Optimized Barrier Location for Barrier Coverage in Mobile Sensor Networks

Transcription

1 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks Optimized Barrier Location for Barrier Coverage in Mobile Sensor Netorks Xiaoyun Zhang, Mathe L. ymore, and Daji Qiao Ioa State University, Ames, IA 5 {zxydut, mlymore, daji}@iastate.edu Abstract Barrier coverage is an important application of sensor netorks. This paper studies ho to build a strong barrier ith mobile sensors in hich the maximum moving distance of sensors is minimized. Our ork differs from others in the ay the y-coordinate of the barrier is determined. e optimize the y-coordinate of the barrier instead of fixing it a priori. An efficient algorithm is proposed, in hich the search space of the y-coordinate of the barrier is first discretized and then searched over iteratively. In the theoretical orst case, O(N 4 ) iterations may be needed to find the optimal barrier location, here N is the number of sensors, but in practice, our algorithm requires less than O(N 2 ) iterations, as confirmed in simulation. I. INTRODUCTION Intruder detection and border surveillance are intuitive applications of sensor netorks. In these applications, sensors are deployed along the perimeter of a protected area such that no intruder can cross the perimeter ithout being detected. The arrangement of sensors for this purpose is referred to as the barrier coverage problem in sensor netorks. Early research in barrier coverage focused on static sensors [] [4]. Different kinds of barrier coverage ere defined and the corresponding critical densities of sensors ere calculated. Also, various schemes ere proposed to identify the barriers formed by static sensors. More recent research has examined barrier coverage ith mobile sensors [5] [8]. The primary focus of these papers has been minimizing the moving distances of sensors hen building the barrier. These orks either restricted the sensors deployment and movement regions to a -dimensional (D) line, or fixed the y-coordinate of the barrier if considering a 2-dimensional (2D) rectangular deployment region ith the minimum number of sensors. In practice, it is difficult to deploy sensors along a D line, and in the 2D case, fixing the y-coordinate of the barrier a priori results in a suboptimal solution, since the y-coordinate of the barrier affects the moving distance of sensors. In this paper, e study the strong barrier coverage problem in a 2D rectangular region, here the goal is to detect intruders that may follo any path to cross the deployment region. This is in contrast to eak barrier coverage, hich deals ith intruders that only perpendicularly cross the deployment region. e assume that a strong barrier is formed ith the minimum number of sensors, along a horizontal line that spans the length of the deployment region. e present an algorithm that decides the optimal y-coordinate of the barrier, and identifies a subset of sensors to move to their corresponding destinations, so that the maximum moving distance of sensors is minimized. To achieve this, our algorithm first discretizes the search space for the y-coordinate of the barrier, then quickly iterates over this discretized search space. Simulation results sho that our algorithm is efficient and scalable. The rest of this paper is organized as follos. In Section II, e discuss related ork in detail. In Section III, the system model is presented and the problem is formally defined. e present our algorithm in Section IV and evaluate it in Section V. Finally, e conclude this paper in Section VI. II. RELATED ORK Various D mobile barrier coverage problems have been studied in [5, 9, ]. In these papers, mobile sensors are initially deployed on the line here the barrier ill be built, then the sensors move along this line to form the barrier. The sensors move according to various objectives, such as minimizing the maximum (min-max) or the sum (min-sum) of moving distances, maximizing the coverage, or minimizing the number of moving sensors. Regardless of the objective, the y-coordinate of the barrier is not considered because the sensors are assumed to already be on the