c3.1, the path, denoted by P
3.1−2, that begins at v
a, arrives at v
z,walong the shortest path between v
aand v
z,w, and stops at v
x,yalong the shortest path between v
z,wand v
x,y, shown in Fig. 3b, is shorter than P
3.1−2. It is a contradiction. Similarly, for c3.2, the path, denoted by P
3.2−2, that begins at v
a, arrives at v
x,yalong the shortest path between v
aand v
x,y, and stops at v
z,walong the shortest path between v
x,yand v
z,w, shown in Fig. 3c, is shorter than P
3.2−2, constituting another contradiction. For c3.3, dist(v
p,q, v
x,y) < 4
RtRs
+ 1 and dist(v
p,q, v
z,w) < 4
RtRs
+ 1 by Lemma 1, the path, denoted by P
3.3−1, that begins at v
a, arrives at v
p,qalong the shortest path between v
aand v
p,q, and stops at v
x,yalong the shortest path between v
p,qand v
x,y, shown in Fig. 3d, is shorter than P
3.3−1, and the path, denoted by P
3.3−2, that begins at v
a, arrives at v
p,qalong the shortest path between v
aand v
p,q, and stops at v
z,walong the shortest path between v
p,qand v
z,w, shown in Fig. 3d, is shorter than P
3.3−2. It is a contradiction.
Theorem 1. Sensors deployed on the grid points in DEP fully cover all critical grids and form a connected wireless sensor network.
Proof. By Lemma 2, each terminal node is a leaf in ST . It implies that sensors deployed on the grid points in DEP are connected, and thus form a connected wireless sensor network. We must show that sensors deployed on the grid points in DEP fully cover all critical grids. Consider the critical grid on which grid point (i, j) is located. There is a terminal node t
i,jin G. Since t
i,jis in node-weighted Steiner tree ST established by STBCGCA, edge (v
x,y, t
i,j) is in ST for some non-terminal node v
x,y. It implies that grid point (x, y) is in DEP . It also implies that (v
x,y, t
i,j) is in G, and thus the sensor deployed on grid point (x, y) fully covers the critical grid on which grid point (i, j) is located.
2.4 Performance Guarantee
Theorem 2 proves the performance guarantee for STBCGCA using the following lemmas:
Lemma 3. The sum of the weights of nodes and edges in the node-weighted Steiner tree ST established by STBCGCA, w(ST ), is (4
RtRs
+
1) |V
2| + |DEP |.
Proof. Since each terminal node is a leaf in ST by Lemma 2, |V
2| edges in ST are in E
2. In addition, the DEP formed by STBCGCA is a set of non-terminal nodes in ST . Since each non-terminal node is assigned weight 1, w(ST ) = (4
RtRs
+ 1)|V
2| + |DEP |.
6
weighted Steiner tree ST
OP T, w(ST
OP T), is (4
RtRs
+ 1)|V
2| + |DEP
OP T|, where DEP
OP Tdenotes the optimal solution to CRITICAL-SQUARE-GRID COVERAGE.
Proof. We first show that each terminal node is a leaf in ST
OP T. Assume that there exists one terminal node t
i,jand two non-terminal nodes v
x,yand v
z,wsuch that edges (t
i,j, v
x,y) and (t
i,j, v
z,w) are in ST
OP T. We can construct a node-weighted Steiner tree, ST
1, by deleting (t
i,j, v
z,w) from ST
OP Tand adding the shortest path between v
x,yand v
z,wto ST
OP T. By Lemma 1, dist(v
x,y, v
z,w) is smaller than the weight of (t
i,j, v
z,w). This implies that w(ST
1) < w(ST
OP T), constituting a contradiction.
Next, we show the number of non-terminal nodes in ST
OP Tis |DEP
OP T|.
Let DEP
STOP Tbe the set of grid points (x, y) such that non-terminal node v
x,yis in ST
OP T. It is clear that sensors deployed on the grid points in DEP
STOP Tfully cover all critical grids. In addition, sensors deployed on the grid points in DEP
STOP Tform a connected wireless sensor network because each terminal node is a leaf in ST
OP T. Therefore, DEP
STOP Tis a solution to CRITICAL-SQUARE-GRID COVERAGE. This implies that |DEP
OP T| ≤ |DEP
STOP T|. Assume that |DEP
OP T| < |DEP
STOP T|.
