Abstract
Providing successful data collection and aggregation is a primary goal for a broad spectrum of critical applications of wireless sensor networks. Unfortunately, the problem of connectivity loss, which may occur when a network suffers from natural disasters or human sabotages, may cause failure in data aggregation. To tackle this issue, plenty of strategies that deploy relay devices on target areas to restore connectivity have been devised. However, all of them assume that either the landforms of target areas are flat or there are sufficient relay devices. In real scenarios, such assumptions are not realistic. In this paper, we propose a hybrid recovery strategy based on random terrain (simply, HRSRT) that takes both realistic terrain influences and quantitative limitations of relay devices into consideration. is proved to accomplish the biconnectivity restoration and meanwhile minimize the energy cost for data collection and aggregation. In addition, both of complexity and approximation ratio of are explored. The simulation results show that performs well in terms of overall/maximum energy cost.
1. Introduction
Wireless sensor networks (s) have raised a great attention thanks to their vast spectrum of industrial and social applications [1, 2], such as biological detection, environment monitoring, and battlefield surveillance. Data collection and aggregation are the first priority of s. The primary objective of such task is to gather sensor readings from field sensors deployed over a geographic area (called the area of interest ()) and then successfully deliver all gathered data to the sink node through multihop paths [3]. That implies the importance of both connectivity maintenance and optimal network topologies discovery for s [4]. However, natural disasters and human sabotages will jeopardize the network connectivity so that the process of data aggregation will be compromised without doubt. For example, when a is carrying out a surveillance mission on the activity of a volcano, if the connectivity is lost, then all gathered data will not be able to reach the sink node for further analysis. Without such important data, volcano eruption can not be predicted. That may cause massive casualties and severe economic losses.
Due to the significance of network connectivity as mentioned above, the problem of connectivity restoration has been receiving increasing attention in recent years. Thus, it is imperative to design both connectivity recovery strategies and routing algorithms for s. All known solutions, which exploit relay devices such as stationary relay nodes (s) and mobile data collectors (s), can be classified into two categories. One is employing s only for the purpose of establishing a connected intersegment topology with stable communication paths between every pair of segments [5–9]. This category of work generally aims to minimize the number of s during the restoration process. The other one relies mainly on s repeatedly visiting each individual’s segments for data collection and aggregation with a few s involved [10–18]. However, all of these works assume that either the terrain of is flat or the number of relay devices is unlimited.
Given that these assumptions are not realistic in real scenarios, our goal in this paper is to develop an efficient connectivity restoration strategy that takes both realistic terrain influences and quantitative limitations of relay devices into consideration.
Our Contribution. This paper presents a hybrid recovery strategy based on random terrain in s, namely, , that establishes a biconnected intersegment topology in a disconnected network with a limited number of s and MDCs; meanwhile, the energy cost for data collection and aggregation is minimized. The accomplishes our goal in this paper as follows:(1)To quantify realistic terrain influences, the area of interest () is mapped into a grid of equal-sized cells. Each cell is associated with a weight that represents the corresponding terrain influence within. We calculate , the weight of each path , by accumulating the weight of each cell along , so that the weighted complete graph is constructed based on minimum weight paths between segments.(2)A path planning algorithm () is developed on to build a Hamilton cycle of minimum weight as the biconnectivity restoration tour for s. And is proportional to the cost for data collection and aggregation during the connectivity restoration.(3)According to different numbers of s, two different relay nodes deployment strategy, and , are devised to merge intersected paths of by carefully choosing candidate positions for s, so that is greatly reduced.
The rest of the paper is organized as follows. Related work is covered in Section 2. The notions and terminologies are introduced in Section 3. The problem description is described in Section 4. The algorithm is elaborated in Section 5. Section 6 gives the theoretical analysis on approximation radio and complexity of . And the validation results are presented in this section as well. We conclude this paper in Section 7.
2. Related Work
There are two categories of approaches pursued for connectivity restoration [19]: the first category is to establish connectivity without terrain influences; the second one is to federate disconnected segments with consideration of terrain influences.