line. Barrier coverage has also been studied for mobile sensors deployed in a 2D region. Bhattacharya et al. [5] solved the min-max and min-sum problems for sensors moving from a circle s interior to its circumference to form a barrier. Their algorithm optimizes an angle variable, analogous to our y- coordinate of the barrier. Hoever, their solution cannot be applied to our problem due to different search spaces of the angle variable and the y-coordinate, and the difference in ho these factors decide sensors final positions. Saipulla et al. [6] studied the min-max problem of forming a barrier ith a minimum number of mobile sensors in a rectangular region. Hoever, they fixed the y-coordinate of the barrier a priori. Shen et al. [7] did not fix the y-coordinate of the barrier in their investigation into the min-sum problem for a rectangular region, but they instead fixed the order of sensors along the barrier. In contrast, our algorithm does not fix either the y- coordinate or the order of the sensors along the barrier, thus alloing for optimal min-max results ith no assumptions about ho the sensors are deployed. Distributed algorithms have also been proposed for the barrier coverage problem [7, 8,, 2], but the goal of these algorithms is usually to reduce the communication beteen sensors. They are suboptimal hen considering objectives such as min-max or min-sum of sensors moving distances. III. SYSTEM MODEL AND PROBLEM STATEMENT A. System Model ) Sensor Netork: e study a netork of N mobile sensors deployed in a long rectangular region of size L, /5/$3. 25 IEEE 572

2 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks here L, as shon in Fig.. Sensors are named s to s N from left to right of the deployment region, and the initial position of sensor s i is denoted by (x i, y i ). The set of all the sensors is denoted by S. e adopt the idely-used disk coverage model and denote the coverage radius as R. An intruder can be detected by a sensor if and only if it is ithin R of the sensor. In addition, e assume sensors can acquire their positions from GPS or another localization scheme. y Exit side s s2 s3 t t2 t3 t4 t5 t6 t47 t48 t49 t5 s4 s6 s49 s5 R s5 Entrance side... s47 intruder path Fig.. An example sensor netork of 5 mobile sensors. 2) Intruder and Barrier: e assume that intruders may take any path to cross the deployment region from bottom to top, as shon in Fig.. In order to detect such intruders, strong barrier coverage is required. This is in contrast to eak barrier coverage, hich assumes intruders only take paths perpendicular to the x-axis of the deployment region. The minimum number of sensors needed to form a strong barrier is N min = L 2R. For simplicity, and ithout loss of generality, e assume that L is a multiple of 2R; hence, N min = L 2R. To form a strong barrier ith N min sensors, hich is the focus of our study in this paper, these sensors must be aligned along a horizontal line parallel to the x-axis of the deployment region. In other ords, the destination positions of these sensors (denoted by t j, j N min ) must have the coordinates of (α j, j ), here α j = (2j )R, j N min, and = 2 = = Nmin. e use T to denote the set of destination positions, and use to denote their common y- coordinate, called the barrier location. 3) System: e assume that the sensor netork remains connected during sensor movement, and that there is a central processing unit hich collects information from sensors, executes the proposed algorithm, and disseminates the movement strategy to sensors. B. Problem Statement Our goal is to minimize the maximum moving distance of mobile sensors that form a strong barrier, ith selected N min sensors from a netork of N (N N min ) sensors ith knon initial positions. More specifically, our algorithm must decide () here the barrier shall be formed, i.e., the optimal barrier location, denoted by opt ; and (2) ho to form the barrier at opt, i.e., the optimal sensor movement strategy, hich can be described as a mapping function M opt from S to T. In general, a mapping function M : S T is defined as follos: M(i) = j means that sensor s i moves to destination t j, hile M(i) = means that