We can construct a tree, ST
2(V
ST2, E
ST2), in the following steps: 1) V
ST2contains each non-terminal node v
x,y, such that the grid point (x, y) is in DEP
OP T, and contains each terminal node in G, and 2) E
ST2contains exactly one edge (t
i,j, v
x,y) for each terminal node t
i,j, where v
x,yis a non-terminal node in V
ST2, and contains each edge in the spanning tree in the subgraph of G induced by V
ST2∩V
1. It is easy to verify that ST
2is a node-weighted Steiner tree containing |V
2| edges in E
2and |DEP
OP T| nodes in V
1. This implies that w(ST
2) < w(ST
OP T). It is a contradiction.
Thus, w(ST
OP T) = (4
RtRs
+1)|V
2|+|DEP
STOP T| = (4
RtRs
+1)|V
2|+
|DEP
OP T|.
Theorem 2. The performance ratio of STBCGCA is bounded in O(
αβ3ln g
), where α =
Rs, β =
Rt, and g
denotes the number of critical grids.
Proof. By Theorem 1.1 of [2],
w(STw(ST )OP T)
≤ 2 ln |V
2|. In addition, w(ST ) = (4
RtRs
+ 1)|V
2| + |DEP | by Lemma 3, and w(ST
OP T) = (4
RtRs
+ 1) |V
2| + |DEP
OP T| by Lemma 4. It is easy to verify that
|DEP|DEP |OP T|≤
(4 Rs
Rt
+1)|V2|(2 ln |V2|−1)
|DEPOP T|
+ 2 ln |V
2|. Because |DEP
OP T| ≥
2πR|V2|2s2
, we have
|DEP |
|DEPOP T|
≤
(4Rs
Rt
+1)|V2|(2 ln |V2|−1)
|V2|2
(2πR
s2) + 2 ln |V
2| = O(
RRts32ln |V
2|) = O(
αβ3ln |V
2|). It is noted that |V
2| = g
.
7
In this project, CRITICAL-SQUARE-GRID COVERAGE was shown to be NP-Complete in [1]. In addition, we proposed an approximation algorithm, STBCGCA, in [3] for CRITICAL-SQUARE-GRID COVERAGE. STBCGCA constructs an auxiliary graph G , uses Klein and Ravi’s approximation algorithm [2] to establish a node-weighted Stenier tree ST in G , and selects a set of grid points based on ST so that sensors deployed on the selected grid points form a connected wireless sensor network and fully cover all critical grids. In [3], to evaluate the performance of STBCGCA, we proposed another three algorithms, C-RNP, MSC-RNP, and MSC-MNWST, and investigated the numbers of sensors deployed by STBCGCA, C-RNP, MSC-RNP, and MSC-MNWST. C-RNP, MSC-RNP, and MSC-MNWST are all two-phase algorithms that deploy sensors to fully cover all critical grids in phase one and deploy sensors to establish the connectivity between sensors in phase two. Simulations show that STBCGCA deploys fewer sensors than C-RNP, MSC-RNP, and MSC-MNWST, in most cases.
4 參 參 參考 考 考文 文 文獻 獻 獻
[1] W. C. Ke, B. H. Liu, and M. J. Tsai, “The critical-square-grid coverage problem in wireless sensor networks is NP-complete,” Computer Networks, vol. 55, no. 9, pp.
2209–2220, 2011.
[2] P. N. Klein and R. Ravi, “A nearly best-possible approximation algorithm for node-weighted steiner trees,” J. Algorithms, vol. 19, no. 1, pp. 104–115, Jul. 1995.
[3] W. C. Ke, B. H. Liu, and M. J. Tsai, “Efficient algorithm for constructing minimum size wireless sensor networks to fully cover critical square grids,” IEEE Transactions on Wireless Communications, vol. 10, no. 4, pp. 1154–1164, 2011.
5 計 計 計畫 畫 畫成 成 成果 果 果自 自 自評 評 評
在此計劃中，我們分析了使用最少感測器佈置在一個被分割成正方形格子的 感測區域並能形成一個連通且覆蓋所有關鍵區域的感測網路的問題困難度，
並將此結果發表在 Computer Networks 國際期刊上 [1] ， 另外，本研究團隊提出了 一接近演算法來解決此問題，並將其成果發表在 IEEE Transactions on Wireless
Communications 國際期刊上 [3] 。在此計劃的相關研究中，有另外三篇相關成果分
別發表在國際 / 國內會議上，詳細資料請參酌附錄，整體而言，本研究計畫之學 術價值已被國內外學者所認可、接受及肯定。
8
相關發表論文陳列如下：
[ 附錄一 ] Wei-Chieh Ke, Bing-Hong Liu, and Ming-Jer Tsai, “Efficient algorithm for constructing minimum size wireless sensor networks to fully cover critical square grids,” IEEE Transactions on Wireless Communications , vol. 10, no. 4, pp. 1154-1164, 2011.