There are many excellent works regardless terrain influences during the connectivity restoration, which fall into the first category. Some of these works that employ s deployment only are listed as follows. Cheng et al. formulate placing the fewest s to connect segments as finding the Steiner minimum tree with minimum number of Steiner points and bounded edge length [5]. Lee and Younis propose grid based approaches, [6] and [7], both of which recursively deploy s until all segments are connected. In [20], recovery algorithms are proposed to minimize the deployment cost of sinks and relays and guarantee all sensors have two length-constrained paths to two sinks. Sitanayah et al. [8] find a minimal set of s which ensures length-bounded vertex-disjoint shortest paths to a sink for each sensor node. Lee et al. [9] focus on achieving a biconnected intercluster topology. Other works that employ s, s, and mobile nodes are listed as follows. In [10], Senel and Younis devise a convex hulk based recovery strategy -. It finds convex hulks of segments; then, each optimal tour for a convex hulk is assigned a to restore the connectivity. In [13], a least-disruptive algorithm is designed, which considers the impact of topology change on network performance through selecting candidate mobile nodes based on routing tables. In [14], a -hop neighboring information based algorithm is presented, which drives the backup mobile nodes to its destination to avoid intersensor collisions. In [15], a localized hybrid timer based cut-vertex node failure recovery approach is proposed, which adopts cascaded movement to relocate the mobile nodes so that the timely restoration is ensured. Joshi and Younis [21] establish balanced and optimized data collection and aggregation tours using the mobile nodes within the network. They first construct a minimum spanning tree then successively split it around the center into partitions such that the segments of each partition form a convex hulk. Eventually, all available s are assigned to partitions to complete the recovery process. In [16], Liao et al. aim at providing target coverage and network connectivity establishment through the minimum movement of mobile sensors. In [11], a delay-conscious recovery strategy, , was proposed to federate disjoint segments with a limited number of relay nodes. In [12], a convex hulk based recovery scheme is designed, which assigns s for intersegment federation and deploys s for intercluster connection. In [18], a distributed algorithm is designed to decompose the deployment area into its corresponding skeleton outline, along which mobile s are placed to finish connectivity restoration.
There are many excellent works with consideration of terrain influences during the connectivity restoration, which fall into the second category. Zhou et al. [22] propose an extended rapidly exploring random tree () based algorithm to initiate a path for a mobile node to reach the intended destination without crashing into any obstacles. Senturk and Akkaya [23] investigate how realistic terrain influences affect network connectivity recovery. Then, they design a terrain based restoration strategy [24] that considers different terrain types, such as forest, hill, swamp, and flat. ReBAT attempts to find the least cost paths between disjoint segments regardless of the subsequent data collection and aggregation. Truong et al. [17] propose a family of algorithms under the consideration of the impact of obstacles on mobility and communication, all of which collaborate to restore the connectivity with the least number of relay nodes and meanwhile minimize the mobility cost of agents. In [25], Mi et al. propose an obstacle-avoiding connectivity restoration strategy to avoid convex obstacles and intersensor collisions; however, they fail to consider realistic terrain influences on the process of connectivity restoration.
In this paper, we focus on establishing a biconnected intersegment topology in a disconnected network with a limited number of s and s and meanwhile minimize the energy cost for data collection and aggregation. It is worth to mention that establishing a biconnected intersegment topology, the quantitative limitation of relay devices, and minimizing the energy cost for data collection and aggregation are seldom considered all at pervious works unlike ours.
3. Preliminary
Some notations used throughout this paper are given first; the important symbols with their definitions are collected in Notations.
Definition 1. A weighted complete graph is a complete graph of vertices with each edge associated with its weight, where and denote the set of vertices and edges in , while stands for the set of weights of edges. To be more specific, the weight of an edge is equal to ).
Definition 2 (see [26]). An Euler closed trail is a closed trail that visits every edge of graph exactly once. A graph that has an Euler closed trail is called an Euler graph.
Definition 3 (see [26]). A Hamilton cycle is a closed trail that visits every vertex of graph exactly once. A minimum weighted Hamilton cycle of graph is denoted by .
Figures 2 and 3 show the examples of a Euler closed trail and a Hamilton cycle , where
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Definition 4 (see [26]). Given a graph , a matching is a set of pairwise nonadjacent edges; that is, any two edges share no common vertex.
Definition 5 (see [26]). A perfect matching is a matching which matches all vertices of the graph . That is, every vertex of the graph is incident to exactly one edge of the matching.
Definition 6. For three edges , , and of a triangle , we define , the weighted triangle inequality, abbreviated as .
Definition 7 (see [26]). A graph is said to be -connected, if for each pair of vertices there exist at least mutually independent paths connecting them; that is, the graph is still connected even after removal of any vertices from . A -connected graph is of -connectivity.