sensor s i remains stationary. Sensors not used in the initial barrier may participate in forming future barriers after the operational lifetime of the current barrier has expired. Formally, the problem e try to address in this paper is described as follos: Given: rectangular deployment region: L coverage radius: R s48 L x total number of deployed sensors: N initial sensor positions: (x i, y i ) for each s i S x-coordinate of the destinations: α j for each t j T Output: optimal barrier location opt and sensor movement strategy M opt : { opt, M opt } = arg min D(M, ) here {,M} (xi ) 2 D(M, ) = max α M(i) + (yi ) 2. s i S,M(i) Constraint: opt Essentially, e solve a min-max problem. Our motivation is the assumption that the sensor ith the maximum moving distance runs out of energy first, hich ould break the barrier. Therefore, our scheme produces a barrier ith the longest operational lifetime. e use D(M, ) to denote the maximum moving distance of the sensors hen they follo strategy M to form a barrier at location. e restrict the barrier location beteen and. IV. PROPOSED SCHEME A. Overvie of the Proposed Scheme To determine the optimal barrier location opt, e first reduce its search space from a continuous range [, ] to a discrete set ith less than N 2 Nmin 2 points, hich are called candidate barrier locations. Afterards, e propose an efficient iterative algorithm to search over this discrete set. The orst-case number of iterations is O(N 2 Nmin 2 ), but in practice, our algorithm uses approximately O(NN min ) iterations. B. Identification of Candidate Barrier Locations Candidate barrier locations are derived from the minimum and intersection points of the group of functions defined belo. Definition (Function of Moving Distance). Suppose a sensor s i moves to a destination t j. The moving distance of s i can be represented as a function of the barrier location, i.e., f i,j () = [x i (2j )R] 2 + (y i ) 2, () here. e use F to denote the set of all f associated ith a sensor netork of N sensors, i.e., F = {f i,j () i N, j N min, }. (2) For simplicity, e also define f i, () =, i,. ith these functions defined, e can no define the candidate barrier locations. Definition 2 (Candidate Barrier Locations). The set of candidate barrier locations is Φ = Φ mins Φ ints, here Φ mins = { arg min f i,j()} includes all the values f i,j F that yield a minimum value for at least one of the f i,j functions in F ; Φ ints = { f i,j () = f m,n()} includes all the f i,j F,f m,n F values at hich to functions in F intersect. Let Φ denote the total number of candidates in Φ. e sort these candidates in ascending order and name them {K i i =,, Φ }. Let Φ mins and Φ ints denote the number 573

3 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks of candidates in Φ mins and Φ ints, respectively. Then, e have O(N 4 ) candidates, as follos: Φ = Φ mins + Φ ints ( NNmin ) NN min + = O(N 2 N 2 2 min ) = O(N 4 ). (3) Fig. 3 plots all the functions in F corresponding to the sensor netork shon in Fig. 2. In this example, Φ mins = 3, Φ ints =, and Φ = 3. s 3 t t 2 s t 3 s 2 L Initial Position Destination Position Fig. 2. An example sensor netork of 3 mobile sensors. { } = min max f i,m(i)(k j ), max f i,m(i)(k j+ ) i N i N = min {D (M, K j ), D (M, K j+ )}. (7) C. Iterative Algorithm ith the candidate barrier locations being identified, e further reduce the search complexity by proposing an iterative algorithm in hich only a small portion of candidates are checked. Fig. 4 shos the flochart of our iterative algorithm, hich is explained in detail next. Initialization: curr =, opt = curr, M opt =, D opt = Customized Bottleneck Bipartite Matching 25 f,3 G(V,E) : V = S T, E = {(s i,t j ) s i S,t j T}, eight(s i,t j ) = f i,j ( + curr ) 2 f 2,3 M =Bottleneck Bipartite Matching(G) f ij () 5 f,2 f 3, f 2,2 i = argmaxf i,m (i)( curr + ) i 5 f, f 2, K K 2 K 3 K 4 K 5 K 6 K 7 K 8 K 9 K K K 2 Fig. 3. For the example netork in Fig. 2, there are 9 functions in F and 3 candidate barrier locations. Φ mins = {K 3, K 4, K 3 } and Φ ints = {K, K 2, K 5, K 6, K 7, K 8, K 9, K, K, K 2 }. The optimal barrier location opt is guaranteed to be at one of the candidate barrier locations, as Theorem belo proves, any movement strategy M for a barrier at any beteen to adjacent candidate locations can alays yield a smaller maximum moving distance at one of these to candidates. Theorem. M, (K j, K j+ ), here j Φ, D(M, )> min {D (M, K j ), D (M, K j+ )}. Proof: Let s i denote the sensor that has the maximum moving distance in M at. This means f 3,2 K 3 i, f i,m(i) () f i,m(i )(). (4) Since K j and K j+ are adjacent candidate locations, no other functions in F intersect beteen them. Therefore, (4) implies that { fi,m(i) (K j ) f i,m(i )(K j ), i, (5) f i,m(i) (K j+ ) f i,m(i )(K j+ ), i. In other ords, f i,m(i )(K j ) = max f i,m(i)(k j ), i N f i,m(i )(K j+ ) = max f i,m(i)(k j+ ). i N Moreover, according to the definition of candidate barrier locations, all the functions in F are monotone beteen to adjacent candidates such as K j and K j+. Therefore, e have: D(M, ) = max f i,m(i)() = f i i N,M(i )() (Definition) > min { f i,m(i )(K j ), f i,m(i )(K j+ ) } (Monotone) (6) D(M, curr ) = max i N f i,m (i)( curr ) D(M, curr ) < D opt? no Continue ith the Next Candidate next = min{k j K j > curr,k j Φ i,m (i )} Does next exist? no STOP, output M opt, opt yes Update: opt = curr, M opt = M, D opt = D(M, curr ) yes curr = next Fig. 4. Flochart of the iterative algorithm (note that + curr = curr + δ). The algorithm starts ith curr =. After initialization ith the optimal min-max moving distance D opt set to infinity, the Bottleneck Bipartite Matching (BBM) algorithm [3] is applied to determine the best movement strategy for the current barrier location as follos. The input to BBM is a eighted bipartite graph G(V, E): V = S T here S is the set of sensors, T is the set of destinations along curr ; E = {(s i, t j ) s i S, t j T }; eight(s i, t j ) = f i,j ( curr) + here curr + = curr +δ and δ is a positive offset hich is sufficiently small so that curr + does not reach the next candidate in Φ. Recall that f i,j is the function of moving distance for sensor s i to reach destination t j. BBM returns a max-cardinality matching hose maximum edge eight is minimized. In other ords, it produces a 574

4 f ij () 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks movement strategy hich minimizes the maximum moving distance of sensors from their initial positions to the destination positions along curr. In this customized BBM algorithm, the offset δ is used as a tie breaker in case there are multiple movement strategies that all yield the same min-max moving distance at curr. For example, in Fig. 5, at curr = K 2, there exist to such movement strategies: M = { 2, 2, 3 3} and M 2 = {, 2 2, 3 3}. By adding a small offset, the tie is broken and M 2 is chosen hich yields a smaller maximum moving distance than M at K 2 +. This serves as an important base for the next round of iteration, as shon in the proof of Theorem 2. As shon in the flochart, e record the output of the customized BBM algorithm as follos: M : the min-max matching at + curr; i : the index of the sensor that has the maximum moving distance in M at + curr; D(M, curr ): the min-max moving distance at curr. The next candidate location to check is next = min{k j K j > curr, Kj Φ i,m (i )}, (8) here Φ i,m (i ) is the set of candidates along f i,m (i ) hich is formally defined belo. Definition 3 (Candidate Barrier Locations along f i,j ). The set of candidate barrier locations along the f i,j function is Φ i,j = Φ mins i,j Φints, here i,j Φ mins i,j = { arg min f i,j()} includes the value at hich f i,j achieves its minimum; Φ ints i,j = { f i,j () = f m,n()} includes all the f m,n F values at hich f i,j intersects ith another function in F. Note that next is the first candidate location along the f i,m (i ) function after curr. There might exist other candidate locations beteen curr and next but belong to a different f function, hich are skipped in our iterative algorithm to reduce the search complexity. For example, in Fig. 5, the next candidate to check after K 2 is K 6. It is determined ith the folloing steps: () the min-max matching at K 2 + is M = {, 2 2, 3 3}; (2) the sensor that has the maximum moving distance in M at K 2 + is s 3; (3) the first candidate along after K 2 is K 6. Comparing Fig. 5 ith Fig. 3, e can see that candidate locations K 3, K 4, and K 5 are skipped. The reason hy these candidate locations can be skipped is that, as e ill sho in Theorem 2, at any beteen curr and next, the maximum moving distance of any movement strategy is alays larger than or equal to that of M at curr or next. Theorem 2. M, [ curr, next ], here curr is the current candidate location in the iterative algorithm, and next is the next candidate location to check as defined in (8), e alays have D(M, ) min {D(M, curr ), D(M, next )}, here M is the min-max matching at curr. + Proof: Recall that i is the index of the sensor that has the maximum moving distance in M at + curr, i.e., i = arg max f i,m i (i)( curr). + (9) Iteration # Iteration #2 Iteration #3 Iteration #4 f,2 f 2,2 f 2,3 f,3 f 3, ( opt,d opt ) f2,2 f 5 f, f, f 2, 3,2 K 2 K 6 K 2 Fig. 5. The iterative algorithm starts ith curr =. Iteration #: The minmax matching at curr + is { 2, 2, 3 3}, in hich s 3 has the maximum moving distance at curr. + The next candidate to check is the first candidate location along after curr =, hich is K 2. Iteration #2: The minmax matching at K + 2 is {, 2 2, 3 3}, in hich s 3 has the maximum moving distance at K + 2. The next candidate to check is the first candidate along after K 2, hich is K 6. The same process repeats for Iteration #3 and Iteration #4 till the algorithm terminates. Therefore, the folloing inequality holds for all i: f 2,2 f, f i,m (i )( + curr) f i,m (i)( + curr). () As next is the first candidate location along the f i,m (i ) function after curr, no other functions in F intersect ith f i,m (i ) beteen curr and next. Consequently, e have: [ curr, next ], i, f i,m (i )() f i,m (i)(). () Hence, e have: [ curr, next ], f i,m (i )() = D(M, ). (2) On the other hand, let i denote the index of the sensor that has the maximum moving distance in M at + curr, i.e., i = arg max f i,m i (i)( curr). + (3) As M is the min-max matching at + curr, e have: f i,m (i )( + curr) f i,m (i )( + curr). (4) Similarly, due to the fact that no other functions in F intersect ith f i,m (i ) beteen curr and next, e have: [ curr, next ], f i,m (i )() f i,m (i )(). (5) As the definition of D(M, ) implies: e have: D(M, ) = max f i,m i N (i)() f i,m (i )(), (6) [ curr, next ], D(M, ) f i,m (i )(). (7) Furthermore, as the f i,m (i ) function is monotone beteen curr and next, (7) implies: [ curr, next ], D(M, ) min{f i,m (i )( curr ), f i,m (i )( next )}. Finally, by combining (2) ith (8), e have: [ curr, next ], D(M, ) min{d(m, curr), D(M, next)}. (8) (9) 575

5 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks The algorithm terminates hen next cannot be found, meaning that e have completed the search over the entire range [, ]. Fig. 5 illustrates the iterative algorithm ith an example and the iteration process is explained in its caption. D. Complexity Analysis In theory, there is a total of O(N 2 Nmin 2 ) candidates in Φ, hich means that, in the orst case, e may need O(N 2 Nmin 2 ) iterations to complete the search. Hoever, in practice, the number of iterations is more comparable to O(NN min ). This is because our algorithm checks the candidate barrier locations iteratively along a continuous function that is composed of multiple sections from different f functions. In other ords, in each iteration, our algorithm only checks the candidate locations along a particular section of a single f function. Thus, the total number of checked locations is on the same order as the number of candidate locations along a single f function, hich is O(NN min ). In Section V, e validate this complexity analysis by simulation. V. EVALUATION RESULTS e evaluate our algorithm by simulating to types of sensor deployment: random uniform deployment, and linebased deployment. In both scenarios, e compare the optimal min-max moving distance found by our algorithm to that of a naive algorithm that fixes the barrier location to 2, hich is the middle point of the deployment region. e refer to our algorithm as Opt and the naive algorithm as Mid. The time complexity of our algorithm is also evaluated in terms of the number of checked candidate locations. A. Uniform Deployment e first evaluate our algorithm ith a random uniform deployment of N sensors in an L region. ) Min-max Moving Distance: Fig. 6 shos the min-max moving distances of the to algorithms varying ith N in the uniform case. e test different sizes of deployment region. For each size, the effect of redundant sensors on the min-max moving distance is also tested. e first note that, regardless of the size of the deployment region, the min-max moving distance of both algorithms decreases as the number of sensors increases. This is an intuitive result, as more sensors deployed in the area means a higher chance that one or more sensors ill already be close to each final destination. e also note that, regardless of the size of the deployment region, Opt outperforms Mid more hen N is larger. This is because the horizontal moving distance tends to dominate the overall 2D moving distance as L, but as the number of sensors increases, the horizontal moving distance decreases. Therefore, the benefit of optimizing vertical moving distance becomes relatively larger. Finally, e note from Fig. 6(b) that, given the same L and N, the difference beteen Opt and Mid is larger hen is larger. This is again due to the increased proportion of vertical distance in the total moving distance. Fig. 6 shos the average min-max moving distances of Opt and Mid. To further illustrate the improvement of Opt over Mid, Fig. 7 presents the cumulative distribution function (CDF) of the absolute and relative improvement of Opt over (m) Opt, L=5, =5 Mid, L=5, =5 Opt, L=, =5 Mid, L=, =5 Opt, L=2, =5 Mid, L=2, = N (a) Fixed and varying L. (m) Opt,L=,=2 Mid,L=,=2 Opt,L=,=5 Mid,L=,=5 Opt,L=,= Mid,L=,= 5 5 N (b) Fixed L and varying. Fig. 6. Min-max moving distance hen N sensors are uniformly deployed in an L region. The coverage radius of each sensor is R = m. Mid in terms of min-max moving distance, hich is a result of trials for each of three different setups shon in the figure. In the scenario here N = 5, Opt has a min-max moving distance hich is on average 2.5 m or.2% less than that of Mid, but in around 4% of the trials, the improvement is greater. In the most extreme case, Opt reduces the min-max moving distance by 9.4 m or 38%. Similar observations can be made for the other to setups..6 L=,=5,N=5 L=,=5,N= L=,=5,N= Absolute improvement (m).5 L=,=5,N=5 L=,=5,N= L=,=5,N= Relative improvement (%) Fig. 7. CDF of the absolute and relative improvement of Opt over Mid in terms of min-max moving distance. 2) Number of Checked Candidate Locations: Table I shos the number of total and checked candidate locations of selected experiments from the uniform scenario. Total is the number of candidates in Φ, hich is up to N 2 Nmin 2, hile check is the number of candidates checked in our iterative algorithm. hen N, L, or increases, the number of total and checked candidates increases accordingly. Hoever, in any scenario, only a small portion of candidates are checked, ith a number even less than NN min, hich indicates that our algorithm is efficient and scalable. For example, hen L = 2 m, = m, N min =, given N = 3 sensors, NN min = 3, but only 783 candidates are checked. TABLE I TOTAL AND CHECKED CANDIDATES FOR THE UNIFORM DEPLOYMENT CASE. R = M AND N MIN = L/2R. L=,=5 N min = 5 L=2,=5 N min = L=2,= N min = N total check N total check N total check B. Line-based Deployment The second deployment scenario is the line-based sensor deployment strategy proposed in [4], here N sensors are deployed in an L region along the line of y = 2. Each sensor s i is deployed at its final position [(2j )R, 2 ] ith a random x-axis error δi x and y-axis error δ y i, here 576

6 25 IEEE ireless Communications and Netorking Conference (CNC): - Track 3: Mobile and ireless Netorks δi x, δy i N(, σ). In practice, this error could be the result of ind or other environmental conditions during an air drop. hen redundant sensors are deployed, sensors are assumed to be dropped in groups, e.g., for N = 2N min, to sensors are dropped at each position. ) Min-max Moving Distance: Fig. 8(a) shos the minmax moving distances of Opt and Mid ith varying σ. As σ increases, the min-max moving distances of both Opt and Mid increase, because the sensors tend to be initially deployed farther from their final positions. Also, as σ increases, Opt outperforms Mid more. This is because hen σ is larger, the sensors ill be scattered in a ider region. It is more necessary to optimize the vertical moving distance in this case. Interestingly, hen σ is fixed, the difference beteen Opt and Mid is larger hen N is smaller, hich is the opposite of the observations in the uniform deployment case. This is due to the folloing reasons. Firstly, hen N increases and multiple sensors are dropped at each point, e tend to have a combination of sensors that yields an optimal barrier location close to 2 ; therefore, the difference beteen the vertical moving distance of Opt and Mid is smaller. Secondly, the vertical and horizontal moving distances are comparable in the line-based deployment case; hence, the reduction of the vertical moving distance can be reflected in the overall 2D moving distance. (m) Opt, N=5 Mid, N=5 Opt, N= Mid, N= Opt, N=5 Mid, N= σ (m) (a) Min-max moving distance vs. σ given three different N values..59 σ=5,n=5 σ=5,n=5 σ=2,n=5 σ=2,n= Absolute improvement (m) (b) CDF of the absolute improvement of Opt over Mid in terms of min-max moving distance. Fig. 8. The min-max moving distance for the line-based deployment strategy, ith L = m, = 5 m, R = m, and N min = 5. Similar to before, e plot in Fig. 8(b) the CDF of the improvement of Opt over Mid in four different setups. e can see that hen σ = 2 m and N = 5, on average, the min-max moving distance of Opt is 4.6 m or 8.7% less than Mid. In the most extreme case, the min-max moving distance of Opt may be up to 9.3 m or 36% less than Mid. Similar observations can be made for the other setups. 2) Number of Checked Candidate Locations: Table II shos the number of total and checked candidate locations of selected experiments from the line-based deployment scenario. Similar to before, hen N or σ increases, the number of total and checked candidates increases, but only a small proportion of the total candidates are checked in any scenario. C. Discussions As shon above, our proposed scheme outperforms the scheme that fixes the barrier location, significantly in terms of the min-max moving distance, thus saving much energy on sensor movement. As a tradeoff, it may increase the computational complexity at the central processing unit, hich usually is much more poerful than sensor nodes and has TABLE II TOTAL AND CHECKED CANDIDATES FOR THE LINE-BASED DEPLOYMENT CASE. L = M, = 5 M, R = M, AND N MIN = 5. N σ = 5 σ = 2 total check total check a sustainable poer supply. On the other hand, these to schemes incur the same communication overhead, as both algorithms are executed by the central processing unit thus all sensors need to report their initial positions to there. VI. CONCLUSIONS AND FUTURE ORK In conclusion, e have shon that our proposed algorithm reduces the min-max moving distance for mobile sensors in a barrier coverage netork by determining the optimal barrier location, instead of fixing it a priori. Depending on the deployment scheme, this reduction can be up to 38% of the min-max moving distance obtained by an algorithm that uses an intuitive fixed barrier location. Additionally, simulation results sho that our proposed algorithm for finding the optimal barrier location is both efficient and scalable. Future ork includes extending the proposed scheme to K-barrier coverage and to use the probabilistic coverage model, in hich sensors may collaborate ith each other to further reduce the moving distance and the number of sensors used. ACKNOLEDGMENT Xiaoyun Zhang ould like to thank China Scholarship Council for the financial support. REFERENCES [] S. Kumar, T. H. Lai, and A. Arora, Barrier coverage ith ireless sensors, in Proc. of ACM MobiCom, Aug. 25. [2] B. Liu, O. Dousse, J. ang, and A. Saipulla, Strong barrier coverage of ireless sensor netorks, in Proc. of ACM MobiHoc, May 28. [3] A. Chen, S. Kumar, and T. H. 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