[ 附錄二 ] Wei-Chieh Ke, Bing-Hong Liu, and Ming-Jer Tsai, “The CRITICAL-SQUARE-GRID COVERAGE problem in wireless sensor networks is NP-complete,” Computer Networks , vol. 55, no. 9, pp. 2209-2220, 2011.
[ 附錄三 ] Bing-Hong Liu, Ying-Hong Jhuang, Li-Ping Tung, and Jyun-Yu Jhang, “An improved method of constructing a data aggregation tree in wireless sensor networks,” ICGEC 2012 , 2012, pp. 344-347.
[ 附錄四 ] Bing-Hong Liu, Jyun-Yu Jhang, and Kuo-Wen Su, “GPS-free event-to-sink routing scheme for data aggregation in wireless sensor networks,” IBICA 2011 , 2011, pp.
25-28.
[ 附錄五 ] 王維聖、莊英宏、蘇國文、張峻瑜、劉炳宏， 2011 ， “ 應用於具障礙物 移動感測網路之利益點覆蓋方式 ” ， 2012 Conference on Information Technology and Applications in Outlying Islands , 2012.
9
附 附 附錄 錄 錄一 一 一
10
Efficient Algorithm for Constructing
Minimum Size Wireless Sensor Networks to Fully Cover Critical Square Grids
Wei-Chieh Ke, Bing-Hong Liu, and Ming-Jer Tsai
Abstract—Wireless sensor networks are formed by connected sensors that each have the ability to collect, process, and store environmental information as well as communicate with others via inter-sensor wireless communication. These characteristics allow wireless sensor networks to be used in a wide range of applications. In many applications, such as environmental mon-itoring, battlefield surveillance, nuclear, biological, and chemical (NBC) attack detection, and so on, critical areas and common areas must be distinguished adequately, and it is more practical and efficient to monitor critical areas rather than common areas if the sensor field is large, or the available budget cannot provide enough sensors to fully cover the entire sensor field. This provides the motivation for the problem of deploying the minimum sensors on grid points to construct a connected wireless sensor network able to fully cover critical square grids, termed CRITICAL-SQUARE-GRID COVERAGE. In this paper, we propose an approximation algorithm for CRITICAL-SQUARE-GRID COV-ERAGE. Simulations show that the proposed algorithm provides a good solution for CRITICAL-SQUARE-GRID COVERAGE.
Index Terms—Wireless sensor network, coverage problem, NP-Complete problem, sensor deployment, approximation algorithm.
I. INTRODUCTION
W
IRELESS sensor networks are formed by connected sensors that each have the ability to collect, process, and store environmental information as well as communicate with others via inter-sensor wireless communication. These characteristics allow wireless sensor networks to be used in a wide range of applications, including health care, environ-mental monitoring, battlefield surveillance, intruder detection, and so on. Recently, the study of wireless sensor networks has become one of the most important areas of research [1]–[6]. A wireless sensor network must achieve the specified coverage level of the application so that the quality of service provided by the wireless sensor network can be guaranteed. Here, we address the coverage problem in wireless sensor networks, the WSN coverage problem.In the art gallery problem, cameras are deployed such that the whole gallery is thief-proof [7]–[9]. The cameras are
Manuscript received January 29, 2010; revised August 15, 2010 and De-cember 9, 2010; accepted January 26, 2011. The associate editor coordinating the review of this paper and approving it for publication was H. Thomas.
W. C. Ke and M. J. Tsai are with the Department of Computer Science, National Tsing Hua University, 101, Kuang Fu Rd., Sec. 2, Hsinchu 30013, Taiwan, ROC (e-mail: [email protected], [email protected]).
B. H. Liu is with the Department of Electronic Engineering, National Kaohsiung University of Applied Sciences, 415, Chien Kung Rd., Kaohsiung 80778, Taiwan, ROC (e-mail: [email protected]).
Digital Object Identifier 10.1109/TWC.2011.021611.100123
assumed to see an infinite distance and 360∘, i.e., the cameras have unlimited sensing ranges. In the WSN coverage problem, however, sensors have limited sensing ranges. In the circle covering problem, overlapping equal circles are used to fully cover rectangles [10], equilateral triangles [11], and squares [12]. The circle covering problem is different from the WSN coverage problem because the circles are independent and can be located anywhere. Because connectivity between sensors must be established, their locations are limited.
