)Program, and in part by the Postgraduate Research & Practice InnovationProgram of Jiangsu Province No. KYCX18 0044.

• X. Wang, H. Dai, W. Wang, J. Zheng, N. Yu, G. Chen, and W.Dou are with the State Key Laboratory for Novel Software Technology,Nanjing University, China, Nanjing 210023. E-mails: {xiaoyuwang18,nanyu}@smail.nju.edu.cn, {haipengdai, gchen, douwc}@nju.edu.cn, wei-junwang [email protected], [email protected].

• X. Wu is with University of Canterbury, New Zealand. E-mail:[email protected].

Corresponding author: G. Chen (E-mail: [email protected]).

only provide (or receive) non-zero power in a sector areacalled (power) charging area (or (power) receiving area). Weargue that, however, this model is not sufficient to fullycapture the charging characteristics in practice. To be spe-cific, although it is a common sense that a device too faraway from a charger receives negligible charging power, adevice too close to a charger may also receive negligibleor zero power for the following practical reasons. First, acharger may cease to work once it detects that a deviceis in close proximity for reasons like security or efficiency.For instance, by our filed test results, the commodity off-the-shelf TX91501 wireless charger produced by Powercast[4] transmits charging power if and only if a device isat least 17 cm away under typical settings, and it has anLED status indicator to show the working status. Second, inpractical deployment, a wireless charger is usually elevatedoff the flat surface, where rechargeable devices locate at,for practical concerns such as electrical safety or largercoverage area. Thus, despite that we can project the 3Dcharging area of chargers to the flat surface for analysis,some devices below chargers may not be covered becauseof the directional charging pattern of chargers. Last but notleast, a wireless charger together with its accessories, suchas a platform it is mounted on, occupies considerable spacein reality. Therefore, after abstracting a wireless charger as apoint in theoretical analysis, it would be an appropriate (notperfect) choice to assume that a device receives zero powerwhen it is too close to a charger.

Consequently, we propose our practical directionalcharging model as shown in Figure 1, which generalizes thetraditional directional charging model [2], [3]. Specifically,the charging area of chargers which are of the same type of




is

jo

ko

is

jo

isr

jor

kor

imind

imaxd

ko

Fig. 1: Charging model withheterogeneity

io

jh

Fig. 2: Obstacles on theplane

charger si is modeled as a sector ring with distance rangesbetween dimin and dimax. Due to geometric symmetry, thepower receiving area of devices is also modeled as a sectorring within range between dimin and dimax. Besides, wefor the first time take into consideration heterogeneity ofchargers and devices as well as obstacles in wireless chargernetworks. Chargers/devices can have different parametersettings such as different types of antennas, which leads todifferent charging/receiving power and charging/receivingareas. Obstacles can be of arbitrary shapes and can blockthe line-of-sight transmitting power without reflection. Forexample, as shown in Figure 1, oj and ok denote two devicesof different types, and they have different power receivingangles due to their distinct hardware parameters. Moreover,in Figure 2, an obstacle lies in the receiving area of device oimakes chargers placed in the two shaded areas (we call thatholes) unable to charge oi.

In this paper, we consider the problem of practicalHeterogeneous wIreless charger Placement with Obstacles(HIPO). Formally, given a number of heterogeneousrechargeable devices with fixed positions and orientationsdistributed on a 2D plane where obstacles of arbitraryshapes exist, deploying heterogeneous chargers with a givencardinality of each type, i.e., determining their positions andorientations, the combinations of which we name as strate-gies, on the plane such that the devices achieve maximizedcharging utility.

The related work of this paper mainly involves wire-less charger placement, wireless sensor placement with ob-stacles, and heterogeneous wireless sensor placement net-works; but none of their solutions can be adapted to addressour problem. Specifically, the first adopts charging modelsless practical than ours, and it does not consider obstaclesor heterogeneity of chargers and devices. The second con-siders the omnidirectional sensing model and its presentedalgorithms are heuristic. The third substantially differs fromours due to its different problem settings, which makes itssolutions not applicable to our problem.

Generally, there are four main challenges in our prob-lem. First, the charging power is nonlinear with distanceand, therefore, the problem cannot be regarded as a sim-ple geometric coverage problem. Second, obstacles are as-sumed to be in arbitrary shapes, which raises challengesin evaluating not only the occupation of solution space ofchargers by obstacles, but also the caused blockage effecton transmitting power of chargers. Third, there are infinitecandidate strategies for chargers to choose, which means thesolution space is unlimited. Moreover, the non-convexityof the sector ring charging area leads to further difficultyin solution space analysis. Fourth, the heterogeneity of the

chargers and devices makes the problem more complicated.We need to enumerate all possible combinations of chargersand devices due to their distinct charging parameters, andjointly consider placing all types of chargers towards anoptimal solution.

To address these challenges, for the first one, we use apiecewise constant function to approximate the nonlinearcharging power, and accordingly divide the whole area intoseveral so-called geometric areas, such that a charger placedanywhere in a geometric area provides the same constantapproximated charging power to a given device. For thesecond challenge, we further divide the geometric areasinto feasible geometric areas by the boundaries of obstaclesand holes corresponding to devices, and thereby, a chargeranywhere in the feasible geometric areas with respect to adevice can charge the device with non-zero power. For thethird challenge, we propose the so-called Practical Dominat-ing Coverage Set (PDCS) extraction algorithm to reduce theunlimited solution space to a limited one in each feasiblegeometric area without performance loss. In particular, thePDCS extraction algorithm leverages the rotational symme-try of the sector ring charging area to deal with its non-convexity. For the last challenge, the whole area is dividedfor multiple times into several versions of feasible geometricareas (multi-feasible geometric areas) corresponding to differ-ent types of chargers independently. Then, we perform thePDCS extraction algorithm in each feasible geometric areaand obtain candidate strategies for chargers. Afterwards,the whole problem is modeled as maximizing a monotonesubmodular function subject to a partition matroid con-straint, which allows a greedy algorithm to solve with 1

2 − εapproximation ratio.

We evaluate our algorithm with simulations and field ex-periments. The results show that our algorithm outperformsthe comparison algorithms by at least 33.49%.

2 RELATED WORK

Wireless charger networks. To the best of our knowledge,all existing works regarding wireless charger networksnever consider the sector ring charging model of chargers,obstacles, or heterogeneity of chargers and/or devices, andnone of them can be applied to address our problem.Generally, there are two commonly used charging models.First, some existing related works adopt the omnidirectionalcharging model for which both of the charging area ofchargers and the power receiving area of devices are disks,i.e., regardless of the charging and receiving direction. Someworks aim to optimize the charging quality, prolong the net-work lifetime, or consider fault tolerance [5]–[15]. In addi-tion, some studies consider the wireless charger placementproblem with low electromagnetic radiation constraints forthe sake of human health. Nikoletseas et al. first proposedthe concept of low radiation efficient charging [16], and Daiet al. proposed a charger placement scheme that guaranteesradiation safety for every location on the plane [17]. Theauthors in [18]–[23] studied the safe charging problem ofscheduling power of chargers so that the radiation any-where will never exceed a given threshold on the consid-ered field. Second, the other works adopt the directionalcharging model for which the charging area of chargers




and/or the power receiving area of devices are sectors. Daiet al. investigated the problem of detecting omnidirectionalcharging as well as the omnidirectional charging probabilityfor randomly placed chargers [2]. They also consideredthe charger deployment problem for obstacle-free areas [3],[24] as well as the charging task scheduling problem [25].Moreover, we launched the first study on heterogeneouswireless charger placement problem considering sector ringcharging model and obstacles in the conference version ofthis paper [26].

Wireless sensor placement with obstacles. The closelyrelated problem of wireless sensor placement with obstaclesis essentially linear and geometric, and thus fundamentallydiffers from ours that is nonlinear. Further, most litera-tures regarding this problem adopt omnidirectional sensingmodel and have no performance guarantee, and thereforecannot be adapted to address our problem with perfor-mance requirement. Agarwal et al. studied the problem ofcovering a 2D spatial region with some occluders usingsensors, and gave a randomized algorithm [27]. In [28],[29], the authors explored the area coverage problem withpolygon obstacles. Chang et al. considered arbitrary shapedobstacles and employed a simple grid division methodfor placement [30]. Saeed et al. developed a system thatprovides visual coverage of wide and oriented targets usingcamera-mounted drones with obstacles on the plane [31]. Inparticular, this system assumes a sector ring sensing modelfor drones rather than sector model adopted by other works.

Heterogeneous wireless sensor networks. There are abunch of related works regarding heterogeneous wirelesssensor networks, but none of them are applicable to ourproblem due to their different problem settings comparedwith ours. Zhang et al. studied two-layered heterogeneoussensor networks which have better scalability and loweroverall cost than homogeneous sensor networks [32]. Lianget al. proposed a heterogeneous and hierarchical wirelesssensor network architecture [33]. Wang et al. investigated thecoverage and energy consumption control issues in mobileheterogeneous wireless sensor networks using omnidirec-tional sensing model [34]. Gupta et al. derived probabilisticexpressions to optimize the cost of random deploymentadopting the 3D heterogeneous and directional sensingmodel for sensors [35]. Guo et al. proposed the necessarycondition of the optimal sensor deployment and studiedthe dynamic sensor deployment in both homogeneous andheterogeneous wireless sensor networks with limited com-munication range for sensor nodes [36].