4. Problem Description
Wireless sensor networks are deployed in realistic environments for data collection and aggregation. Therefore, the connectivity of a is easily compromised due to nature disasters. Under such circumstances, the recovery strategy is required to restore the connectivity in a realistic environment. That implies that the terrain influences need to be taken into consideration. It is worth mentioning that the terrain influences on connectivity restoration are closely related to the landforms of , such as forest, hill, swamp, and flat. Especially in some realistic scenarios, there exists a shortcut between two cites and ; however, taking such shortcut will incur a significant energy cost. Intuitively, the possibility of taking a detour should be considered. To quantify the terrain influences, the area of interest () is mapped into a grid of equal-sized cells (squares) with side length . Besides, both segments and s are assumed to center at cells. The rationale is that s placed at eight neighboring cells are reachable for . More importantly, sensing data from eight neighboring sensors can be collected by a while travelling through .
As shown in Figure 4(a), a set of segments or s, for example, , is located at eight neighboring cells of . It is clear that , where . Therefore, they are neighbors. Figure 4(b) shows that sensing data of , , , , , and are collected by a while travelling through , where the dotted line represents the data collection tour.
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Similar to ReBAT [24], risk and elevation are taken into account while determining the optimal path besides the distance. Specifically, each cell is associated with a random terrain type, the corresponding risk factor, and an elevation. Accordingly, we use the weight functionto represent the terrain influence on travelling through . In addition, Manhattan distance is used to accurately estimate the cost of paths and 4 directions (north, east, west, and south) are considered directly accessible while moving s to an adjacent cell. We then measure the weight of through the sum of the weight of all visiting cells except two specific cells, for example, and , where and are located. According to (2), we give the weight function of a path as follows:
Figures 5(a) and 5(b) give the examples of and , respectively. Note that if there are several minimum weighted paths s, then choose the one with the shortest as . Furthermore, we assume in this paper. Let be a data collection and aggregation tour; according to (3), is given as follows:In [24], Senturk et al. give the energy cost function for travelling through a cell as follows, where is a constant value:According to (4) and (5), it is easy to deduce that the energy cost is proportional to the terrain influence. For simplicity, we use and to represent the energy cost of the path and the tour , respectively. The performance comparison in energy cost will be conducted using (8) in Section 6.2.
In addition, the required restoration strategy should be allowed to use only a limited number of s and s due to the fact that relay devices could be expensive. We then give the formal problem definition as follows.
Given nodes with a transmission range of that form disjoint partitions in a squared region which consists of cells of size , the goal is to provide a random terrain based solution (distributed/centralized) which ensures that partitions and the sink node will be biconnected by deploying a limited number of s and s and meanwhile minimize the cost of data collection and aggregation.
This paper is dedicated to solving such problem by proposing a polynomial time algorithm, named . It is worth mentioning that the the tour constructed by is not only the connectivity restoration tour but also the data collection and aggregation tour (see Figure 6).
5. The HRSRT Approach
is a random terrain based recovery strategy. It aims to ensure all disjoint segments including the sink node are biconnected; meanwhile, the energy cost of data collection and aggregation is minimized. During the restoration process, only s and s are employed. In fact, the energy cost is tremendously reduced with more s involved. Thus, according to different values of , HRSRT adopts corresponding approaches to achieve the connectivity restoration as follows:(i): If there is only one available, then it needs to tour around all disjoint segments and the sink to collect and aggregate data.(ii): If the number of s is more than one, then the corresponding tour for each should be carefully chosen so that total energy cost for data collection and aggregation is minimized.
The framework of is shown in Figure 7. In this paper, we first introduce as a centralized procedure; then, the distributed is elaborated in Section 5.3. It is worth mentioning that the theoretical proof on the biconnectivity of tour established by is given in Section 6.1. That implies can restore the connectivity with the consideration of terrain influences.
5.1. HRSRT with
For , works in two phases. First, a random terrain based path planning () is implemented to initiate a connectivity restoration tour in phase one. Then, an Optimized Relay Node Deployment () is adopted to reduce in phase two. The pseudocode for HRSRT with is shown in Algorithm 1.
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5.1.1. Random Terrain Based Path Planning (RTPP)
As we explained above, various terrains have significant influences on the cost of data collection and aggregation for s. To quantify such influences, the cost for travelling through a unit area(cell) is represented by a weight value. Therefore, the priority of a path planning strategy on realistic terrains is to establish a tour of minimum weight for s. In this section, a random terrain based path planning strategy, called , is proposed to accomplish the goal under the constrain that only s are available.