In a dense wireless sensor network, many methods only activate a subset of sensors responsible for surveillance in order to prolong the network lifetime. A centralized method is proposed to partition sensors into mutually exclusive sets such that sensors in each set fully cover the entire sensor field [13]. OGDC is a distributed algorithm used to activate a subset of sensors to fully cover the entire sensor field at one time [14]. In order to ensure that each point in a sensor field is covered by at least𝑘 sensors, a subset of sensors are selected for ensuring𝑘-coverage of a wireless sensor network [15]. In addition, a method is proposed to select a subset of senors for the construction of a wireless sensor network with𝑘-coverage and𝑘′-connectivity [16], where a𝑘′-connected wireless sensor network is disconnected only if at least𝑘′sensor failures exist.
Many sensor deployment algorithms attempt to fully cover a sensor field using the minimum sensors or the minimum cost of sensors. A method is proposed to deploy sensors to provide full coverage on a sensor field with obstacles [17]. Methods of deploying directional sensor networks are proposed to fully cover a sensing field [18]. Given heterogeneous sensors, sensor deployment algorithms achieve full coverage [19] or k-coverage [20] on a sensor field using near-minimum cost of sensors. A simulated annealing algorithm deploys𝑘 mutually exclusive sets of sensors such that sensors in each set fully cover the entire sensor field in an attempt to prolong the network lifetime [21]. In addition, an optimal regular pattern is proposed to deploy sensors for the construction of a wireless sensor network that is 2-connected and provides full coverage on the sensor field [22].
Sometimes, however, the sensor field is large, or the avail-able budget cannot provide enough sensors to fully cover the entire sensor field. Given a certain number of sensors, many sensor deployment algorithms attempt to maximize the area covered by sensors in a sensor field. An incremental deployment algorithm uses the data gathered from previously-deployed sensors to deploy subsequent sensors [23]. Some
1536-1276/11$25.00 c⃝ 2011 IEEE
methods consider the components in the sensor field (sensors, obstacles, and preferential fields) as sources of virtual forces and deploy sensors so that virtual forces in the field are balanced [24]–[26]. In addition, it is shown that if sensors are deployed based on the pattern of the equilateral triangle, the area covered by sensors is maximized [27].
In many applications, sensors are required to monitor specific targets or given points in a sensor field. CCANS determines a connected dominating set in a dense wireless sensor network such that the coverage probabilities of the given points each are larger than a given parameter [28].
MC-MIP partitions sensors into mutually exclusive sets and activates sensors in sets alternately to monitor given points [29]. In addition, Greedy-MSC improves on MC-MIP by prolonging the network lifetime with sensors that are allowed to participate in multiple sets [30].
In many applications, critical and common areas must be adequately distinguished; it is more practical and efficient to monitor critical areas than common areas. For example, in a wilderness ecological observation network, the “hot spots,”
such as nests of animals may be assigned to critical areas.
Because infinite points exist in the critical area, the method of covering given points cannot be used directly in these applications. This introduces the problem of deploying the minimum sensors on grid points to construct a connected wire-less sensor network able to fully cover critical square grids, termed CRITICAL-SQUARE-GRID COVERAGE. So far, CRITICAL-SQUARE-GRID COVERAGE has been shown to be NP-Complete [31]. However, to the best of our knowledge, no efficient algorithms for CRITICAL-SQUARE-GRID COV-ERAGE exist to date, thereby providing motivation for this paper. The remainder of this paper is organized as follows.
An approximation algorithm for CRITICAL-SQUARE-GRID COVERAGE, termed STBCGCA, is introduced in Section II, and analyzed in Section III. In Section IV, the problem of deploying heterogeneous sensors with minimum cost on grid points to construct a connected wireless sensor network able to fully cover critical square grids, termed CRITICAL-SQUARE-GRID COVERAGE-H, is introduced and an exten-sion of STBCGCA, termed STBCGCA-H, is proposed for CRITICAL-SQUARE-GRID COVERAGE-H. We evaluate, by simulations, the performance of STBCGCA and STBCGCA-H in Section V. Finally, we conclude this paper in Section VI.
II. STEINER-TREE-BASEDCRITICALGRIDCOVERING
ALGORITHM(STBCGCA)
We first illustrate MINIMUM NODE-WEIGHTED STEINER TREE [32], an NP-Complete problem, and introduce Klein and Ravi’s algorithm for MINIMUM NODE-WEIGHTED STEINER TREE [33], where Klein and Ravi’s algorithm is used to design STBCGCA. Subsequently, we describe CRITICAL-SQUARE-GRID COVERAGE [31].
Finally, we present STBCGCA, which is used to select the grid points for sensor deployment.
A. MINIMUM NODE-WEIGHTED STEINER TREE The problem is illustrated as follows:
INSTANCE: A graph𝐺 with nodes and edges assigned by nonnegative weights and a set of terminal nodes in 𝐺.
QUESTION: Find a tree𝑆𝑇 in 𝐺 such that each terminal node is in𝑆𝑇 , and the sum of the weights of nodes and edges in𝑆𝑇 is minimum.
Klein and Ravi propose an algorithm with approximation ratio 2 ln 𝑇 to construct a node-weighted Steiner tree [33], where 𝑇 is the number of terminal nodes. The algorithm maintains a set of disjoint trees containing all terminal nodes and iteratively merges the trees until only one tree exists.
Initially, each terminal node is in a tree by itself. In each iteration, at least two trees are merged into one by the following two steps. Firstly, the node that has the minimum quotient cost is selected. The quotient cost of a node is the minimum ratio of the sum of the weight of the node and the distances to at least two trees to the number of trees, where the distance to a tree denotes the minimum sum of weights of nodes and edges in the path, excluding its endpoints, to the tree. Secondly, the shortest paths, the paths having the minimum distances, between the node and the trees selected in the first step are used to merge the selected trees into one. Take Fig. 1a, for example. In graph𝐺, each of terminal nodes 𝑡1,1, 𝑡2,3, and𝑡3,2 is assigned weight 0; each non-terminal node is assigned weight 1; each edge between two non-terminal nodes is assigned weight 0; and each of the other edges is assigned weight 9. Klein and Ravi’s algorithm initially constructs three one-node trees𝑇1,𝑇2, and𝑇3containing nodes𝑡1,1,𝑡2,3, and 𝑡3,2, respectively. In iteration one, each node first computes its quotient cost. Take node𝑣2,2, for example. The distances from 𝑣2,2 to𝑇1,𝑇2, and𝑇3 are 10, 9, and 9, respectively. Because 𝑣2,2 has weight 1, the quotient cost of𝑣2,2 is 1+9+92 = 9.5, which is the ratio of the sum of the weight of 𝑣2,2 and the distances to 𝑇2 and 𝑇3 to the number of trees, due to the selection of 𝑇2 and 𝑇3. Similarly, the distances from 𝑡1,1 to 𝑇1,𝑇2, and 𝑇3 are 0, 20, and 20, respectively. Because𝑡1,1 has weight 0, the quotient cost of 𝑡1,1 is 0+0+202 = 10. It is easy to verify that 𝑣2,2 has the minimum quotient cost.
Therefore,𝑇2 and𝑇3are merged into one, denoted by 𝑇2+3, using the shortest path between 𝑣2,2 and𝑇2 and the shortest path between𝑣2,2 and𝑇3. In iteration two, the distances from 𝑣1,1 to𝑇1and𝑇2+3are 9 and 0, respectively, and it is easy to verify that𝑣1,1 has the minimum quotient cost. Therefore,𝑇1 and𝑇2+3are merged into one using the shortest path between 𝑣1,1 and𝑇1 and the shortest path between𝑣1,1 and𝑇2+3. Fig.
1b shows a node-weighted Steiner tree constructed by Klein and Ravi’s algorithm.
B. CRITICAL-SQUARE-GRID COVERAGE
The unit disk graph model [34], in which a sensor can send messages to another sensor if the transmission range 𝑅𝑡reaches that sensor, is used as the communication model.
The binary sensor model [13], [19], [35] is also employed.
In the binary sensor model, the probability of detecting an event by a sensor is 1 if the event is within the sensing range 𝑅𝑠; otherwise, the probability of detecting the event is 0. A sensor field, denoted by 𝐹 𝑖𝑒𝑙𝑑, is divided into grids of squares having length ℓ. A sensor is deployed on a grid point located on the center of a grid. A grid is a critical one
if the grid has to be fully covered, and𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 denotes the set of critical grids. We assume that𝑅𝑠 ⩾ √12ℓ and 𝑅𝑡 ⩾ ℓ so that a sensor deployed on a grid point can fully cover at least one grid and can communicate with sensors deployed on neighboring grids, where neighboring grids are defined as grids sharing a common boundary. CRITICAL-SQUARE-GRID COVERAGE is illustrated as follows:
INSTANCE:𝑅𝑠,𝑅𝑡,𝐹 𝑖𝑒𝑙𝑑, and 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙.
QUESTION: Find a connected wireless sensor network𝑊 constructed by deploying minimum sensors on the grid points in𝐹 𝑖𝑒𝑙𝑑 such that 𝑊 fully covers all critical grids in 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙.
Take Fig. 2a, for example, where 𝑅𝑠 = √210ℓ, 𝑅𝑡 =√ 𝐹 𝑖𝑒𝑙𝑑 is the sensor field, and 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 contains 3 grids on2ℓ, which grid points (1, 1), (2, 3), and (3, 2) are located. It is easy to verify that the connected wireless sensor network formed by two sensors deployed on grid points (1, 1) and (2, 2), shown in Fig. 2b, fully covers all critical grids.
C. STBCGCA
In MINIMUM NODE-WEIGHTED STEINER TREE, the terminal node must be in the established node-weighted Ste-nier tree. Because the critical grid must be fully covered in CRITICAL-SQUARE-GRID COVERAGE, our idea is to construct an auxiliary graph𝐺, use an existing algorithm to establish a node-weighted Stenier tree 𝑆𝑇 in 𝐺, and deploy sensors based on𝑆𝑇 . In the construction of 𝐺, each square grid is denoted by a non-terminal node, each critical square grid is denoted by a terminal node, there is a link between two non-terminal nodes if two sensors on the square grids denoted by the two non-terminal nodes can communicate with each other, and there is a link between a terminal node and a non-terminal node if the sensor on the square grid denoted by the non-terminal node can fully cover the square grid denoted by the terminal node. STBCGCA deploys sensors on all square grids denoted by non-terminal nodes in the established node-weighted Stenier tree. To obtain the minimum size set of non-terminal nodes, each non-non-terminal node is assigned weight 1 and each terminal node is assigned weight 0. To guarantee the connectivity of the constructed wireless sensor network, each terminal node must be a leaf of the established node-weighted Stenier tree. For this purpose, because in𝐺 the hop distance between two non-terminal nodes that have a common terminal neighbor is no more than 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉, as described in Lemma 1, each edge between two non-terminal nodes is assigned weight 0 and each edge between a terminal node and a non-terminal node is assigned weight 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉+1. Given 𝐹 𝑖𝑒𝑙𝑑, 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙, 𝑅𝑠, and 𝑅𝑡, STBCGCA constructs a set of grid points for sensor deployment,𝐷𝐸𝑃 , by the following three steps:
1) Construction of Graph 𝐺(𝑉, 𝐸) and Terminal Set 𝑇 𝑒𝑟𝑚: Let 𝑉1 be the set of nodes 𝑣𝑥,𝑦 for all grid points (𝑥, 𝑦) in 𝐹 𝑖𝑒𝑙𝑑, and let 𝑉2 be the set of nodes 𝑡𝑖,𝑗 for all grid points (𝑖, 𝑗) located on critical grids in 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙. Then, 𝑉 = 𝑉1∪ 𝑉2. Let𝐸1 be the set of edges (𝑣𝑥,𝑦, 𝑣𝑧,𝑤) for all grid points (𝑥, 𝑦) and (𝑧, 𝑤) with a distance not greater than 𝑅𝑡, and let𝐸2be the set of edges (𝑣𝑥,𝑦, 𝑡𝑖,𝑗) for all grid points (𝑥, 𝑦) and (𝑖, 𝑗), such that the sensor deployed on grid point (𝑥, 𝑦) fully covers the critical grid on which grid point (𝑖, 𝑗)
is located. Then, 𝐸 = 𝐸1∪ 𝐸2. Each node in 𝑉1 is assigned weight 1; each node in 𝑉2 is assigned weight 0; each edge in𝐸1 is assigned weight 0; and each edge in 𝐸2 is assigned weight 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉ + 1. 𝑇 𝑒𝑟𝑚 is set to 𝑉2.
2) Establishment of Node-Weighted Steiner Tree 𝑆𝑇 : Based on 𝐺 and 𝑇 𝑒𝑟𝑚, Klein and Ravi’s algorithm [33] is used to establish𝑆𝑇 .
3) Formation of Set𝐷𝐸𝑃 : 𝐷𝐸𝑃 contains grid point (𝑥, 𝑦) if 𝑣𝑥,𝑦 is in𝑆𝑇 .
Take Fig. 2a, for example. Firstly,𝑣1,1is in𝑉1and assigned weight 1 because grid point (1, 1) is in 𝐹 𝑖𝑒𝑙𝑑; 𝑡1,1 is in 𝑉2 and assigned weight 0 because grid point (1, 1) is located on a critical grid in 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙; (𝑣1,1, 𝑣2,1) is in 𝐸1 and assigned weight 0 because the distance between grid points (1, 1) and (2, 1) is smaller than 𝑅𝑡; and (𝑣2,2, 𝑡2,3) is in 𝐸2and assigned weight 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉+1 = 9 because the sensor deployed on grid point (2, 2) fully covers the critical grid on which grid point (2, 3) is located. The constructed graph 𝐺 is shown in Fig.
1a. Secondly, a node-weighted steiner tree𝑆𝑇 is established, as shown in Fig. 1b. Finally,𝐷𝐸𝑃 = {(1, 1), (2, 2)} because 𝑣1,1 and𝑣2,2 are in 𝑆𝑇 .
III. ANALYSIS OFSTBCGCA
Sensors deployed on the grid points selected by STBCGCA are first shown to form a connected wireless sensor network and fully cover all critical grids. Subsequently, the perfor-mance guarantee for STBCGCA is proved.
A. Correctness of STBCGCA
Theorem 1 shows that sensors deployed on the grid points in 𝐷𝐸𝑃 fully cover all critical grids and form a connected wireless sensor network based on the following lemmas:
Lemma 1: If two sensors deployed on grid points (𝑥, 𝑦) and (𝑧, 𝑤) each fully cover the grid on which grid point (𝑖, 𝑗) is located, the distance between nodes 𝑣𝑥,𝑦 and 𝑣𝑧,𝑤 in 𝐺, 𝑑𝑖𝑠𝑡(𝑣𝑥,𝑦, 𝑣𝑧,𝑤), is no more than 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉.
Proof: Let 𝑆1 and 𝑆2 be two sensors deployed on grid points (𝑥, 𝑦) and (𝑧, 𝑤), respectively. It is easy to verify that the number of sensors deployed on grid points required to connect𝑆1and𝑆2is no more than⌈⌊∣𝑥−𝑧∣𝑅𝑡
ℓ ⌋ℓ⌉ + ⌈∣𝑦−𝑤∣⌊𝑅𝑡
ℓ ⌋ℓ⌉. Since 𝑆1 and𝑆2each fully cover the grid on which grid point (𝑖, 𝑗) is located, the Euclidean distance between 𝑆1 and 𝑆2 is no more than 2𝑅𝑠. Therefore, the number of sensors deployed on grid points required to connect𝑆1and𝑆2is no more than 2⌈⌊2𝑅𝑅𝑡𝑠
ℓ ⌋ℓ⌉. It implies that there exists a path, containing no more than 2⌈⌊2𝑅𝑅𝑡𝑠
ℓ ⌋ℓ⌉ internal nodes, between 𝑣𝑥,𝑦 and𝑣𝑧,𝑤 in the subgraph of𝐺 induced by 𝑉1. Therefore,𝑑𝑖𝑠𝑡(𝑣𝑥,𝑦, 𝑣𝑧,𝑤), the sum of the weights of internal nodes and edges in the shortest path between 𝑣𝑥,𝑦 and 𝑣𝑧,𝑤 in 𝐺, is no more than 4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉.
Lemma 2: Each terminal node is a leaf in the node-weighted Steiner tree𝑆𝑇 established by STBCGCA.
Proof: Assume that there exists one terminal node 𝑡𝑖,𝑗
and two non-terminal nodes 𝑣𝑥,𝑦 and 𝑣𝑧,𝑤 such that edges (𝑡𝑖,𝑗, 𝑣𝑥,𝑦) and (𝑡𝑖,𝑗, 𝑣𝑧,𝑤) are in 𝑆𝑇 . Let the tree containing
Fig. 1. Example of MINIMUM NODE-WEIGHTED STEINER TREE. (a) A graph𝐺 with nodes and edges assigned nonnegative weights, shown in parentheses, where𝑡1,1,𝑡2,3, and𝑡3,2 are terminal nodes. (b) A node-weighted Steiner tree𝑆𝑇 (𝑉𝑆𝑇, 𝐸𝑆𝑇), where 𝑉𝑆𝑇 = {𝑣1,1, 𝑣2,2, 𝑡1,1, 𝑡2,3, 𝑡3,2} and𝐸𝑆𝑇 = {(𝑣1,1, 𝑡1,1), (𝑣1,1, 𝑣2,2), (𝑣2,2, 𝑡2,3), (𝑣2,2, 𝑡3,2)}, shown in red.
𝑡𝑖,𝑗merge with𝑣𝑥,𝑦and𝑣𝑧,𝑤in iterations𝑎 and 𝑏, respectively, in step Establishment of Node-Weighted Steiner Tree 𝑆𝑇 . There are three cases: c1)𝑎 < 𝑏; c2) 𝑎 > 𝑏; and c3) 𝑎 = 𝑏.
We only adhere to the proof of c3 and omit the proofs of c1 and c2 due to their similarities. Let 𝑣𝑎 be the node having the minimum quotient cost in iteration𝑎. We first show that 𝑣𝑎∕= 𝑡𝑖,𝑗. Assume that𝑡𝑖,𝑗 has the minimum quotient cost in at least 1. This implies that the quotient cost of𝑣𝑥,𝑦 is smaller than that of𝑡𝑖,𝑗, constituting a contradiction. Therefore, there are three cases such that (𝑡𝑖,𝑗, 𝑣𝑥,𝑦) and (𝑡𝑖,𝑗, 𝑣𝑧,𝑤) are inserted into𝑆𝑇 in iteration 𝑎: c3.1) the shortest path between 𝑣𝑎 and 𝑡𝑖,𝑗, denoted by 𝑃3.1−1, begins at 𝑣𝑎, arrives at 𝑣𝑧,𝑤 along shown in Fig. 3c; and c3.3) the shortest path between𝑣𝑎 and 𝑣𝑥,𝑦, denoted by 𝑃3.3−1, begins at 𝑣𝑎, arrives at 𝑣𝑝,𝑞 along
3b, is shorter than𝑃3.1−2. It is a contradiction. Similarly, for
c3.2, the path, denoted by 𝑃3.2−2′ , that begins at 𝑣𝑎, arrives at 𝑣𝑥,𝑦 along the shortest path between 𝑣𝑎 and 𝑣𝑥,𝑦, and stops at 𝑣𝑧,𝑤 along the shortest path between𝑣𝑥,𝑦 and 𝑣𝑧,𝑤, shown in Fig. 3c, is shorter than𝑃3.2−2, constituting another contradiction. For c3.3,𝑑𝑖𝑠𝑡(𝑣𝑝,𝑞, 𝑣𝑥,𝑦) < 4⌈⌊𝑅𝑡𝑅𝑠 begins at 𝑣𝑎, arrives at 𝑣𝑝,𝑞 along the shortest path between 𝑣𝑎 and𝑣𝑝,𝑞, and stops at𝑣𝑧,𝑤 along the shortest path between 𝑣𝑝,𝑞 and 𝑣𝑧,𝑤, shown in Fig. 3d, is shorter than𝑃3.3−2. It is a contradiction.
Theorem 1: Sensors deployed on the grid points in 𝐷𝐸𝑃 fully cover all critical grids and form a connected wireless sensor network.
Proof: By Lemma 2, each terminal node is a leaf in𝑆𝑇 . It implies that sensors deployed on the grid points in 𝐷𝐸𝑃 are connected, and thus form a connected wireless sensor network. We must show that sensors deployed on the grid points in 𝐷𝐸𝑃 fully cover all critical grids. Consider the critical grid on which grid point (𝑖, 𝑗) is located. There is a terminal node𝑡𝑖,𝑗 in𝐺. Since 𝑡𝑖,𝑗 is in node-weighted Steiner tree 𝑆𝑇 established by STBCGCA, edge (𝑣𝑥,𝑦, 𝑡𝑖,𝑗) is in 𝑆𝑇 for some non-terminal node 𝑣𝑥,𝑦. It implies that grid point (𝑥, 𝑦) is in 𝐷𝐸𝑃 . It also implies that (𝑣𝑥,𝑦, 𝑡𝑖,𝑗) is in 𝐺, and thus the sensor deployed on grid point (𝑥, 𝑦) fully covers the critical grid on which grid point (𝑖, 𝑗) is located.
B. Performance Guarantee
Theorem 2 proves the performance guarantee for STBCGCA using the following lemmas:
Lemma 3: The sum of the weights of nodes and edges in the node-weighted Steiner tree𝑆𝑇 established by STBCGCA, 𝑤(𝑆𝑇 ), is (4⌈⌊𝑅𝑡𝑅𝑠
ℓ ⌋ℓ⌉ + 1)∣𝑉2∣ + ∣𝐷𝐸𝑃 ∣.
Proof: Since each terminal node is a leaf in 𝑆𝑇 by Lemma 2, ∣𝑉2∣ edges in 𝑆𝑇 are in 𝐸2. In addition, the
Proof: Since each terminal node is a leaf in 𝑆𝑇 by Lemma 2, ∣𝑉2∣ edges in 𝑆𝑇 are in 𝐸2. In addition, the