3 PROBLEM FORMULATION

3.1 Network Model and Charging Model with ObstaclesSuppose there are No heterogeneous directional recharge-able devices with fixed positions and orientations O ={o1, o2, ..., oNo} on the 2D plane γ. Moreover, we have Nsheterogeneous directional chargers S = {s1, s2, ..., sNs} tobe placed on the plane γ. Without confusion, we also useoi and si to represent their positions, respectively. Supposethere exist Nh static obstacles H = {h1, h2, ..., hNh

} that canbe in arbitrary shapes on γ. Any charger or device cannotbe placed inside them, and charging power cannot penetratethese obstacles or reflect from the obstacles’ surface. We also

TABLE 1: NotationsSymbol Meaning

si ith wireless charger, or its positionoi ith wireless rechargeable device, or its positionhi ith obstacle, or the set of points inside itNs Number of wireless chargers to be deployedNqs Number of q-th type wireless chargers to be deployed

No Number of rechargeable devices to chargeNh Number of obstaclesαis Charging angle of charger siαio Receiving angle of device oi~rsi Unit vector of the orientation of charger si~roi Unit vector of the orientation of device oiφis Orientation of charger siφio Orientation of device oi

Pw(.) Charging power functionP jth Power threshold of oj for charging utility function

aij , bij Constants in the charging model for si and ojdimin Nearest distance charger si can reachdimax Farthest distance charger si can reachUj(.) Charging utility function for device oj

use hi to denote the set of points inside the i-th obstacle.Some of the notations in this paper are shown in Table 1.

We establish our charging model based on our empiricalstudies and observations of real scenarios. According towork [2], [3], the charging and power receiving area can bemodeled as sectors. However, the practical scenarios showthat when a device is too close to a charger, the charger willstop to emit power for security reasons. For example, thecommodity off-the-shelf TX91501 wireless charger producedby Powercast [4] can only provide charging power withdistance at least 17 cm away when it is put at the height of14 cm from the ground. Consequently, the (power) chargingand receiving areas of chargers and devices can be modeledas sector rings in this study as shown in Figure 1. In thisfigure, the charging area of charger si with unit orientationvector ~rsi is in the shape of a sector ring with a near radiusof dimin and a far radius of dimax as well as a central angleof αis. Similarly, the device oj with unit orientation vector~roj has a power receiving area in the shape of a sector ringwith a near (far) radius of dimin (dimax) due to geometricsymmetry and a central angle of αjo. Note that the aboveparameters may vary across different chargers and differentdevices due to their heterogeneity, e.g., αko for device ok isdifferent from αjo for device oj in Figure 1. By incorporatingthe widely accepted empirical charging model proposed in[2], [3] and following our experimental results, the chargingpower from charger si to device oj considering obstacles isgiven by

Pw(si, φis, oj , φ

jo) =

aij(‖sioj‖+ bij)2

, dimin ≤ ‖sioj‖ ≤ dimax,

−−→sioj · ~rsi − ‖sioj‖ cos(αis/2) ≥ 0,

−−→ojsi · ~roj − ‖ojsi‖ cos(αjo/2) ≥ 0,

and sioj ∩ hk = ∅,∀k ∈ {1, 2, ..., Nh},0, otherwise,

(1)where φis and φjo are orientations of si and oj , respectively,aij and bij are two constants decided by charger/devicehardware and surrounding environment, ‖sioj‖ denotes thedistance between si and oj , αis and αjo denote the chargingand receiving angles, ~rsi and ~roj are the unit vectors denot-




ing the orientations of si and oj , respectively. Note that thecondition sioj ∩hk = ∅ reflects the requirement that the lineconnecting si and oj should not cross any obstacle, becauseotherwise the charging power from si could be blocked bysome obstacle and no power is received by oj .

When a device is charged by multiple chargers, weassume that the charging power is additive [2], [3], i.e.,

Pw(oj) =

Ns∑i=1


jo). (2)

3.2 Charging Utility ModelAs any device has a power saturated state, we assume thatthere is a power threshold P jth for device oj , i.e., the harvestpower by device oj must be no more than P jth regardless ofthe charging power of chargers. Accordingly, we define thecharging utility model for device oj as follows.

Uj(x) =

1

P jth· x, x ≤ P jth,

1, x > P jth,

(3)

where x denotes the received charging power by device oj .

3.3 Problem FormulationOur target is to decide the strategies of chargers to maximizethe overall charging utility for all devices on the plane γ. Byassigning a uniform weight 1

Noto the utility of each device

for normalization, HIPO can be formalized as the followingP1 problem.

(P1) maxsi,φi

s

1

No

No∑j=1

Uj(Ns∑i=1


jo)),

s.t. si ∈ γ and φis ∈ [0, 2π).

(4)

We have the following theorem to indicate the hardness ofour problem HIPO.Theorem 3.1. The HIPO problem is NP-hard.

Proof: Consider the special case in HIPO where αis =αio = 2π, dimin = 0, dimax = D (D is a positive constant)for all chargers and devices as well as there is no obstacle,so that each charger has a disk-shaped charging area witha constant radius D. Moreover, we suppose that once thedevice is covered by a charger, i.e., the device falls in thecharging disk of a charger, the charging utility for the devicebecomes 1. Therefore, each device can be seen as a point, andHIPO changes to the problem of covering most points forNopoints in the area by Ns disks with the same radius D. Thisproblem is exactly the partial disk covering problem whichis proved to be NP-complete [37]. In general, we can provethe NP-hardness of HIPO by reducing from the partial diskcovering problem.

4 SOLUTION

In this section, we propose a 12 − ε approximation algorithm

to address HIPO, which mainly contains three steps. First,as the charging power is nonlinear with distance, blocked byobstacles, and varies for heterogenous chargers or devices,given a type of chargers, we approximate the chargingpower of chargers with respect to a device by a piecewise

O d

ijwP

ijl k ijl Kijl kis

imind

imaxd

Fig. 3: Piecewise constant function approximation

constant function, and divide the whole area into severalfeasible geometric areas by considering the blockage effectof obstacles. By doing so, the approximated power at anypoint in a feasible geometric area is constant. Further, byenumerating all types of chargers, we divide the area formultiple times, and obtain the so-called multi-feasible ge-ometric areas. Second, we propose a Practical DominatingCoverage Set (PDCS) extraction method to confine the con-tinuous solution space for strategies in each feasible geomet-ric area, so that the number of candidate strategies becomeslimited. Third, we reformulate the problem into maximizinga monotone submodular optimization problem subject to apartition matroid, which allows a greedy algorithm to solvewith performance guarantee.

4.1 Area Discretization with Obstacles and Hetero-geneity of Chargers and Devices4.1.1 Piecewise Constant Function ApproximationLet P ijw (d) denote the charging power from si to oj atdistance d. We use a piecewise constant function to approx-imate the power as follows.›P ijw (d) =

P ijw (l(k)), l(k − 1) < d ≤ l(k)

(k = kij0 , kij0 + 1, ...,Kij),

0, d < dimin or d > dimax,

(5)

where l(Kij) = dimax and kij0 is a positive integer such thatl(kij0 − 1) < dimin ≤ l(k

ij0 ).

We have the following lemma to bound its error.Lemma 4.1. For charger si and device oj , setting

l(Kij) = dimax, l(k) = bij((1 + ε1)k/2 − 1), where

k = kij0 , kij0 + 1, ..., Kij − 1 (therefore Kij =⌈ln(aij/(b

2ijPw(dimax)))

ln(1+ε1)

⌉, and kij0 =

⌈2 ln(dimin/bij+1)

ln(1+ε1)

⌉), we

have the approximation error as

1 ≤ P ijw (d)›P ijw (d)≤ 1 + ε1, dimin ≤ d ≤ dimax. (6)

As shown in Figure 3, the charging area of charger si isdivided into three subareas, each in which the chargingpower is approximated as a constant as the horizontalsegments show.

4.1.2 Area DiscretizionIn this subsection, we show how to divide the whole 2Darea into multi-feasible geometric areas. We first introducegeometric area discretization and feasible geometric areadiscretization to assist understanding.




o

o

o o

h h

o

ss

o

Fig. 4: Area discretization

Geometric area discretization. First, we discuss the caseof only a single type of chargers to be placed on the planewithout obstacles. Due to geometric symmetry, if a device isfacing a charger at distance d, then the charger facing the de-vice is also at distance d. Thus, we can divide the area withthe power receiving area of devices by l(kij0 ), ..., l(Kij),and each subarea is called a geometric area. For example, asshown in Figure 4(a), the power receiving area is dividedinto 12 geometric areas for the charger type of s1 (s1 is notdrawn in this subfigure).

Feasible geometric area discretization. Then we furtherconsider the area including obstacles. If the obstacles arepositioned in the power receiving area, the area is furtherdivided by the obstacles and corresponding holes of devices,that is, there are two more cases which are infeasible toplace chargers for the specified device: the area inside theobstacles and the area in which the placed chargers cannotcover the device. We define the feasible geometric area fordevice oi as the area in which the placed chargers can providedevice oi with constant non-zero charging power. We saythe area is feasible (or infeasible) for oi to describe the factthat the placed chargers in that area can (or cannot) chargeoi with non-zero power for simple. Moreover, it should benoticed that the infeasible area for oi may be feasible for oj ,thus the feasible geometric area discretizing for the wholearea also requires the boundaries of holes and obstacles. Asshown in Figure 4(b), the power receiving area of o1 ando2 is further divided by the boundaries of h1 and holes ofthe devices based on Figure 4(a) according to the chargertype of s1. Since h1 completely shields the further powerreceiving area of o1 and o2, these geometric areas can beignored. Thus, we get only six feasible geometric areas intotal for simple. Moreover, the feasible geometric area 3 is ahole of device o1. It should be considered as there may existpower receiving areas of other devices cover it. Charger s1can provide device o1 with power Pw(l(k110 + 1)).

Multi-feasible geometric area discretization. Next, weconsider the heterogeneity of chargers. Clearly, the areadiscretization varies for different types of chargers. Sincethe charging power is linearly additive, we can dividethe area into feasible geometric area for several times bydifferent charging parameters, and consider the strategiesin each feasible geometric area independently. Thus, multi-feasible geometric area discretization is to discretize the area toget feasible geometric areas for several versions based ondifferent parameters of heterogeneous chargers. Figure 4(c)shows another division of receiving area for the charger typeof s2 different from 4(b). In this case, s2 can provide o1 withpower Pw(l(k210 )).

We have the following lemmas for multi-feasible geo-metric area discretization.Lemma 4.2. Let ›Pw(oj) denote the approximated charging

power received by device oj in its multi-feasible geo-metric areas. Then,›Pw(oj) = 0 if Pw(oj) = 0; otherwise,the approximation error is

1 ≤ Pw(oj)

Pw(oj)≤ 1 + ε1. (7)

Proof: According to Lemma 4.1, 1 ≤ Pw(oj)

Pw(oj)=∑Ns

i=1 Pw(si,φis,oj ,φ

jo)∑Ns


jo)≤ (1+ε1)

∑Nsi=1 Pw(si,φ

is,oj ,φ

jo)∑Ns


jo)

≤ 1 + ε1.

Lemma 4.3. Let Uj(x) denote the utility function for deviceoj as Equation (3) shows. Then, Uj(›Pw(oj)) = 0 ifUj(Pw(oj)) = 0; otherwise, the approximation error is

1 ≤ Uj(Pw(oj))Uj(Pw(oj))

≤ 1 + ε1. (8)

Proof: Since Pw(oj) ≥ ›Pw(oj), there are only threecases to be considered:

1) ›Pw(oj) ≤ Pw(oj) ≤ P jth;2) ›Pw(oj) ≤ P jth ≤ Pw(oj);3) P jth ≤›Pw(oj) ≤ Pw(oj).

For Case 1), Uj(Pw(oj)) = Pw(oj)/Pjth and Uj(›Pw(oj)) =›Pw(oj)/P jth. It is obvious that the conclusion stands accord-

ing to Lemma 4.2. For Case 2), Uj(›Pw(oj)) = ›Pw(oj)/P jthand Uj(Pw(oj)) = 1 ≤ Pw(oj)/P

jth, thus, the conclusion

still stands. For Case 3), Uj(›Pw(oj)) = Uj(Pw(oj)) = 1, sothe conclusion also comes. In all, the conclusion stands.

Suppose that each obstacle can be expressed by a poly-gon with no more than c edges and we have the followinglemma to describe the number of feasible geometric areas.Lemma 4.4. The number of feasible geometric areas for

each type of chargers is O(No2ε−21 Nh

2c2), where c is themaximum number of edges of these obstacles.

Proof: We consider a relaxed bound which also in-cludes the number of divided infeasible areas. In general,there are O(ε−11 ) disjoint sector rings in the power receivingarea of one device. Suppose the worst case that all theobstacles are in the power receiving area of one device,and it generates many holes. Since the holes are generatedby connecting the device and the vertices of the obstacles,we connect the device with all the vertices and extend theline to intersect with the farthest boundary of the powerreceiving area. Thus, the power receiving area is dividedinto O(ε−11 (1 + Nhc)) sector rings. Moreover, there areNo devices, which generates O(Noε

−11 (1 + Nhc)) areas in

total. Plus considering Nh obstacles, the total number ofshapes to intersect is O(Noε

−11 (1 + Nhc) + Nh). There-

fore, the number of the divided subareas is squared, i.e.,O((Noε

−11 (1 +Nhc) +Nh)

2), and this can be simplified as

O(No2ε−21 Nh

2c2).Note that if the boundaries of obstacles are continuous

curves rather than segments, our algorithm still works andits achieved approximation ratio remains, but its time com-plexity is no longer bounded. As this is out of the focus ofthis paper, we omit it here to save space.




i 1o2o3o4o

5o 6o

i1o2o

3o4o

5o 6o

i 1o2o

3o4o

5o 6o

i 1o2o3

o4o

5o 6o

(a) (b) (c) (d)

Fig. 5: A toy example of point case

4.2 Practical Dominating Coverage Set (PDCS) Extrac-tionAfter area discretization, the whole 2D area is divided intomulti-feasible geometric areas where the placed chargersemit constant approximated charging power to devices any-where in each feasible geometric area. Thus, we only needto consider strategies in each feasible geometric area. In thissubsection, we propose the Practical Dominating CoverageSet (PDCS) extraction algorithm to extract strategies in eachfeasible geometric area. Essentially, the PDCS extractionalgorithm exploits the rotational symmetry of the sectorring charging area to address its non-convexity. Note thatwe perform the algorithm multiple times for multi-feasiblegeometric areas.

4.2.1 PreliminariesWe first give some definitions.Definition 4.1. Dominance: Suppose there are two strategies〈si, φis〉 and 〈sj , φjs〉 of the same type of chargers, andtheir corresponding covered device sets are Oi and Oj ,respectively. If Oi ⊂ Oj , we say 〈sj , φjs〉 dominates〈si, φis〉.

Definition 4.2. Practical Dominating Coverage Set: Supposethere is a strategy 〈si, φis〉 with covered device set Oi. Ifthere doesn’t exist a covered set Oj with strategy 〈sj , φjs〉such that 〈sj , φjs〉 dominates 〈si, φis〉, then we say Oi is aPractical Dominating Coverage Set (PDCS).

Definition 4.3. Candidate Covered Set of Devices: Thedevices in the Candidate Covered Set of Devices O ofsubarea γi are those devices which can be charged bychargers located in γi with non-zero power.

4.2.2 PDCS Extraction for Point CaseFirst, we consider a special case where a feasible geometricarea γi is reduced to a point which is still denoted by γi.Algorithm 1 gives the details of the algorithm, and its basicidea is to rotate the charger at point γi for 360◦ and extractthe PDCSs. Figure 5 shows a toy example for Algorithm1. First, the charger is initialized at the position γi withorientation 0◦. We rotate the charger anticlockwise and addo2, o3 in the charging area with o1 going to fall out, andobtain the PDCS {o1, o2, o3} as shown in Figure 5(b). Next,continue to rotate the charger and add new devices into thecharging area. Rotate it until a device, say o3, is going to fallout, and obtain the PDCS {o3, o4} as shown in Figure 5(c).Repeat the above operations and we can get the covereddevice set {o5, o6} in Figure 5(d) as a PDCS. When thecharger has rotated for 360◦, this operation terminates.

Algorithm 1: PDCS Extraction for Point CaseInput: Reduced point γi and its candidate covered set

of devices OOutput: PDCSs and their corresponding strategies

1 Place a charger at the point γi and compute the anglebetween the line connecting the charger and eachcandidate device and 0◦ orientation. Sort the devicesby their angles.

2 Initialize the orientation of the charger θ = 0◦.3 while θ < 360◦ do4 Rotate the charger anticlockwise until a device is

going to fall out of the charging area, and add thePDCS and corresponding strategy into thecandidate solution set.

5 Rotate the charger anticlockwise until a device isadded to the charging area.

(a) (b) (c) (d)

1o2o5o

4o 6o3o

i i

1o2o 4o

3o

i

5o6o

i

22 , ss

11, ss

33 , ss

44 , ss

Fig. 6: A toy example of area case

4.2.3 PDCS Extraction for Area Case

Next, we discuss the general area case, and show thealgorithm in Algorithm 2. See Figure 6, suppose thereare six devices in the candidate covered set of devices offeasible geometric area γi, which are classified into threetypes: {o1, o2}, {o3, o4}, and {o5, o6}. First, we draw astraight line through each pair of devices, say o1 and o2,and put a charger at the intersection points of the feasiblegeometric area boundaries with the charger’s clockwiseboundary crossing the two devices as shown in Figure 6(b),and thus obtain two candidate PDCSs {o1, o2, o3, o6} and{o1, o2, o3, o4, o5, o6} as well as their corresponding strate-gies 〈s1, φ1s〉 and 〈s2, φ2s〉. Second, we draw arcs througheach pair of devices, say o3 and o4, with circumferentialangle being the charging angle of the current type of charg-ers, say αks , and put a charger at the intersection points ofthe feasible geometric area boundaries with the charger’stwo line boundaries crossing the two devices, respectively,as shown in Figure 6(c). We then obtain candidate PDCS{o3, o4, o5, o6} and strategies 〈s3, φ3s〉 and 〈s4, φ4s〉. Note thatthe gray charging area in all the subfigures in Figure 6 is




Algorithm 2: PDCS Extraction for Area CaseInput: Feasible geometric area γi and its candidate

covered set of devices OOutput: PDCSs and their corresponding strategies

1 for all pairs of devices in O, say oi and oj do2 Draw a straight line crossing oi and oj , and intersect

the boundaries of the feasible geometric area.3 Put the charger at the intersection point, and let the

clockwise boundary cross oi and oj .4 Add the PDCS and the corresponding strategy

under this setting into the solution set.5 Draw arcs crossing oi and oj with circumferential

angle αks , and intersect the boundaries of thefeasible geometric area.

6 Put the charger at the intersection point, and let thetwo line segment boundaries cross oi and oj .

7 Add the PDCS and the corresponding strategyunder this setting into the solution set.

8 Select a point on the boundaries of the feasiblegeometric area randomly and perform PDCSextraction algorithm for point case in Algorithm 1.

9 Filter the PDCSs and remove the subsets and theircorresponding strategies.

only for the current discussed type of devices, thus theremay exist devices in other types like o5 and o6 which arenot included in the gray charging area in Figure 6(c) but canstill be charged. Third, we randomly select a point on theboundary of the feasible geometric area and perform PDCSextraction algorithm for point case, as shown in Figure 6(d).Finally, we check all the obtained candidate PDCSs and theircorresponding strategies, and only retain the true PDCSs bycomparing them and their corresponding strategies. In thisexample, we reserve {o1, o2, o3, o4, o5, o6}.

Next, we define a transformation of the strategy.Definition 4.4. Projection: Keep the orientation of a strategy〈si, φis〉 fixed and move the strategy’s position along thereverse direction of its orientation until it reaches theboundary of the current feasible geometric area.

The projection operation is shown in Figure 7(a). We can seethat after projection, the strategy can cover not only o1 buto2 and o3 as well. Generally, we have the following lemma.Lemma 4.5. If 〈sj , φjs〉 is the projection of 〈si, φis〉, then〈sj , φjs〉must either dominate or be equivalent to 〈si, φis〉.

By Lemma 4.5, we have the following corollary.Corollary 4.1. Considering PDCSs with corresponding

strategies on the boundaries of a feasible geometric areais equivalent to considering that in the whole area.

Further, let Γ denote the output set of strategies of Algo-rithm 2. We have the following theorem.Theorem 4.1. Given any strategy 〈si, φis〉, there must exist a

strategy 〈sj , φjs〉 ∈ Γ such that 〈sj , φjs〉 either dominatesor is equivalent to 〈si, φis〉.

Proof: For an arbitrary strategy 〈si, φis〉 in a feasiblegeometric area, we do the following three operations andobtain a new strategy 〈sj , φjs〉:

1) Perform the projection transformation until thestrategy’s position reaches the boundary of the area.

2) Keep the strategy’s position fixed and rotate itsorientation anticlockwise until a device is going tofall out of the clockwise boundary of the strategy’scharging area.

3) Keep the clockwise boundary of the charger cross-ing the device which is going to fall out andmove the charger along the feasible geometric area’sboundaries until another device is going to fall outof the charging area. If no other device is going tofall out, then stop the operation.

It is obvious that 〈sj , φjs〉 either dominates or is equiva-lent to 〈si, φis〉 according to Corollary 4.1.

Next, we will prove that the solution set obtained byAlgorithm 2 corresponds to the set after the above threeoperations. The cases of the covered device set and corre-sponding strategy include five critical conditions after thosethree operations:

1) Another device touches the clockwise boundary ofthe charging area (Figure 7(b)).

2) Another device touches the anticlockwise boundaryof the charging area (Figure 7(c)).

3) Another device touches the arc with distance diminto the charger (Figure 7(d)).

4) Another device touches the arc with distance dimaxto the charger (Figure 7(e)).

5) None of the other devices touches the boundary ofthe charging area (Figure 7(f)).

Cases 1) and 2) are those cases that the device is going tofall out of the charging area, while the device in the leftthree cases will never fall out. Cases 3) and 4) are not criticalconditions that the device is going to fall out, since if thedevice falls out, the feasible geometric area must be furtherdivided into more feasible geometric areas. As shown inFigure 8(a) and (b), the devices o3 and o4 fall out of thecharging area by the nearest arc boundary and the farthestarc boundary, respectively, when the charger moves on theboundaries of feasible geometric area γi. We show that thefeasible geometric area γi can be divided into two feasiblegeometric areas by the arc centering at o3 and o4 with radiusdimin and dimax, say ıA1 and ıA2, respectively.

We can see that Step 2-4 and 5-7 in Algorithm 2 cor-respond to the first and the second cases, respectively. Andfor the last three cases, arbitrary positions on the boundariesof the feasible geometric area are equivalent, thus, leads toStep 8 of the PDCS extraction algorithm for point case inAlgorithm 2. Therefore, the set of strategies obtained byAlgorithm 2 is just the set of 〈sj , φjs〉. Since 〈sj , φjs〉 eitherdominates or is equivalent to 〈si, φis〉, the result follows.

4.3 Problem Reformulation

After performing Algorithm 2 for several times for multi-feasible geometric areas, we obtain the PDCSs and theircorresponding strategies. For each strategy, we can computethe charging power and charging utility for each device.Let Γ denote the strategy set, Γq denote the strategy setof q-th type of chargers, and xi be the indicator whichdenotes whether the i-th strategy is selected. The problem




1o2o

3o

(a)

i

(b)

1o2o

3o

i

(c)

1o2o

i

3o

(d)

i

1o2o3o

(f)

i

1o2o

3o

(e)

i

1o2o

3o

Fig. 7: Critical conditions

(a)

1o2o

3o

i1A

(

(b)

1o

2o

4o

i2A

(

Fig. 8: Arc explanation

P1 in Section 3.3 can be reformulated as the combinatorialproblem as follows:

(P2) max1

No

No∑j=1

Uj(∑

〈si,φis〉∈Γ

xiPw(si, φis, oj , φ

jo)),

s.t.∑

〈si,φis〉∈Γq

xi = Nqs , q = 1, ...,Q, xi ∈ {0, 1}.

(9)

We will further reformulate the problem and obtain the finalsolution with a constant performance guarantee. First, wegive some definitions.Definition 4.5. [38] Monotone submodular set function:

Let S be a finite ground set. A real-valued set functionf : 2S → R is normalized, monotonic and submodu-lar if and only if it satisfies the following conditions,respectively: (1) f(∅) = 0; (2) f(A ∪ {e}) − f(A) ≥ 0for any A ⊆ S and e ∈ S\A; (3) f(A ∪ {e}) − f(A) ≥f(B ∪ {e})− f(B) for any A ⊆ B ⊆ S and e ∈ S\B.

Definition 4.6. [38] Matroid: A Matroid M is a strategyM = (S,L) where S is a finite ground set, L ⊆ 2S isa collection of independent sets, such that: (1) ∅ ∈ L;(2) if X ⊆ Y ∈ L, then X ∈ L; (3) if X,Y ∈ L, and|X| < |Y |, then ∃y ∈ Y \X , X ∪ {y} ∈ L.

Definition 4.7. [38] Partition matroid: Given S =⋃ki=1 S

′i

is the disjoint union of k sets, l1, l2, . . . , lk are positiveintegers, a partition matroid M = (S, I) is a matroidwhere I = {X ⊂ S : |X ∩ S′i| ≤ li for i ∈ [k]}.Generally, the obtained strategy set Γ by Algorithm 2

applying to multi-feasible geometric areas can be defined asthe disjoint union of the Q strategy sets of different types ofchargers, i.e., Γ =

⋃Qq=1 Γq , and thereby, define the partition

matroidM = (Γ, I) with I = {X ⊂ Γ : |X ∩ Γq| ≤ Nqs for

i ∈ [Q]}. Based on these definitions, problem P2 in Equation(9) can be rewritten as

(P3) maxsi,φi

s

f(X) =1

No

No∑j=1

Uj(∑

〈si,φis〉∈X


jo)),

s.t. si ∈ γ, φis ∈ [0, 2π),

X ∈ L,L = {X ⊆ Γ : |X ∩ Γq| ≤ Nq

s }. (10)

For the problem P3 shown in Equation (10), we have thefollowing critical lemma.

Lemma 4.6. The objective function f(X) in Equation (10) isa monotone submodular function, and the constraint isa partition matroid constraint.

Proof: To check whether f(X) in Equation (10) is amonotone submodular function, we can check whether itsatisfies the three requirements in Definition 4.5. First, itis obvious that f(X) = 0 when X = ∅ since there isno charger to provide charging utility. Second, it is alsoclear that when a new strategy is selected, the chargingutility increases since the charging utility function Uj(·)defined in Equation (3) is non-decreasing. Third, we de-fine g(X, j) =

∑〈si,φi

s〉∈X›Pw(si, φis, oj , φjo) and obviously

g(·, j) is non-decreasing since the charging power is non-decreasing with more chargers. Thus, for strategy setsA ⊆ B ⊆ Γ and a strategy e ∈ Γ\B, it can be seen thatg(A, j) ≤ {g(A ∪ {e}, j), g(B, j)} ≤ g(B ∪ {e}, j). As aresult, we have

[Uj(g(A ∪ {e}, j))− Uj(g(A, j))]− [Uj(g(B ∪ {e}, j))− Uj(g(B, j))] ≥ 0,

(11)

since for any 0 ≤ x1 ≤ x2 and ∆x ≥ 0,

[Uj(x1 +∆x)− Uj(x1)]− [Uj(x2 +∆x)− Uj(x2)] ≥ 0, (12)

and

g(A ∪ {e}, j)− g(A, j) = g(B ∪ {e}, j)− g(B, j) = g({e}, j).(13)

Therefore, we have

[f(A ∪ {e})− f(A)]− [f(B ∪ {e})− f(B)]

=1

No

No∑j=1

{[Uj(g(A ∪ {e}, j))− Uj(g(A, j))]

− [Uj(g(B ∪ {e}, j))− Uj(g(B, j))]}≥0.

(14)

Moreover, it is obvious that the constraint in Equation (10)is a partition matroid constraint.

Thus, we use the algorithm described in Algorithm 3 toselect strategies for heterogeneous chargers. Algorithm 3 isessentially a greedy algorithm that goes through all typesof chargers, and greedily selects the strategy that leads tomaximum charging utility increment on global.

Theorem 4.2. Setting ε1 = 2ε1−2ε , our algorithm to HIPO

achieves an approximation ratio of 12 − ε and its time

complexity is O(NsNo4ε−2Nh

2c2).

Proof: First, it is proved that the greedy algorithmsolves the monotone submodular function with partition




Algorithm 3: Strategy Selection for HeterogeneousChargers

Input: Number of chargers Nqs for the q-th type,

candidate strategy set Γq for the q-th type,objective function f(X)

Output: Selected strategy set Xq (1 ≤ q ≤ Q)1 Xq = ∅ (1 ≤ q ≤ Q).2 for all q ∈ [Q] do3 while |Xq| ≤ Nq

s do4 X =

⋃Qq=1Xq .

5 e∗ = arg maxe∈Γq\Xq

f(X ∪ {e})− f(X).6 Xq = Xq ∪ {e∗}.

matroid constraint with 12 -approximation ratio in [38]. Tak-

ing the utility approximation in Lemma 4.3 into consider-ation, the total approximation ratio is 1

2(1+ε1)= 1

2 − ε bysetting ε1 = 2ε

1−2ε .Then, according to Algorithm 2, we should enumerate

every pair of devices in each feasible geometric area, ofwhich the number isO(N2

o ). Thus, there areO(N4o ε−21 N2

hc2)

strategies in the candidate solution set, according to thenumber of feasible geometric areas in Lemma 4.4. More-over, Algorithm 3 requires O(Ns) iterations for all theelements in the candidate solution set, so the time com-plexity is O(NsNo

4ε−21 Nh2c2). Since ε1 and ε are equiv-

alent infinitesimals, the final result of time complexity isO(NsNo

4ε−2Nh2c2).

Note that we can improve the approximation ratio from12−ε to 1−1/e−ε by adopting the algorithm in [39], which is,however, too computationally demanding to use in practice.

5 DISTRIBUTED ALGORITHM FOR PDCSS EX-TRACTION

In this section, we consider the distributed algorithm forPDCS extraction. The detailed algorithm for a single taskis described in Algorithm 4. The basic idea is to divideextracting PDCSs corresponding to each set of neighboringdevices into several independent tasks. We first calculatethe set of devices for each device within distance 2dkmaxas the neighboring device set in terms of the charger typeof sk. Then, for each set, Algorithm 2 can be conductedindependently for different tasks. As shown in Figure 9,for device o1, we calculate its neighboring device set, say{o2, o3, o4, o5, o6}, and draw the line and arcs through eachpair of devices including o1. For example, we draw a lineand two arcs with circumferential angle αks through o1 ando2, and intersect with the feasible geometric area boundariesgenerated by the common neighbors and related obstacles.To avoid repeated calculating, these tasks only conductAlgorithm 2 on the devices with larger indices j > i inthe neighboring device set Oki as Algorithm 4 shows.

Moreover, we use Longest Processing Time (LPT) al-gorithm [40] to assign the tasks to different machinessince the time span of tasks varies, which achieves 4/3-approximation ratio to minimize the longest time span ofmachines. The final distributed algorithm is described inAlgorithm 5.

Algorithm 4: PDCS Extraction for Single Neighbor-ing Device Set

Input: The index of device to be computed i,parameters of devices, and the charger type of sk

Output: PDCSs and corresponding strategies1 Compute the neighboring device set Oki of device oi in

terms of the charger type of sk.2 for each device in Oki with index j > i, say oj do3 Draw a straight line crossing oi and oj , and

intersect the boundaries of feasible geometric areagenerated by neighboring devices and relatedobstacles.

4 Put the charger at the intersection point, and let theclockwise boundary cross oi and oj .

5 Add the PDCS and the corresponding strategyunder this setting into the candidate solution set.

6 Draw arcs crossing oi and oj with circumferentialangle αks , and intersect the boundaries of thefeasible geometric area generated by neighboringdevices and related obstacles.

7 Put the charger at the intersection point, and let thetwo line segment boundaries cross oi and oj .

8 Add the PDCS and the corresponding strategyunder this setting into the candidate solution set.

9 Perform PDCS extraction algorithm for point casein Algorithm 1 at the intersection points of theapproximated power receiving area of oi and oj . Ifthe point is not in the feasible geometric area,ignore it.

10 Perform PDCS extraction algorithm for point case inAlgorithm 1 at the intersection points of theapproximated power receiving area of oi and therelated obstacles and holes.

11 Filter the PDCSs and remove the subsets and theircorresponding strategies.

kmaxd

oo

o o

o

o

Fig. 9: Distributed PDCS extraction

One may argue that why not establish the distributedalgorithm on the original feasible geometric areas. This isbecause for programming, it is hard to obtain the feasiblegeometric areas and it consumes much more time complex-ity. Moreover, there might be a large number of feasiblegeometric areas, which makes it challenging and almostimpossible to process all the information about the feasiblegeometric areas.

6 SIMULATION RESULTS

In this section, we conduct simulation experiments to eval-uate our algorithm.

In the simulations, we randomly distributed some het-erogeneous devices of four types on a 40m × 40m squarearea where two obstacles exist as shown in Figure 10(a).Moreover, there are three types of chargers for placement




Algorithm 5: Distributed HIPOInput: The number of parallel machines n, all the

parameters of chargers and devices, objectivefunction f(X)

Output: PDCSs and corresponding strategies1 if n ≥ No then2 Assign the task (Algorithm 4) with device index i

and all the charger types to parallel machine i.

3 else4 Apply LPT algorithm [40] to assign tasks to

different parallel machines.

5 Execute Algorithm 3 when all the parallel machineshave done their tasks.

TABLE 2: Default charger parametersCharger type 1 Charger type 2 Charger type 3

αis π/6 π/3 π/2dimin 5 3 2dimax 10 8 6

whose detailed hardware parameters are shown in Table 2-4. The initial number of chargers are one, two, and threefor charger type 1, 2, and 3, respectively, while that ofdevices are four, three, two, and one for device type 1, 2,3, and 4. The default setting for charger number is threetimes of initial setting while that for device number is fourtimes of initial setting. The default value of P jth for alldevices and ε are set to 0.05 and 0.15, respectively. Notethat if the randomly generated position happens to be insidean obstacle and is thus infeasible, we repeat the processuntil a feasible position is obtained. Besides, each point inthe evaluation figures indicates the average value of 100experiments of random device topologies.

As there is no existing algorithm for our consideredproblem, we propose eight algorithms for comparison asfollows. Randomized Position with Angular Randomization(RPAR) randomly generates charger positions and orien-tations. Randomized Position with Angular Discretization(RPAD) improves RPAR by enumerating the orientationof a charger on each position with an angle of value in0, αis, ..., and (d2π/αise − 1)αis. Grid Point with AngularRandomization (GPAR) and Grid Point with Angular Dis-cretization (GPAD) improves RPAR and RPAD, respectively,by placing chargers on grid points, and Grid Point withPractical Dominating Coverage Set Extraction for point case(GPPDCS) further improves GPAD by replacing the aboveorientation selection method by our practical dominatingcoverage set extraction algorithm for point case. Further,each of the above three algorithms has two versions: trianglegrid points (GPAR Triangle, GPAD Triangle, GPPDCS Tri-angle) and square grid points (GPAR Square, GPAD Square,GPPDCS Square), both with grid length

√2/2·dimax for each

charger with specified charging radius dimax.

6.1 Performance Comparison6.1.1 Instance IllustrationWe show the solutions for all algorithms for an instancewith sensors shown in Figure 10(b). We set the numberof chargers four times of the initial setting, that is, thereare 12, 8, and 4 chargers for type 1, 2, and 3, respectively.The charging utility for our algorithm is 0.8495, while

the others are 0.1000, 0.4046, 0.4605, 0.4867, 0.6006, 0.6191,0.6348, and 0.6932 for RPAR, RPAD, GPAR Square, GPARTriangle, GPAD Square, GPAD Triangle, GPPDCS Square,and GPPDCS Triangle, respectively. We can see that theplacement of our algorithm can charge all the devices whileothers cannot. In particular, for the RPAR solution as shownin Figure 10(d), only few devices are charged due to therandomness of positions and orientations, and many strate-gies share the same positions and orientations to charge thelimited covered devices. In the RPAD solution shown inFigure 10(e), the result improves due to the enumerationand selection of orientations, but still almost half of thedevices are not charged. The algorithms based on gridpoints perform better, as shown in Figure 10(f)-10(k), butmost of the devices in the right down corner are not charged.

6.1.2 Impact of Number of Chargers NsOur simulation results show that on average, HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,GPAD Square, GPAR Triangle, GPAR Square, RPAD, and RPARby 33.49%, 38.32%, 43.43%, 47.65%, 116.60%, 144.15%,166.85%, and 970.37%, respectively, in terms ofNs. Figure 11(a)shows that the charging utility increases monotonically withNs. The charging utility of our algorithm first increases ata high rate and becomes almost 1 when Ns is five times ofinitial setting; then, the increasing rate tends to be gentle,while that of comparison algorithms still remain low. Incontrast, the charging utilities of the comparison algorithmsare limited because the positions or the orientations ofchargers are predetermined or randomly generated for thesealgorithms. Note that RPAD increases at a relatively higherrate and even performs better than the two GPAR algo-rithms when the number of chargers is larger than five timesof the default setting. It is because there are more betterchoices of charger orientations in RPAD when number ofposition choices increases, while GPAR may generate trivialorientations with low charging utility.

6.1.3 Impact of Number of Devices NoOur simulation results show that on average, HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,GPAD Square, GPAR Triangle, GPAR Square, RPAD, and RPARby 37.13%, 42.84%, 49.87%, 55.50%, 124.50%, 141.66%,197.85%, and 1106.68%, respectively, in terms of No. It canbe seen from Figure 11(b) that the charging utility monoton-ically decreases with the number of devices. Our algorithmperforms well when the numbers of devices are one and twotimes of the initial setting, but degrades relatively fast whenthe number of devices becomes larger. The charging utilitiesof the four grid points based algorithms with orientationsselected decrease at nearly the same rate with the GPPDCSalgorithm gaining a bit higher utility, while the two GPARalgorithms and RPAD algorithm gain low charging utilitywith relatively slower decreasing rate. As a charger is ex-pected to cover more devices when devices become dense,the charging utility decreases more slowly when the numberof devices becomes larger.

6.1.4 Impact of Charging Angle αisOur simulation results show that on average HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,




TABLE 3: Default device parametersDevice type 1 Device type 2

αio π/2 2π/3Device type 3 Device type 4

αio 3π/4 π

TABLE 4: Correlated parametersDevice type 1 Device type 2 Device type 3 Device type 4

Charger type 1 a = 100, b = 40 a = 130, b = 52 a = 160, b = 64 a = 190, b = 76Charger type 2 a = 110, b = 44 a = 140, b = 56 a = 170, b = 68 a = 200, b = 80Charger type 3 a = 120, b = 48 a = 150, b = 60 a = 180, b = 72 a = 210, b = 84

(a) Simulation scenario

0 10 20 30 40x (m)

0

10

20

30

40

y (m

)

(b) Sensors instance

0 10 20 30 40 x (m)

0

10

20

30

40

y (m

)

(c) HIPO instance

0 10 20 30 40x (m)

0

10

20

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y (m

)

(d) RPAR instance

0 10 20 30 40 x (m)

0

10

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y (m

)

(e) RPAD instance

0 10 20 30 40x (m)

0

10

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y (m

)

(f) GPAR Square instance

0 10 20 30 40x (m)

0

10

20

30

40

y (m

)

(g) GPAR Triangle instance

0 10 20 30 40 x (m)

0

10

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30

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y (m

)

(h) GPAD Square instance

0 10 20 30 40 x (m)

0

10

20

30

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y (m

)

(i) GPAD Triangle instance

0 10 20 30 40 x (m)

0

10

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30

40

y (m

)

(j) GPPDCS Square instance

0 10 20 30 40 x (m)

0

10

20

30

40

y (m

)

(k) GPPDCS Triangle instance

Fig. 10: Instances

GPAD Square, GPAR Triangle, GPAR Square, RPAD, and RPARby 38.54%, 42.64%, 51.86%, 55.94%, 109.53%, 124.91%,198.37%, and 997.82%, respectively, in terms of αis. Figure 11(c)shows that the charging utility increases slowly with charg-ing angle, while RPAD remains relatively stable. For ouralgorithm and grid points based algorithms, chargers withlarger charging angles generally cover more devices, whileRPAD may select relatively bad positions around whichthere are only a few devices. Moreover, the influence ofcharger orientations decreases with larger charging angle, sothe performance of algorithms with angular randomizationis approaching that of according algorithms with angulardiscretization. Our algorithm always gains much highercharging utility than the other algorithms.

6.1.5 Impact of Receiving Angle αioOur simulation results show that on average, HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,GPAD Square, GPAR Triangle, GPAR Square, RPAD, andRPAR by 33.03%, 36.59%, 45.72%, 49.85%, 110.05%, 25.88%,189.07%, and 1016.13%, respectively, in terms of αio. Figure11(d) shows the trend of charging utility with receivingangles of devices. The charging utilities of all the algorithmsincrease when receiving angles of devices becomes larger.

6.1.6 Impact of Power Threshold P jthOur simulation results show that on average, HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,GPAD Square, GPAR Triangle, GPAR Square, RPAD, and RPARby 36.21%, 39.73%, 50.24%, 55.33%, 111.64%, 131.15%,192.40%, and 1089.49%, respectively, in terms of P jth. Figure




PDCS GPPDCS Triangle GPPDCS Square GPAD Triangle

GPAD Square GPAR Triangle GPAR Square RPAD RPAR

1 2 3 4 5 6 7 8Number of Chargers (Times)

0

0.2

0.4

0.6

0.8

1C

harg

ing

Util

ity

(a) Ns vs. charging utility

1 2 3 4 5 6 7 8Number of Devices (Times)

0

0.2

0.4

0.6

0.8

1

Cha

rgin

g U

tility

(b) No vs. charging utility

0.6 0.8 1 1.2 1.4 1.6 1.8 2Charging Angle (Times)

0

0.2

0.4

0.6

0.8

1

Cha

rgin

g U

tility

(c) αis vs. charging utility

0.6 0.8 1 1.2 1.4 1.6 1.8 2Receiving Angle (Times)

0

0.2

0.4

0.6

0.8

1

Cha

rgin

g U

tility

(d) αio vs. charging utility

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Power Threshold

0

0.2

0.4

0.6

0.8

1

Cha

rgin

g U

tility

(e) P jth vs. charging utility

0 0.2 0.4 0.6 0.8 1 1.2 1.4(Times)

0

0.2

0.4

0.6

0.8

1

Cha

rgin

g U

tility

dimin

(f) dimin vs. charging utility

Fig. 11: Simulations

11(e) demonstrates that the charging utility first remainsstable, then gradually decreases when P jth becomes larger.The reason is that with a higher value of P jth, all the algo-rithms need to select more chargers to charge a device. Ouralgorithm performs much better than the other algorithms.

6.1.7 Impact of Nearest Distance diminOur simulation results show that on average, HIPO outper-forms GPPDCS Triangle, GPPDCS Square, GPAD Triangle,GPAD Square, GPAR Triangle, GPAR Square, RPAD, and RPARby 40.38%, 43.93%, 53.65%, 58.12%, 117.69%, 136.21%,188.26%, and 1024.88%, respectively, in terms of dimin. The val-ues of x-axis in Figure 11(f) means the times of the originalsetting of dimin. It shows that the charging utility graduallydecreases when dimin becomes larger, since the charging areabecomes smaller. Moreover, the charging utility decreasesfaster when dimin is larger, since the charging area decreasesmore when dimin becomes larger. Still, our HIPO algorithmoutperforms the other algorithms.

6.1.8 Impact of Number of Devices No and Number ofParallel Machines on Time ConsumptionOur simulation results show that on average, 5-distributed,10-distributed, 15-distributed, 20-distributed, and 25-distributedreduce the time consumption by 80.10%, 88.79%, 91.05%,92.32%, and 92.39%, respectively, in terms of No. We conductthe simulation on time consumption comparison betweennon-distributed and distributed HIPO algorithm. Figure 12shows the results, in which we plot data with logarithmicscale for the y-axis to show the differences between thesesettings more clearly. All the values of time consump-tion are divided by the value of non-distributed at onetime of number of devices to eliminate the influence ofdifferent platforms. We just show the time consumptionof the parallel-processing part. We can see that the dis-tributed algorithm consumes much less time than the non-

distributed algorithm, and the larger the number of devices,the more reduced time consumption. Note that accordingto Algorithm 5, the time consumption will not continue toreduce when the number of machines becomes no smallerthan that of devices. We can see that in Figure 12, whenthere are more machines, the time consumption reducesmore slowly or even does not reduce since the numberof machines is approaching the number of devices, thus,the time consumption is approaching the time span of thelongest task.

6.1.9 Impact of Different Power Thresholds P jths

Our simulation results show that in HIPO, the changing trendsof charging utility are almost the same with different powerthresholds for different types of devices, and the difference is3.20% on average, in terms of No. In Figure 13, the legendmeans the power threshold difference of each two adjacentdevice types, and we always keep the power threshold ofdevice type 2 as 0.05. For example, the legend −0.01 meansthat the power thresholds for device types 1-4 are 0.06,0.05, 0.04, and 0.03, respectively. We also change the defaultnumber of devices as the same number 2 for all four typesof devices to better show the impact of different powerthresholds. The x-axis in Figure 13 shows the multiple ofthe default number of devices. As shown in Figure 13, thecharging utility of all different settings of power thresholdsgo with the same pattern as No increases, just like that inFigure 11(b). Moreover, according to the parameters shownin Table 2 and 4, the received charging power of devices oftype 1-4 monotonically decreases with the same charger atthe same distance. Thus, if the power threshold for a largerdevice type number is larger, more chargers are neededto get the power threshold for these types of devices, sothe charging utility decreases, which explains the differencebetween the different settings in Figure 13.





100

102Ti

me

Con

sum

ptio

n (T

imes

, Log

Sca

le) Non-Dis

Dis-10

Dis-15Dis-20Dis-25

Dis-5

Fig. 12: Comparison of time complexity


0.5

0.6

0.7

0.8

0.9

1

Cha

rgin

g U

tility

-0.01-0.0050+0.005+0.01

Fig. 13: Comparison of different P jths

0.4

0.5

0.6

2.5

0.7

0.8

0.9

Cha

rgin

g U

tility

1

2 00.21.5

dimax (Times)

0.4

dimin /di

max

0.61 0.80.5 1

Fig. 14: Impact of dimin and dimax

0 0.2 0.4 0.6 0.8 1Charging Utility

0

0.2

0.4

0.6

0.8

1

CD

F

PDCSGPPDCS TriangleGPPDCS SquareGPAD TriangleGPAD SquareGPAR TriangleGPAR SquareRPADRPAR

Fig. 15: Charging utility CDF of different devices

6.2 Insights

In this subsection, we study the impact of dimax and diminand the charging utility distribution of all the devices toreveal the advantages of our algorithm. First, we study theimpact of dimax and dimin. We set the number of chargersto be two times of the initial setting, and the multiple ofdimax varies from 0.6 to 2 while dimin/d

imax varies from 0 to

0.9. Note that each obtained date point denotes the averagevalue of 100 experimental results. Figure 14 shows that ifdimin tends to be zero, the charging utility increases muchfaster, while dimin/d

imax remains high. That is, if the charg-

ing area is relatively small, the charging utility increasesvery slow with dimax. The other comparison algorithmssuffer from dimin because the predetermined positions maycause some devices within the distance of dimin not charged.

Figure 15 shows the cumulative distribution function(CDF) of charging utilities of all the 40 devices in onetopology. We can see that no device obtains charging utilityunder 0.5 in our algorithm while a large amount of devicesin other comparison algorithms do not harvest any chargingutility. Therefore, the charging utility gained by devices inour algorithm is relatively balanced at a high rate, whichcontributes to the good performance of our algorithm.

7 FIELD EXPERIMENTS

In this section, we conduct field experiments to evaluate ourproposed algorithm.

Our testbed consists of six chargers with three TB-Powersource power adjustable wireless power transmitters[41] as shown in Figure 16 and 17 with one tuned to 1Wworking power and two of 2W, and three TX91501 wirelesspower transmitters [4] with working power of 3W asshown in Figure 17. Thus, there are three types of chargers.Moreover, there are two types of rechargeable sensor nodesequipped with P2110 power receivers both produced byPowercast [42] as shown in Figure 18 and 19, respectively.Each type has five nodes. An AP is connected to the laptop

to report the collected data from sensor nodes as shown inFigure 20. The rechargeable sensor nodes are placed withstrategies 〈(20, 15), 200◦〉, 〈(47, 20), 350◦〉, 〈(113, 65), 20◦〉,〈(20, 85), 140◦〉, 〈(13, 95), 40◦〉, 〈(7, 115), 190◦〉,〈(27, 110), 310◦〉, 〈(47, 100), 150◦〉, 〈(50, 118), 160◦〉, and〈(60, 93), 270◦〉 in a square area of 120 cm × 120 cm. Thissquare area is bounded by the dotted square includingthree obstacles as shown in Figure 24. The layout scenes ofHIPO, GPPDCS triangle, and GPAD triangle are shown inFigure 21, 22, and 23, respectively.

The placement of all the three algorithms can be seendirectly from Figure 24. We can see that in our algorithm,the chargers are deployed around the sensors closely anduniformly, while the chargers are placed a little far from theplacing field of sensors in other two algorithms. This leadsto the results in Figure 25 that all the devices can receivecharging utility from the chargers in our algorithm, whilethat of the other algorithms cannot. Although comparisonalgorithms have more charging utility for device #2, #3, and#4, but this is not the case for other devices. The CDF ofcharging power is depicted in Figure 26, and it shows thatthe line of HIPO approaches 1 at the lowest speed, whichindicates that HIPO generally leads to more charging powerfor the devices.

8 DISCUSSION

8.1 Charger RedeploymentIn this subsection, we discuss the charger redeploymentproblem when the topology of devices dynamically change,that is, how to schedule the chargers so that the incurredoverhead of switching from their previous deploymentscheme to a new redeployment scheme for the chargers,such as moving and rotating cost, is minimized.

Suppose that we know the original and new devicetopologies. Naturally, we can perform our HIPO algorithmtwo times for the two different topologies and obtain theircorresponding solutions. In the following, we consider twodifferent optimization problems, i.e., minimizing the overallswitching overhead and minimizing the maximum switch-ing overhead for all chargers.

8.1.1 Minimizing Overall Switching OverheadGenerally, the problem can be formulated into multipleweighted bipartite graph perfect matching subproblems,that is, given complete bipartite graphs Gq(Uq, Vq, Eq), q =1, ...,Q and a weight wqij for the edge connecting the i-th vertex in Uq and the j-th vertex in Vq , where Q is the




Fig. 16: Charger type 1 Fig. 17: Charger type 2 Fig. 18: Sensor type 1 Fig. 19: Sensor type 2 Fig. 20: AP

Fig. 21: HIPO testbed Fig. 22: GPPDCS Triangle testbed Fig. 23: GPAD Triangle testbed

Fig. 24: Positions & orientations of chargers & sensors

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

Cha

rgin

g U

tility

Device Index

HIPO GPPDCS Triangle GPAD Triangle

Fig. 25: Charging power of each device

number of charger types, Uq and Vq are the original and thenew strategies sets, respectively, we need to find matchingsfor all the graphs which match all the vertices such that thetotal value of weights of the selected edges is minimized.These subproblems can be then easily addressed by usingthe well-known Hungarian algorithm [43], [44].

We take a toy example shown in Figure 27 for illus-tration. Suppose there are two types of chargers to bedeployed: s1, s2, and s3 are of type 1 while s4 and s5 areof type 2. Suppose the strategies obtained by HIPO are asshown in Figure 27(a), and the new obtained strategies fora new device topology are shown in Figure 27(b). And weneed to determine how to transfer the original strategies tothe new ones with the minimum overhead. For example, s1has three choices of transferring to s′1, s′2, or s′3, while s4 hastwo choices of transferring to s′4 or s′5. Each transformationleads to different switching overhead. The set of the originalstrategies and that of the new strategies of each charger typeconstitute a weighted bipartite graph, in which the weightof an edge denotes the switching overhead of the associatedtransformation, as shown in Figure 28. Finally, we can applythe Hungarian algorithm to optimally solve the obtainedtwo weighted bipartite matching subproblems.

0 10 20 30 40Charging Power (mW)

0

0.2

0.4

0.6

0.8

1

CD

F

PDCSGPPDCS TriangleGPAD Triangle

Fig. 26: Charging power CDF of different devices

(a) Original scenario (b) Redeployment

1s

4s1s

3s

2s4s

5s

1s

2s

3s

4s

5s

Fig. 27: A toy example of redeployment

8.1.2 Minimizing Maximum Switching Overhead

Clearly, for this problem, we also need to find perfect match-ings in multiple weighted bipartite graphs, though the finalobjective function is changed. Besides, we stress that we takeour study one step further by continuing minimizing theoverall switching overhead after the maximum switchingoverhead is minimized.

In particular, our proposed algorithm consists of twosteps: the minimum maximum weight searching step andthe perfect matching generating step. In the first step, wesort all the weights and apply binary search to determine theminimum maximum weight. In each iteration of the search-ing process, we first remove all the edges with weightslarger than the current selected weight whose initial value isset to the maximum value of weights in the graph. Then, weuse the Hall’s Theorem [45] to check whether the remainedbipartite graph can induce a perfect matching, and accord-ingly adjust the searching range. We record the final selectedweight as the minimum maximum weight when the search-ing process terminates. In the second step, we remove allthe edges with weights larger than the minimum maximumweight, and apply the Hungarian algorithm [43], [44] tofurther optimize the overall switching overhead. Obviously,the obtained solution has the minimum overall switching




1s

2s

3s

4s

5s

1s

2s

3s

4s

5s

ChargerType 1

ChargerType 2

111w

112w1

13w

211w212w

Fig. 28: Bipartite graphs of Figure 27

overhead given that the maximum switching overhead ofchargers is minimized.

8.2 Deployment CostsIn this subsection, we discuss the charger deployment costs.We first introduce means to deploy chargers, then measurethese costs, and finally jointly consider charging utility anddeployment costs by formulating the whole problem.

In general, there are two methods to deploy chargers: bymanual work or by machines. In the case of small rangewalking available area, the chargers can be deployed bymanual work easily. In more complicated cases, the chargerscan be transported by machines such as mobile cars, aero-planes, and robots. The transportation cost can be formu-lated as the sum of functions of travel distance and rotatingangles of all the chargers, since traveling and rotating arethe only two ways for energy consumption no matter howto deploy these chargers. The other part of deployment costis how the chargers are charged. For the traditional chargerssuch as TX91501 wireless power transmitters [4] and TB-Powersource power adjustable wireless power transmitters[41], they are charged by cables, while for future chargers,they may be charged by solar or wind energy. Moreover, asthe working charging power of the chargers is fixed, thispart of deployment cost can be described as a function ofthe working charging power. Thus, the overall deploymentcosts can be formulated as

c(S) =∑si∈S

fd(di) + fθ(θi) + fP (Pi)

where di, θi, and Pi denote the traveling distance (fromthe former position), the rotating angle, and the workingcharging power of charger si when deploying it, respec-tively; fd(·), fθ(·) and fP (·) are three monotone increasingfunctions of the traveling distance, the rotating angle, andthe working charging power, respectively; and c(·) is thecost function.

To combine the original HIPO problem in Section 3.3,one way is to limit the deployment costs not to exceed acertain level B. Note that it is more beneficial to obtainmore charging utility in the long term since the chargingscenario is static. Thus, the whole optimization problem canbe formalized as follows:

maxsi,φi

s

1

No

No∑j=1

Uj(Ns∑i=1


jo)),

s.t. c(S) ≤ B, si ∈ S, si ∈ γ, and φis ∈ [0, 2π).

Note that the sum of the functions of traveling distance androtating angle can be formalized as a TSP problem (chargersin one base station initially) or an m-TSP problem (chargers

in m base stations initially), while the sum of the function ofconsuming power can be seen as the nodes placing cost. Af-ter performing our PDCS extraction algorithm (Algorithm2), we obtain the whole candidate strategy set. Then, we cansolve this problem by referring to the algorithm in [46] toget the final solution with 1

2 (1− e−1) approximation ratio.

8.3 Charging Utility BalancingRather than maximizing the overall charging utility, it isalso important to consider the problem of charging util-ity balancing of all the devices, that is, to guarantee thefairness of charging utility. The traditional and the mostcommonly discussed fairness criterion is max-min fairness[47]. The max-min fairness in our problem, i.e., maximizingthe minimum charging utility of the devices, is formulatedas follows:

maxsi,φi

s

minjUj(

Ns∑i=1


jo)),


(15)

Unfortunately, to the best of our knowledge, there is noefficient approximation algorithm for the max-min fairnessproblem of the original submodular optimization as prob-lem P3, but it can be solved by heuristic algorithms such asParticle Swarm Optimization [48], Ant Colony Optimization[49], and Simulated Annealing Algorithm [50].

Moreover, proportional fairness [47] is another fairnesscriterion. It optimizes the sum of individual utility whichis an increasing, strictly concave, and continuously dif-ferentiable function. In fact, the charging utility model inEquation (3) has made HIPO formalized in Equation (4)become an approximated proportional fairness problem ofcharging power since it is a concave function. To furtherachieve proportional fairness of charging utility, we canmaximize the sum of logarithmic of the individual chargingutility [47]. We formulate the proportional fairness HIPOproblem as follows:

maxsi,φi

s

No∑j=1

log

(Uj(

Ns∑i=1


jo)) + 1

),


(16)

After PDCS extration, the objective function can still bereformulated as a monotone submodular function, so wecan obtain the final strategies by Algorithm 3 with 1

2 − εapproximation ratio.

9 CONCLUSION

In this paper, we deal with the problem of practical het-erogeneous wireless charger placement with obstacles. Ourkey contributions are building the practical charging model,proposing an approximation algorithm, and conductingboth simulation and field experiments. The key technicaldepth of this paper is to reduce the infinite solution spaceto a limited one by using multi-feasible geometric area dis-cretization and PDCS extraction algorithm, plus proving theproblem as maximizing a submodular function subject to apartition matroid constraint. The experimental results showthat our algorithm outperforms comparison algorithms byat least 33.49%.




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Xiaoyu Wang received the B.S. degree in theDepartment of Computer Science and Technol-ogy from Soochow University, Suzhou, Jiangsu,China, in 2016. She is studying towards the Ph.Ddegree in the Department of Computer Scienceand Technology in Nanjing University. Her re-search interests focus on wireless charging anddata mining.

Haipeng Dai received the B.S. degree in the De-partment of Electronic Engineering from Shang-hai Jiao Tong University, Shanghai, China, in2010, and the Ph.D. degree in the Departmentof Computer Science and Technology in NanjingUniversity, Nanjing, China, in 2014. His researchinterests are mainly in the areas of wirelesscharging, mobile computing, and data mining.He is a research assistant professor in the De-partment of Computer Science and Technologyin Nanjing University. His research papers have

been published in many prestigious conferences and journals such asACM MobiSys, ACM MobiHoc, ACM VLDB, ACM SIGMETRICS, ACMUbiComp, IEEE INFOCOM, IEEE ICDCS, IEEE ICNP, IEEE SECON,IEEE IPSN, IEEE JSAC, IEEE/ACM TON, IEEE TMC, IEEE TPDS,and IEEE TOSN. He is an IEEE and ACM member. He serves/ed asPoster Chair of the IEEE ICNP’14, Track Chair of the ICCCN’19, TPCmember of the IEEE INFOCOM’20, IEEE IWQoS’19, IEEE ICNP’14,IEEE ICC’14-18, IEEE ICCCN’15-18 and the IEEE Globecom’14-18.He received Best Paper Award from IEEE ICNP’15, Best Paper AwardRunner-up from IEEE SECON’18, and Best Paper Award Candidatefrom IEEE INFOCOM’17.

Weijun Wang received the B.S. degree in theDepartment of Computer and Software fromNanjing University of Post and Telecommunica-tion, Nanjing, China, in 2014 and M.E. degree incomputer technology from PLA University of Sci-ence and Technology, Nanjing, China, in 2017.He is studying towards the Ph.D degree in theDepartment of Computer Science and Technol-ogy in Nanjing University. His research interestsfocus on UAV monitoring, MAC protocols in UAVnetworks and ad hoc networks.

Jiaqi Zheng is currently an assistant researcherfrom Department of Computer Science andTechnology, Nanjing University, China. His re-search area is computer networking, particularlydata center networks, SDN, and NFV. He re-ceived Ph.D. degree from Nanjing University in2017. He was an assistant researcher in theCity University of Hong Kong in 2015, and avisiting scholar in Temple University in 2016.He received the best paper award from IEEEICNP 2015 and Doctorial Dissertation Award

from ACM SIGCOMM China 2018. He is a member of ACM and IEEE.

Nan Yu received the B.S. degree in the De-partment of Computer Science and Technologyfrom Jilin University, Changchun, Jilin, China, in2015. She is studying towards a Ph.D degree inthe Department of Computer Science and Tech-nology in Nanjing University, Nanjing, Jiangsu,China. Her research interests are mainly inthe areas of wireless charging, and device-freesensing.

Guihai Chen received B.S. degree in computersoftware from Nanjing University in 1984, M.E.degree in computer applications from SoutheastUniversity in 1987, and Ph.D. degree in com-puter science from the University of Hong Kongin 1997. He is a professor and deputy chair of theDepartment of Computer Science, Nanjing Uni-versity, China. He had been invited as a visitingprofessor by many foreign universities includingKyushu Institute of Technology, Japan in 1998,University of Queensland, Australia in 2000, and

Wayne State University, USA during Sept. 2001 to Aug. 2003. He hasa wide range of research interests with focus on sensor networks,peer-to-peer computing, high-performance computer architecture andcombinatorics.

Wanchun Dou received the Ph.D. degree inmechanical and electronic engineering from theNanjing University of Science and Technology,China, in 2001. He is currently a Full Profes-sor of the State Key Laboratory for Novel Soft-ware Technology, Nanjing University. From April2005 to June 2005 and from November 2008 toFebruary 2009, he respectively visited the De-partment of Computer Science and Engineering,Hong Kong University of Science and Technol-ogy, Hong Kong, as a Visiting Scholar. Up to

now, he has chaired three National Natural Science Foundation of Chinaprojects and published more than 100 research papers in internationaljournals and international conferences. His research interests includeworkflow, cloud computing, and service computing.

Xiaobing Wu is with Wireless Research Cen-tre, University of Canterbury, New Zealand. Hereceived his PhD degree in 2009 from NanjingUniversity, ME degree in 2003 and BS degreein 2000 from Wuhan University, all in ComputerScience. His research interests are in the fieldsof wireless networking and communications, In-ternet of Things and cyber physical systems.His publications appeared at IEEE TPDS, ACMTOSN, IEEE INFOCOM, IEEE ICDCS, etc. Hewon Honoured Mention Award in ACM MobiCom

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