Suppose there are disjoint segments. We take six steps to build a minimum weighted tour as follows:(1)According to the weight of each unit area, construct a complete weighted graph over the set of segments remaining disjoint. Each edge of , for example, , is the minimum weighted path from to , that is, . Note that if there are several s, then choose with the shortest Manhattan Distance between and . The rationale is that the movement toward an adjacent cell is in one of four directions as we mentioned above.(2)Discover a minimum spanning tree () of .(3)Find the set of odd-degree vertices in ; that is, . Construct , an introduced graph of , to establish a minimum weighted perfect math . Then, initiate an Euler graph .(4)Randomly choose a vertex and draw an Euler closed trail that begins with .(5)According to the order of , consistently visit all vertices starting from . If a vertex has already been visited, then directly go to next vertex until all vertices are visited. Then, all the visited s are put into .(6)Locate and check whether each segment is on . If there is at least one segment, for example, , not on , then find the edge closest to and create the tour . Otherwise, let . Next, calculate , and choose the corresponding tour as the data collection and aggregation tour.
As shown in Figure 8, there are 4 s and one available for a set of disjoint segments, where . Figure 8(b) shows the weighted complete graph over . And each edge is associated with a weight value that represents the minimum cost of travelling from to . A of is built in Figure 8(c). And there are only two odd-degree vertices , so the perfect matching of is . As shown in Figure 8(d), the trail is the Euler closed trail of . Then, according to the order of , continuingly visit all vertices starting from . The solid lines consist of the Hamilton cycle . And is chosen as the initial data collection and aggregation tour . It is worth mentioning that the tour is 2-connected and the corresponding theoretical proof is given in Section 6.1. The pseudocode for RTPP is shown in Algorithm 2.
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5.1.2. Optimized Relay Node Deployment (ORND)
is a highly effective algorithm that aims to improve the initial data collection and aggregation tour established by . Although there are only s available, attempts to place s at the optimal positions such that a number of intersected paths are merged to reduce and the final data collection and aggregation tour is built (see Figure 9). Now we introduce how works:(1)Check all paths on and find out if there exist at least two paths, for example, and , such that they have consecutive cells in common except the cell where is located. That is, and .(2)Start to deploy s along ; then, let .(3)Merge paths and to reconstruct the tour as follows:
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If , then . Otherwise, .
Step () implies that if there exists only one pair of paths, for example, and , that have consecutive cells in common, then can be reduced by at most due to the merging process of . More specifically, if the s deployed along reach the cell , then the can directly collect the data sensed by by simply travelling through the merged path , as shown in Figure 9, instead of travelling along paths and sequentially.
In Figure 10, it is obvious that although the initial tour is a Hamilton cycle and , the does not have to go directly from to along the path due to the fact that there may exist at least two paths, that is, and , that can merge as one. In addition, a common cell within the communication range of is the only cell shared by both of and . Thus, such two paths merge as by deploying a at cell and all data sensed by will be collected while a is travelling through . Figure 10(b) shows the final tour , with . Note that although there are sufficient s, only one is needed. This is attributed to the fact that even if more s are deployed, remains the same in this example. The pseudocode for ORND is shown in Algorithm 3.
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5.2. HRSRT with
For , the same path planning algorithm elaborated in Section 5.1.1 is employed. It takes original disjoint segments as the input to establish a data collection and aggregation tour . Then, we devise a new s deployment strategy () to determine candidate positions for s on such that the energy cost is minimized. Finally, a Path Allocation () approach is developed to locate optimal tours for all s (see Figure 11). Now we elaborate how a minimum weighted tour is built as follows:(1)For each , deploy s along path and populate s along path , where . We use symbols and to mark the latest deployed s along and , respectively.(2)If , then choose as the optimal tour of a for all paths . Otherwise, use the same way to choose optimal tours for all s. For the last , the tour is assigned to it.(3)For all , let , where .
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We call the strategies, described in steps and (), and , respectively. In fact, if there exists a path such that the s are deployed along as step (), then is reduced by . This is because is the tour allocated to a , instead of . Besides, all data sensed by and can be collected, while both ends of the tour are reached by the . As shown in Figure 12(c), since and have a common cell , with deploys a at . The reason for with deploying a at the same location is that and refer to the same cell for paths and , respectively. Figures 12(b) and 12(d) represent final data collection and aggregation tours and of for and , respectively. It is obvious that is established with less weight than . That is, .
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It is worth mentioning that adopts different strategies based on different values of . The rationale is that if employs , , and sequentially with only one available, then the still needs to tour around the entire set of segments to collect data along the same path. That implies the s deployment incurs no reduction in energy cost. Unlike and , is designed to merge intersected paths into one through deploying s at optimal positions such that the weight of is minimized, regardless of how many s are available. Thus, is an ideal choice for the single case. The pseudocode for HRSRT with is shown in Algorithm 4.
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5.3. Distributed Implementation
This section describes how is implemented in a distributed manner. When the network is partitioned into disjoint segments, each segment first chooses a sensor as its representative and then broadcasts its location . We assume that there are some mobile agents in this network. These mobile agents will be sent to those segments that lost contacts based on their original positions. Eventually, all s share each other’s s after mobile agents return. Each calculates the coordinate of the of using (6) [27]. Note that and are coordinates of the , where and are coordinates of the segment .
Then, starts to populate relays toward . Note that each placed by will become new representative of . While two or more s are within each other’s communication range, the corresponding ’s merge as a new segment and the closest to the is chosen as the representative of such a newly established segment. Then, ’s (where ) will stop to deploy s toward the . Each will recursively deploy relays until all s assigned to it are placed.
The path planning is followed by the s deployment. Since the position of the is calculated by each , final positions of each segment, for example, s, are known by . Then, all ’s are calculated and stored in the database in increasing order, where . Next, the shortest s are iteratively added to the data collection and aggregation tour , which does not create a cycle, unless it completes the tour. Note that if the number of mobile data collectors is sufficient, then each segment is assigned a . Otherwise, only segments are assigned s, where . Finally, each tours around all existing disjoint segments along to collect and aggregate data (see Figure 13). The pseudocode for the distributed HRSRT is shown in Algorithm 5.
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6. Performance Evaluation
6.1. Theoretical Analysis
The correctness, complexity, and approximation ratio of are analyzed in this subsection. First we give the following theorems.
Theorem 8. All paths of a data collection and aggregation tour established by RTPP satisfy WTI.
Proof. We prove this theorem by contradiction. Suppose there are three paths , , and , for a travelling among three segments , , and , such that . That implies that the cost of directly travelling from to is higher than travelling from to via , which contradicts the fact that only chooses minimum weighted paths for s. Therefore, should start from and end at via . That implies . So the theorem holds.
As shown in Figure 14, we have , , and . Therefore, the proper choice for travelling from to is via . That makes .
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To illustrate the following theorem clearly, we call a path a direct path, if there is no segment on other than and .
Theorem 9. The tours constructed by RTPP is either a Hamilton cycle or a closed trail.
Proof. RTPP strives to locate the minimum weighted closed trail of a complete weighted graph . Without loss of generality, we set a minimum weighted closed trail. Note that each edge is a , which represents the minimum weighted path from to . Then, we are going to distinguish between two cases to prove this theorem.
If all edges of are direct paths such that each segment is on exactly once for any , then, according to Definition 2, is a minimum weighted Hamilton cycle (see Figure 3).
If there is at least a path not a direct path, that is, is not a direct path, then it is easy to verify that is a closed trail. We assume that the path starts from and ends at via . According to algorithm RTPP, there may exist two other paths and to complete the trail , such that . Intuitively, is on at least twice, which implies is a closed trail (see Figure 2).
Theorem 10. The weighted complete graph established by consists of vertices.
Proof. RTPP employs an introduced graph of a complete weighted graph to construct a perfect matching . When the of is built, the set of odd-degree vertices are employed to construct for the establishment of a perfect matching. For simplicity, let denote the number of edges of . Intuitively, the overall degrees of vertices in is even, because of the fact [26] , where . It is easy to deduce that is even.
For the rest of the paper, we use to represent graph .
Note that a series of based connectivity recovery algorithms, such as [12], - [10], and [21], make s travel along the convex hulk of disjoint segments. Specifically, if there is at least one segment not on , then segments will be clustered into several disjoint groups so that each group can form a convex hulk with each segment of on it. For simplicity, we call such algorithms -s. And we call the tour along the convex hulk of disjoint segment a tour.
Theorem 11. HRSRT establishes a data collection and aggregation tour with less energy cost than -s.
Proof. Let and denote the tours established by the RTPP and a -, respectively. According to RTPP, the selection of perfect matchings of could result in two different types of tours. That is, is either a tour or a non- tour. We distinguish the following two cases to prove that the energy cost of tour , Cost, is less than that of , Cost, regardless of the number of MDCs.
Case 1 ( is a non- tour). If there is at least one segment not on , then is a directed graph with either one of the following two structures. One is that a graph consists of at least two connected components bridged with two directed paths. The other is composed of at least two subtours that share a common vertex. We thus take Figure 15 as an example to explicate this case. Let the tour in Figure 15(d) represent and two tours, for example, and , in Figures 15(b) and 15(c) represent two types of s, respectively. According to Theorem 8, we have and . That implies . According to (5), it is intuitive that .
If all segments are on , then it is easy to get . According to the tour selection of , we have , because is a non- tour. Again, (5) guarantees that .
Case 2 ( is a tour and all segments are on ). It is easy to get , since each edge is the minimum weighted path from to , while the edge is a straight line directly from to regardless of terrain influences. As shown in Figure 16, it is clear that . According to (5), it is intuitive that .
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Theorem 12. RTPP establishes a close trail , the weight of which is less than times that of the least weight Hamilton cycle .
Proof. Without loss of generality, we assume the close trail established by is a Hamilton cycle. For simplicity, let represent the perfect matching of graph . Then, the theoretical proof on the approximation ratio of to is as follows.
First, we set the minimum spanning tree of a weighted complete graph . Let represent the introduced graph of , where denotes an edge of . Obviously, is still a spanning tree of such that . That implies , because of .
Second, obviously is an introduced graph of , where . Suppose and represent the minimum weighted Hamilton cycles of and , respectively. Without loss of generality, we assume that at least one edge of is not in ; for example, . According to Definition 6 and Theorem 8, we have , where . Intuitively, . As shown in Figure 17, we have and such that . And it also holds for the case that there are more than two edges of not in .
Third, by carefully choosing the edges from , we can find a minimum weighted such that . As shown in Figure 18, we have and such that .
Finally, let be an Euler closed trail of established by RTPP. According to Theorem 8, the weights of all paths established by satisfy . That is, , for every three segments . In addition, a path of consecutively adjacent edges in could be replaced by one edge that directly connects two end points of that path during the construction of . By recursively taking such steps, is built. Hence, .
Summing up all four steps above, the theorem holds because the following inequation holds:
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Theorem 13. The data collection and aggregation tour construct by HRSRT is 2-connected.
Proof. We distinguish between two cases to prove this theorem.
Case 1 (). adopts first to calculate a minimum weight Hamilton cycle as the connectivity restoration tour. Theorem 9 proves that is either a Hamilton cycle or a closed trail. According to Definitions 3 and 7, it is intuitive that is 2-connected. Then, is employed to merge any pair of intersected paths so that through s deployment. It is easy to verify the fact that there still exist 2 mutually independent paths which implies the biconnectivity of tour . As shown in Figure 19, there exist pairs of independent paths between any pair of segments. Even if a path is cut off, that is, is disconnected, and are still connected via and .
Case 2 (). Similar to Case , is responsible for establishing a 2-connected tour . Then, for each path , deploys s to build subtour . And is assigned to a by . It is obvious that is still 2-connected.
Summing up the two cases above, it is easy to deduce that the data collection and aggregation tour construct by is 2-connected.
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Theorem 14. The approximation ratio of HRSRT is .
Proof. We distinguish between two cases to prove this theorem.
Case 1 (there is only one MDC available). In this case, (4), (5), and Theorem 12 guarantee that the energy cost of tour established by is less than times that of the optimal tour. In addition, can be reduced due to s deployment by .
Case 2 (there are at least two MDCs available). In this case, ensures the sum of weights of all tours is less than that of the tour established by .
Summing up the two cases above, it is intuitive that is a 1.5-approximation algorithm.
Theorem 15. The complexity of HRSRT is .
Proof. We distinguish the following two cases to prove this theorem.
Case 1 (there is only one MDC available). In this case, consists of algorithms and . We then analyze the complexity of them, respectively. takes four steps to build the data collection and aggregation and aggregation tour: the construction of a spanning tree, the establishment of an Euler graph through calculating the perfect matching, the localization of an Euler closed trail, and the discovery of a Hamilton cycle. The cost of obtaining a spanning tree of a graph is less than . And the construction of an Euler graph through locating a perfect matching will not cost more than . In addition, the construction of an Euler closed trail requires an algorithm, where is a constant. Furthermore, the discovery of a Hamilton cycle is accomplished by depth first search such that its complexity will not exceed . Intuitively, is a polynomial time algorithm with the complexity of . discovers the intersected paths of a data collection and aggregation tour established by . Then, it locates the optimal positions for s. Since both of the intersected paths discovery and the optimal positions localization can be done in a constant time, is a algorithm, where is a constant.
Case 2 (there are at least two MDCs available). In this case, is composed of , , and . It is intuitive that both of and are algorithms.
Summing up two cases above, it is intuitive that the complexity of is .
6.2. Validation Experiments
The simulation environment, performance metrics, and experimental results are discussed in this subsection.
6.2.1. Experiment Setup
We consider a disconnected deployed in an application area a region of size . The sensing range and transmission range of a sensor or a relay are set to and , respectively. In order to represent different terrain types over the deployment region, it is divided into cells (i.e., squares of a certain size) where each cell is associated with a terrain type picked randomly from Table 1. In addition, the cell size is determined on the basis of the application area and the application requirements. We choose the size of for the cells. Then, topologies with varying number of sensors and segments are generated and topologies for each test case are considered. For each topology, terrain features are randomly added. Since obstacle-free environments are assumed for all baseline approaches, terrain features without obstacles are added.
6.2.2. Performance Metrics and Baseline Approaches
In our experiments, a partitioned with varying numbers of segments has been considered. In addition, the parameters that affect the network characteristics are listed as follows.
Number of Relay Nodes . Since s are deployed to federate disjoint segments, a great number of s will significantly shorten intersegment distances, so that the energy cost for data collection and aggregation is mitigated. In this paper, is assumed insufficient to connect all disjoint segments.
Number of Disjoint Segments . Intuitively, the number of communication links between segments raises with . Therefore, a larger results in higher energy cost for data collection and aggregation.
Number of Mobile Data Collectors . s are employed to replace stationary s, due to being insufficient. If there are plenty of s for connectivity recovery, then travelling distance of data collection and aggregation is minimized [12], which implies lower energy cost.
Communication Range of a Relay Node . With the growth of , - distances are reduced effectively that contributes to a low cost data collection and aggregation tour.
We use the following two metrics to evaluate the overall performance of .
Total Energy Cost. Energy cost incurred because of movement is considered. Our goal is to minimize this cost to extend the network lifetime.
Maximum Energy Cost. This metric shows the maximum energy cost of a . This is directly related to the survival of a single , which affects the lifetime of the repaired network.
Recovery Time. This is the time required to complete network-wide recovery. Minimizing the recovery time is not a goal, but it is affected by the path planning. The velocities of s are set to m/s.
We compare the performance of with the following three baseline approaches.
FeSMoR [11]. This algorithm is designed to minimize the average end-to-end delay between every pair of segments in a damaged with a limited number of relay nodes. works in two phases. The first phase is to construct an Euclidian Steiner Minimal Tree () of segments to balance data traffic through stationary deployment. In the second phase, finds the edges that require multiple stationary relays and do serve on the least number of - paths. Then, mobile relays are employed to replace stationary relays on those edges. If there are two adjacent edges that have leaf segments and each of which is assigned with a mobile relay, then such two edges merge as one data collection and aggregation tour, referred to as a Steiner triangle, that requires only one mobile relay.
MiMSI [12]. It is a mixed recovery strategy that utilizes s and stationary s for connecting a set of partitions. first builds an in terms of the average node degree. And Steiner Points of and segments are grouped into clusters based on proximity. Then, gateway nodes between every pair of neighboring clusters are determined. After that, populates stationary relays at the positions where gateways are located for connecting two adjacent clusters. Finally, each cluster will employ a that tour around segments for data collection and aggregation. If there is only one available, then segments of each cluster are federated by stationary s and each gateway node will be repeatedly visited by the only .
ReBAT [24]. It is a connectivity restoration strategy that considers realistic terrain influences. operates in two phases. The first phase is to seek the set of locations for the mobiles sensors to the ensure connectivity. In the second phase, a greedy-based heuristic constructs a as the connected backbone of the network. Then, some dominatee nodes of the are relocated to maintain the - connectivity. And during movement of the nodes, different terrain types associated with different frictions (i.e., risk values) are considered, such that each least cost path to the destination is found. Note that only considers establishing a - network. In our experiments, as a baseline approach, is slightly modified to build a data collection and aggregation tour through planning at least an extra path to the resulting - network after connectivity restoration. It is worth mentioning that if can obtain the least cost path that contains all segments, then in the worst scenario the weight of data collection and aggregation tour is less than or equal to two times that of ; that is, . According to Theorem 12, it is easy to deduce that the extended is a - algorithm. Henceforth, we use - to denote the extended .
In summary, and MiMSI do not consider different terrain types. They attempt to minimize movement distance considering constant energy cost per meter. strives to reestablish a - network, while establishes a bi-connected inter-segment topology and a least cost tour for data collection and aggregation. In addition, we use to represent the optimal energy cost for connectivity restoration.
6.2.3. Energy Cost Formulation
We use the same formula as [24] to measure the energy cost as follows: Note that represents a tour continuingly visiting cells, is total distance that travelled from to , is the constant value referring to the cost for movement per meter on a flat topology which is taken as joules/meter, and and denote the product of risk and elevation, respectively, for the cell shown in Table 1. In addition, the distance of travelling from a cell to its neighboring cell in one of the four directions equals ; that is, . According to (4), (5), and (8), it is intuitive that the energy cost of a tour equals . The performance comparison of FeSMoR, MiMSI, ReBAT-EX, and HRSRT using (8) in terms of energy cost is presented in Section 6.2.4.
6.2.4. Simulation Results
Several configurations with different combinations of , , , and are simulated. We change the value of and from 4 to 20 with increment of 4, respectively, while and vary from 6 to 14 with increment of 2 and from 50 to 250 with increment of 50, respectively. For each individual experiment, we average the results over 30 runs. Note that our experiments employ s instead of mobile sensors for connectivity restoration. For all baseline approaches, if there are insufficient s to establish connectivity, then s are utilized to tour around all disjoint segments for data collection and aggregation, even if there is only one available. Accordingly, the deployment of s will shorten the - distances. That implies the reduction in energy cost for s.
Total Energy Cost. It can be observed from Figure 20 that the energy cost of all approaches for varying declines when there are more relays deployed. The reason for that is that deployment continuously shortens the - distances. This eventually reduces the energy cost for baseline approaches due to a shorter distance of the , while the s deployment of is responsible for the drop of its energy cost. It is clear that consumes significantly less cost than and , because of the effectiveness of the path planning. Furthermore, incurs less than of the cost of as expected.
Figures 21 and 22 give performance comparisons of with baseline approaches for varying with/without deployment. As increases, more energy is consumed for all approaches. This is attributed to the fact that there are more - links required recovery. However, if there are more and more disjoint segments located in the area, then the - distances are shorten significantly. That eventually mitigates the energy cost due to the movement of a shorter distance. It is worth mentioning that - is expected to have more energy cost than , since plans a data collection and aggregation tour through discovering a Hamilton cycle of a weighted complete graph over disjoint segments, while - constructs the tour considering an extra path over the repaired - network. Intuitively, outperforms all the baseline approaches with/without deployment.
The adverse impact on the energy cost is observed with the growth of as seen in Figure 23. Obviously, the expansion of of deployed s contributes to shorter - distances. Consequently, overall movement energy cost of the will be reduced for all approaches. Again, the effectiveness of the path planning is the key for to succeed in establishing a data collection and aggregation tour of the least energy consumption, compared with and .
The performance comparisons of with baseline approaches for varying without deployment are shown in Figure 24. The energy cost of all three baseline approaches falls with the increases. The reason for that is that more s employed will effectively shorten the total travel distance, which implies the fall in total energy cost. Although and consume constant energy due to the lack of RNs deployment, the paths planned by consume less energy cost than , , and ReBAT-EX. The reason for that is that works on discovering the minimum cost data collection and aggregation tour over the weighted complete graph of disjoint segments. More importantly, the effectiveness of becomes the major factor in reduction of energy cost. Furthermore, the energy cost of is less than times that of Opt as expected.
Maximum Energy Cost. As shown in Figure 25, the maximum energy costs of all approaches decline with varying . Because if there are plenty of MDCs employed for Steiner triangles of FeSMoR, clusters of MiMSI, and - link of ReBAT-EX, respectively, then overall travelling distance will be shorten. That results in the drop in maximum energy cost. For HRSRT, although the growth of does not contribute to the decline in total energy cost due to the lack of RNs deployment, the maximum energy cost is reduced. That is attributed to more MDCs involved for sharing the responsibility of connectivity recovery and data collection and aggregation. It is obvious that HRSRT outperform all baseline approaches.
Recovery Time. Figure 26 shows the recovery time comparison of all approaches for varying . The results indicate that recovery times of all approaches increase first with more segments involved and then decrease when the deployment of segments is getting dense. This is because that the intersegment distances increase when more segments are required to be connected which results in the increase in recovery time; however, densely populated segments shorten the intersegment distances that contribute to the decrease in recovery time. It can be observed that nonterrain-aware approaches, FeSMoR and MiMSI, require less recovery time than terrain-aware approaches, HRSRT and ReBAT-EX. This is expected because direct paths are followed from source to destination when nonterrain-aware approaches are applied. