Near-Optimal Online Algorithm for Data Collection by ... - IEEE Xplore

IEEE ICC 2014 - Mobile and Wireless Networking Symposium

Near-Optimal Online Algorithm for Data Collection by Multiple Sinks in Wireless Sensor Networks Ruilong Deng, Shibo He and Jiming Chen Abstract—Data collection by multiple sinks is a fundamental problem in wireless sensor networks. Existing work focused on designing optimal offline algorithms provided that the number and positions of sensors and sinks are predetermined. This may not be practical as, though sensors are cheap, sinks are quite expensive in reality. A more practical scenario is that sinks are deployed step by step during the network operation due to the budget constraint, and we do not know the number, positions and capacities of sinks in prior. In this paper we investigate such an optimal data collection problem by multiple sinks, and design a near-optimal online algorithm via primal-dual approach, requiring very little priori knowledge. We theoretically derive the competitive ratio and show how to improve it by finding the optimal sink location region with an approximation ratio. Extensive simulations are conducted to verify the performance of the proposed online algorithm.

I. I NTRODUCTION With the explosive development of microelectronics and wireless communication in recent decades, wireless sensor networks (WSNs) have been one of the most active fields in computer network research. WSNs are constituted by a large number of sensors deployed in the monitoring area, which form self-organizing networks through wireless communication, in order to sense and report the information of the coverage area in real time. WSNs can make people obtain the reliable and timely information at any time and any place, which have become increasingly popular in many fields, including health care, traffic monitoring, battlefield surveillance, environmental protection, space exploration and many other applications [1]. One of the fundamental issues for WSNs is data collection, where sensing data are collected at sensors and forwarded to sinks for further processing. Since sensors are powered by batteries, which are usually impossible or impractical to replace; therefore, energy should be spared, so as to collect as more data as possible. Sensors can send data directly to sinks, or through multi-hop routing [2]. In both cases, energy consumption depends on the communication distance. To reduce such a distance, one way is to have a mobile sink that moves around to collect data from sensors it passes by [3]. This approach, however, could not fit real-time applications, where data has to reach the sink immediately. Another possibility is to deploy multiple sinks; this would decrease the average R. Deng and J. Chen are with the State Key Laboratory of Industrial Control Technology, Zhejiang University, Hangzhou, 310027, China [email protected], [email protected] S. He is with the School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ, 85287, USA [email protected] Research was supported in part by NSFC under grants 61222305, NCET11-0445, and the National Program for Special Support of Top-Notch Young Professionals.

978-1-4799-2003-7/14/$31.00 ©2014 IEEE

communication distance as data will be sent to the nearest sink. In this paper we present a problem that builds on this approach. Once we decide to use multiple sinks for data collection, the problem comes where to deploy them and how much data to be transmitted from each sensor to each sink. Existing work focused on designing optimal offline algorithms provided that the number and positions of sensors and sinks are predetermined. This may not be practical as, though sensors are cheap, sinks are quite expensive in reality. A more practical scenario is that sinks are deployed step by step during network operation due to the budget constraint, and we do not know the number, positions and capacities of sinks in prior. In this paper we investigate the optimal data collection problem by multiple sinks in an online fashion. This problem is not trivial since sensors do not know how many sinks will be deployed, where they are, and how much data capacity of each sink is, until they are deployed one by one. The sink data capacity means that this sink cannot receive the sensor data more than such a threshold. As aforementioned, from the energy saving point of view, data will be sent to the nearest sink. The unknown factor is that each sensor does not know whether the current available sink is nearer or farther than the future sinks which may be deployed somewhat later. It is possible that a nearer sink will be available at later time slot, but this is not guaranteed. Another issue is that, once a sensor decides to send amount of data to a sink at current time slot, this sensor cannot change this decision in future since at that time this action has already happened. In this paper, we design a near-optimal online algorithm via primal-dual approach to solve the data collection problem by multiple sinks. Our focus is on scheduling data collection to maximize the total transmitted data of all sensors. Our online algorithm makes real-time decision based on the position and data capacity of the newly deployed sink. This algorithm only requires very little priori knowledge of sinks, i.e., each sensor knows how far the nearest sink (no matter it will be deployed or never) is. We theoretically derive the competitive ratio of the online algorithm and show how to improve it by finding the optimal sink location region with an approximation ratio. Extensive simulations are conducted to verify the performance of the proposed online algorithm. The main contributions of this paper are summarized as follows: 1) We investigate the data collection problem by multiple sinks, and design a near-optimal online algorithm via primal-dual approach, requiring little priori knowledge; 2) We theoretically derive the competitive ratio of the online

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algorithm and rigorously prove it; 3) We improve the competitive ratio by finding the optimal sink location region with an approximation ratio; 4) Extensive simulations are conducted to verify the performance of the proposed online algorithm. The remainder of this paper is organized as follows. In Sec. II we formulate the optimal data collection problem by multiple sinks. We design a near-optimal online algorithm to solve the problem via primal-dual approach and derive the competitive ratio in Sec. III. Then in Sec. IV we provide some guidelines on solving two relevant problems. Numerical results are provided in Sec. V, and conclusions are drawn in Sec. VI. We focus on a simple scenario where each sensor sends data directly to sinks. Note that most of the sensor energy is used for data transmission. Let e (i, j) denote the energy consumption of sensor i for sending a unit of data to sink j which is dij apart: (1)

where γ1 , γ2 > 0 are constant coefficients, dij is the Euclidean distance between sensor i and sink j, and δ is the path loss exponent (2 ≤ δ ≤ 4 for the free-space and short-to-mediumrange radio communication) [4]. This means that in a one-hop network the energy cost of sending data to a sink increases with the distance that the data has to travel. For the problem of optimal data collection by multiple sinks, consider a set N = {1, . . . , N } of sensors encounter a set M = {1, . . . , M } of sinks in an online fashion. Let x (i, j) denote the amount of data transmitted from sensor i to sink j. Assume sensor i has an initial energy E (i), and sink j has a data capacity C (j). A formulation of the dual problem (offline, with full priori knowledge of sinks) is shown as follows. The objective function (2) is to maximize the total transmitted data of all sensors. The first constraint (3a) guarantees that each sensor never runs out of energy. The second constraint (3b) means that each sink cannot receive data more than its capacity. Dual Problem (offline): x (i, j) (2) max X

i∈N j∈M

⎧ e (i, j) x (i, j) ≤ E (i) ∀i ∈ N ⎪ ⎪ ⎪ ⎪ ⎨ j∈M s.t. x (i, j) ≤ C (j) ∀j ∈ M ⎪ ⎪ ⎪ ⎪ i∈N ⎩ x (i, j) ≥ 0 ∀i ∈ N , ∀j ∈ M

y,z

(3a) (3b)

(4)

j∈M

∀i ∈ N , ∀j ∈ M (5a)

III. O NLINE S OLUTION We now present the near-optimal online algorithm [5]: Algorithm 1: online algorithm 2 3 4 5 6 7

Initialization: y (i)1 ← 0, ∀i ∈ N ; while a new sink k is deployed (at the k th iteration) do 1 for sensors with y (i)k < e(i,k) do consider the sensor i whose e (i , k) y (i )k is the minimum; x (i , k) ← C (k); z (k) ← 1 − e (i , k) y (i )k ; y (i )k+1 ← e(i ,k)C(k) e(i ,k) C(k) + min e(i ,j) · (α−1)E(i y (i )k 1 + E(i ) ) ; j∈M

8 9

end for end while

Theorem 1 Algorithm 1 is

1−

β α+β−1

1 1+R2max

-competitive, max e(i,j) j∈M where α = (1 + R1max ) , β = max min e(i,j) , i∈N j∈M

e(i,j)C(j) max , and R R1max = max = 2 E(i) max e(i,j) i∈N ,j∈M j∈M · max C (j). max E(i) 1 max R1

i∈N

j∈M

The intuition of Theorem 1 is as follows. Since the comβ , then we need petitive ratio we are aiming for is 1 − α+β−1 to guarantee that at each iteration the change in the primal β objective function is at most 1 + α−1 times the change in the dual objective function, while the primal and dual solutions remain feasible. The proof is given in Appendix A. Note that the online algorithm only needs a little priori knowledge of sinks, i.e., minj∈M e (i, j), ∀i ∈ N (each sensor knows how far the nearest sink is). For simplicity, we relax some parameters of the competitive ratio: β ≤ maxi∈N ,j∈M e(i,j) max C(j) max ≤ minj∈M ·maxi∈N ,j∈M e (i, j), mini∈N ,j∈M e(i,j) , R1 i∈N E(i) max

We use online primal-dual approach [5] to solve the problem. In the primal problem (online, with a little priori knowledge of sinks), there is a variable y (i) associated with each sensor i, and z (j) for each sink j. Consider an online setting in which sinks (new primal constraints) are deployed one by one in an online fashion. Upon arrival, each primal constraint (5a) should be satisfied.

i∈N

⎧ ⎪ ⎨ e (i, j) y (i) + z (j) ≥ 1 s.t. y (i) ≥ 0 ∀i ∈ N ⎪ ⎩ z (j) ≥ 0 ∀j ∈ M

1

II. P ROBLEM F ORMULATION

e (i, j) = γ1 + γ2 dδij

Primal Problem (online): E (i) y (i) + C (j) z (j) min

C(j)

·maxi∈N ,j∈M e (i, j). Assume each and R2max ≤ minj∈M i∈N E(i) sensor’s initial energy E (i) and each sink’s data capacity C are fixed. We define the following two constants: (j)min = mini∈N E (i) E . From the above, the competitive C max = maxj∈M C (j) ratio is relaxed as: Theorem 2

β 1 Algorithm 1 is 1 − α+β−1 1+Rmax -competitive, 1

where α = (1 + Rmax ) Rmax , β =

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max

e(i,j)

min

e(i,j) ,

i∈N ,j∈M i∈N ,j∈M

and


Rmax =

C max E min

·

max

i∈N ,j∈M

e (i, j).

Note that α decreases with Rmax (by proving the function 1 (1 + x) x decreases on R+ ), so the competitive ratio increases when β, Rmax decrease. For the extreme case, when β tends to 1, and Rmax tends to 0, the competitive ratio of the online algorithm tends to 1 − 1e (upper bound). IV. R ELEVANT P ROBLEMS Here rise two relevant problems: one is that, given the sink location region, how to calculate the competitive ratio of the online algorithm; the other is finding the optimal sink location region such that the competitive ratio of the online algorithm is maximized.

imum distance between sensor and sink, i.e., 2 2 (xj − xi ) + (yj − yi ) max min i∈N ,j∈M

(xj ,yj ),∀j∈M

Inspired by Welzl’s work on the smallest enclosing disk [6], we observe that the search space of sink locations can be first reduced to a SES, which is the smallest circular disk that covers all sensors. Lemma 1 To minimize the maximum distance between sensor and sink, sinks should be located within the SES that covers all sensors. The proof is given in Appendix B. Based on Lemma 1, the min-max problem (6) is reformulated as: 2 2 (xj − xi ) + (yj − yi ) max min (xj ,yj )∈SES,∀j∈M

dio r o

diomin

Sink Location Region

SES Center

min

Source

Γ,(xj ,yj )∈SES,∀j∈M

s.t. Fig. 1.

Online data collection, given sink location region

As shown in Fig. 1, we observe that the sink location region can be covered by a smallest enclosing space (SES), which is the smallest circular disk that covers this area. Assume the SES disk is centered at o with a radius of r. For any sink j located within the sink location region, we maxi∈N ,j∈M dij ≤ maxi∈N dio + r have: by the “trimini∈N ,j∈M dij ≥ |mini∈N dio − r| angle” theorem. Combining with (1), we can obtain the upper bound of maxi∈N ,j∈M e (i, j) and the lower bound of mini∈N ,j∈M e (i, j). Based on Theorem 2, we can calculate the lower bound of the competitive ratio of the online algorithm, given the sink location region. For the problem of finding the optimal sink location region such that the competitive ratio of the online algorithm is maximized, we need to minimize β, Rmax , specifically, to minimize maxi∈N ,j∈M e (i, j) and maximize mini∈N ,j∈M e (i, j). Combining with (1), we need to minimize maxi∈N ,j∈M dij and maximize mini∈N ,j∈M dij . In the following we consider to minimize maxi∈N ,j∈M dij , which is dominant in the competitive ratio, and to maximize mini∈N ,j∈M dij can follow the same way. Let (xi , yi ) denote the (fixed, known) location of sensor i, and (xj , yj ) for the (adjustable, unknown) location of sink j. The distance between sensor i and sink j is given by dij = 2

i∈N ,j∈M

(7) However, this problem is still difficult to solve in its current form due to the “max” term in the objective function. This can be resolved by introducing a new auxiliary variable Γ and rewriting (7) in an equivalent form as a nonlinear programming (NLP) problem:

i

diomax

(6)

Γ≥

Γ

2

(8)

(xj − xi ) + (yj − yi )

2

∀i ∈ N , ∀j ∈ M

Although Lemma 1 has reduced the search space of sink locations from two-dimensional space to a smaller enclosing circular area; however, the potential sink locations within SES are still infinite such that we cannot directly apply NLP methods to find the optimal one(s). Following the thought introduced in [7], we can discretize the area within SES, into a limited number of potential sink locations, with an approximation ratio. Due to the space limitation, we do not provide the detail process of space discretization here. After discretizing SES into K zones, the distance from sensor i to sink j deployed within zone k is bounded by: (ki − 1) ε ≤ dij ≤ ki ε

∀j ∈ k

Consequently, all such zones within SES can serve as a finite number of potential sink locations. The former NLP problem (8) is transformed into a linear programming (LP) problem: min

Γ,k∈K

x∗j , yj∗

s.t.

Γ Γ ≥ ki ε

(9) ∀i ∈ N

denote one of the theoretically optimal sink Let locations (unknown), and Γ∗ for the corresponding minimum peak distance between sensor and sink. By solving the LP problem (9) we can obtain Γapprox for the approximated minimum peak distance between sensor and sink. Theorem 3

2

(xj − xi ) + (yj − yi ) . Our goal is to minimize the max-

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The approximated minimum peak distance between sensor


V. N UMERICAL R ESULTS In this section, we provide numerical results to demonstrate the performance of the proposed online algorithm. Consider sensors and sinks are randomly deployed in a 10×10 m2 square area. For an offline setting, all information about sinks are assumed to be known by sensors in prior. In an online manner, sensors have no priori knowledge until sinks are deployed one by one. Each sensor has an initial energy E (i) =304 mW [8]. The energy consumption rate for each sensor to transmit data over a physical distance d is 0.003 + 0.0002 × d3.14 mW/KB [4]. Each sink has a data capacity C (j) =10 M B. One Sensor

One Sensor

50 40 30 20

0 0

4 6 8 Number of Sinks

10

Competitive Ratio

10

12

Simulation Result Lower Bound

10

0 0

12

2

4 6 8 Number of Sensors

10

12

Sensor number impacts on network performance

Five Sensors, Five Sinks

1

1.4

40 30 20

1 0.8 0.6 0.4

10 0.2

Online Algorithm Offline Solution 0 4

0.6


1.2

0.8

6 8 10 12 14 Deployment Area Side Length (m)

0 4

16

6 8 10 12 14 Deployment Area Side Length (m)

16

0.4

20 0.2 2


10

0 0

12

Fig. 4. 2

Ten Sensors or more


10

Ten Sensors or more Five Sensors, Five Sinks

1.4

1 0.8 0.6 0.4

20 Online Algorithm Offline Solution 2


Fig. 2.

10

12

0.2 0 0

2


10

12

40 30 20

1 0.8 0.6 0.4

10 0.2

Online Algorithm Offline Solution 200

Fig. 5.

We first simulate a WSN with ten sinks, while the number of sensors is set at one, five, and ten or more. Each case is run 100 times with randomly deployed sensors and sinks. The corresponding results with mean and variance are shown in Fig. 2. It is observed that, firstly, both online and offline collection data increase with the number of sinks; both increase faster when sinks are less than sensors, and slower when sinks are more than sensors. Offline solutions are always better than online results, and increase faster too. Secondly, as the number of sinks increases, the competitive ratio lower bound (from Theorem 2) always decreases in each case. This is because, with more sinks, maxi∈N ,j∈M e (i, j) may increase and mini∈N ,j∈M e (i, j) may decrease, rising β and Rmax ,


1.2

0 100

Network performance with ten sinks

1.4

Competitive Ratio

40

Total Data Collection (MB)

Competitive Ratio

80 60


50


1.2

300 400 500 Sensor Energy (mW)

550

0 100

100

200


100

1.4 Online Algorithm Offline Solution


1.2

60 50 40 30 20

1 0.8 0.6 0.4 0.2

10 0 4

550


80 70

300 400 500 Sensor Energy (mW)

Sensor initial energy impacts on network performance

Competitive Ratio

100

0 0

Deployment area size impacts on network performance

12




80

0 0


4 6 8 Number of Sensors


1.2

40

2

50 2

Five Sensors

60

0.4 0.2

Online Algorithm Offline Solution

0.4

1.4 Online Algorithm Offline Solution

0.6

10

Fig. 3.

Five Sensors 100

1 0.8

0.6

0 0

12

20

1

0.2 2

30

0.8


10

40


1.2


1.2

Competitive Ratio

Online Algorithm Offline Solution

Five Sinks 1.4

0 0

1.4

Competitive Ratio


70 60

Five Sinks 50

Competitive Ratio

Proof: Due to the space limitation, we omit the proof.

and thus the lower bound of the competitive ratio declines. Finally, for the simulation results of the competitive ratio, when sinks are less than sensors, it remains almost the same; when sinks are more than sensors, the simulated competitive ratio declines. This is because, when sinks are less, sensors still have remainder energy and can transmit data to the newly deployed sink; when sinks are more, sensors are almost out of energy, and they cannot change their previous decisions although the future sink may be better than before. Since the offline method assumes all these information are known in prior, it yields better performance than the online algorithm.


and sink after discretization is bounded by Γ∗ ≤ Γapprox ≤ Γ∗ + ε, where Γ∗ is the theoretically optimal solution, and ε is the approximation ratio.

6

Fig. 6.

8 10 12 Sink Capacity (MB)

14

16

0 4

6

8 10 12 Sink Capacity (MB)

14

16

Sink data capacity impacts on network performance

We then investigate how the sensor number, deployment area size, sensor initial energy, and sink data capacity im-

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pact on the network performance. As shown in Fig. 3, we simulate a WSN with five sinks, while the number of sensors increases from one to ten. It is seen that offline collection data increases with the number of sensors. Online collection data increases when sensors are less than sinks, remains almost the same when sensors are more than sinks, and so is the simulated competitive ratio. As the sensor number increases, the competitive ratio lower bound decreases. As shown in Fig. 4, we simulate a WSN with five sensors and five sinks, while the deployment area side length increases from 5 m to 15 m. It is observed that, as the deployment area becomes larger, both online and offline collection data decrease. This is because the average distance between sensor and sink becomes larger. As the deployment area increases, the competitive ratio lower bound and the simulated competitive ratio also decline. This is because maxi∈N ,j∈M e (i, j) may increase in a larger deployment area, rising β and Rmax , and thus the competitive ratio drops. As shown in Fig. 5, as the sensor energy becomes larger, both online and offline collection data increase. The simulated competitive ratio also increases, while the competitive ratio lower bound is almost unaffected. The competitive ratio rises because as E (i) increases, Rmax will drop, increasing α, which in turn increases the competitive ratio. As shown in Fig. 6, as the sink capacity becomes larger, both online and offline collection data increase. The simulated competitive ratio decreases, while the competitive ratio lower bound is almost unaffected. The competitive ratio declines because as C (j) increases, Rmax will rise, decreasing α, which in turn decreases the competitive ratio. VI. C ONCLUSION In this paper, we investigate the optimal data collection problem by multiple sinks. Instead of traditional offline solutions with all information about sinks known in advance, we consider that sensors encounter sinks in an online manner with very little priori knowledge. Firstly, a near-optimal online algorithm is proposed via primal-dual approach, with the competitive ratio derived. Besides, we provide some guidelines on solving two relevant problems. Finally, extensive simulations are conducted to verify our theoretical analysis. A PPENDIX A P ROOF OF T HEOREM 1 Proof: We need to prove the following three claims: 1) The algorithm produces a feasible primal solution; 2) The ratio between the change in the primal and dual β ; objective functions is bounded by 1 + α−1 3) The algorithm produces an almost feasible dual solution. Proof of 1): To show the feasibility of the primal solution, look at the primal constraint (5a). Consider at the k th iteration, 1 , ∀i ∈ N , when a new sink k is deployed. If y (i)k ≥ e(i,k) then the primal constraint is intuitively satisfied. Otherwise, the algorithm considers the sensor i whose e (i , k) y (i )k is the minimum. Setting z (k) = 1 − e (i , k) y (i )k guarantees that the primal constraint is satisfied.

Proof of 2): If the algorithm updates the primal and dual solutions at the k th iteration, then the dual objective function increases by C (k), and the increase in the primal objective function is: E (i ) y (i )k+1 − y (i )k + C (k) z (k) ⎡ e (i , k) C (k) e (i , k) =E (i ) ⎣ y (i · ) + k E (i ) min e (i , j) j∈M C (k) + C (k) 1 − e (i , k) y (i )k (α − 1) E (i ) β ≤ 1+ C (k) α−1 Thus, the ratio between the change in the primal and dual β . objective functions is at most 1 + α−1 th Proof of 3): Note that at the k iteration, the dual constraint (3b) is intuitively satisfied since i∈N x (i, k) = 1 , ∀i ∈ N 0 if y (i)k ≥ e(i,k) . In order to x (i , k) = C (k) otherwise show the feasibility of the dual solution, look at the dual constraint (3a). The algorithm never updates the dual so1 . We prove that for lution for sensors with y (i)k ≥ e(i,k) k−1 any sensor i, whenever j=1 e (i, j) x (i, j) ≥ E (i), then 1 1 ≥ y (i)k ≥ minj∈M e(i,j) e(i,k) . This is done by proving that: ⎡ k−1 ⎤

1 ⎢ j=1 y (i)k ≥ ⎣α α−1

e(i,j)x(i,j) E(i)

⎥ − 1⎦

1 min e (i, j)

(10)

j∈M

We prove (10) by induction on the (relevant) iterations of the algorithm. Initially, this assumption is trivially true. Consider at the k th iteration, when a new sink k is deployed. If no sensor is updated, then (10) is intuitively satisfied. Otherwise, the algorithm updates the sensor i as (11) (12). The inequality (11) follows the induction hypothesis. The inequality (12) holds since, for any 0 < a ≤ b, there is ln(1+a) ln(1+x) ≥ ln(1+b) (by proving the function decreases a b x +

on R ). That is, we have

ln 1+

e(i ,k)C(k) E (i )

e(i ,k)C(k) E (i )

≥

ln(1+R1max ) , R1max

and

e(i ,k)C(k)

e(i ,k)C(k) then 1 + E(i ) ≥ α E (i ) . In practice, whenever a sensor runs out of energy, it stops transmission immediately. Hence, there is at most one iteration, say k, at which a sensor may transmit data less than E(i)− j∈M\{k} e(i,j)x(i,j) C (k). That is, x∗ (i, k) = . So the e(i,k) actual transmitted data of sensor i is at least: x (i, j) + x∗ (i, k) j∈M\{k} x (i, j) x (i, j) + x∗ (i, k) + max C (j)

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j∈M

j∈M

j∈M\{k}

E (i) x (i, j) E (i) + max e (i, j) · max C (j) j∈M j∈M j∈M 1 ≥ x (i, j) 1 + R2max

≥

j∈M

(13) (14)


C (k) e (i , k) C (k) e (i , k) · = y (i )k 1 + + E (i ) min e (i , j) (α − 1) E (i ) j∈M ⎡ k−1 ⎤ e(i ,j )x(i ,j ) e (i , k) C (k) C (k) 1 1 ⎢ j=1 E(i ) e (i , k) ⎥ · 1 + · (11) − 1⎦ ≥ + ⎣α α−1 min e (i , j) E (i ) min e (i , j) (α − 1) E (i )

y (i )k+1

j∈M

⎧ ⎪ ⎨

k−1

j=1 1 α = α−1⎪ ⎩

⎡ ≥

1 ⎢ ⎣α α−1

k j=1

e(i ,j )x(i ,j ) E (i )

1+

e(i ,j )x(i ,j ) E (i )

⎤ ⎥ − 1⎦

1 e (i , k) C (k) −1 ⎪ E (i ) min e ⎭ j∈M (i , j) 1 min e (i , j)

j∈M

j∈M

max e (i, j) j∈M

≥

k

dik

i

j∈M\{k}

j∈M\{k}

R

!

O

x (i, j) + x∗ (i, k)

SES Center

!

Source

x (i, j) + x∗ (i, k) + max C (j)

Sink

j∈M

E (i) E (i) + max e (i, j) · max C (j) j∈M

1

Fig. 7.

j∈M

Inequality (14) holds since 1+

dij

x (i, j) + x∗ (i, k) + max C (j)

max e (i, j) =

j

j∈M\{k}

j∈M\{k}

(12)

j∈M

where the inequality (13) holds since, for any 0 ≤ c < a < b, there is ab ≥ a−c b−c . That is, x (i, j) + x∗ (i, k)

j∈M

⎫ ⎪ ⎬

maxj∈M e(i,j)·maxj∈M C(j) E(i)

E(i) E(i)+maxj∈M e(i,j)·maxj∈M C(j) ≥ 1+R1max . 2

=

By the second claim that (the change in the primal objective function) / (the change in the dual objective function) ≤ 1 + β β α−1 , such that the dual is at least 1− α+β−1 times the primal, and thus (by weak duality) we conclude

that the competitive β 1 ratio of the algorithm is 1 − α+β−1 1+Rmax . 2

A PPENDIX B P ROOF OF L EMMA 1 Proof: We prove this lemma by contradiction. As shown in Fig. 7, assume there is an arbitrary sink j that is located outside SES. We prove that there always exists a location within SES which can lead to shorter distance between sink j and any sensor i. Let the center of SES be O, and k denote the intersecting point where the line segment [O, j] cuts the SES circle. For any sensor i within SES, we always have dik < dij (by “larger angel has longer opposite side in triangle” theorem). Therefore, by relocating the sink from location j to k, we can always decrease distance between sink j and any sensor i, and thereby decrease the maximum distance between sensor and sink. Consequently, we conclude

Sinks should be located within SES

that any location j outside SES is not possible to be an optimal location, and this completes the proof of Lemma 1. R EFERENCES [1] X. Cao, J. Chen, Y. Zhang, and Y. Sun, “Development of an integrated wireless sensor network micro-environmental monitoring system,” ISA Transactions, vol. 47, no. 3, pp. 247–255, 2008. [2] A. E. Abdulla, H. Nishiyama, J. Yang, N. Ansari, and N. Kato, “Hymn: a novel hybrid multi-hop routing algorithm to improve the longevity of wsns,” IEEE Transactions on Wireless Communications, vol. 11, no. 7, pp. 2531–2541, 2012. [3] L. He, L. Cai, J. Xu, Z. Yang, J. Pan, and Y. Gu, “Evaluating service disciplines for on-demand mobile data collection in sensor networks,” IEEE Transactions on Mobile Computing, to appear, DOI: 10.1109/TMC.2013.62. [4] B. Gaudette, V. Hanumaiah, S. Vrudhula, and M. Krunz, “Optimal range assignment in solar powered active wireless sensor networks,” in Proc. IEEE INFOCOM, pp. 2354–2362, 2012. [5] N. Buchbinder, K. Jain, and J. S. Naor, “Online primal-dual algorithms for maximizing ad-auctions revenue,” in Proc. Springer-Verlag European conference on Algorithms, pp. 253–264, 2007. [6] E. Welzl, Smallest enclosing disks (balls and ellipsoids). Springer, 1991. [7] L. Fu, P. Cheng, Y. Gu, J. Chen, and T. He, “Minimizing charging delay in wireless rechargeable sensor networks,” in Proc. IEEE INFOCOM, pp. 2922–2930, 2013. [8] R. Deng, Y. Zhang, S. He, J. Chen, and S. Shen, “Globally optimizing network utility with spatiotemporally-coupled constraint in rechargeable sensor networks,” in Proc. IEEE GLOBECOM, pp. 5121–5126, 2013.

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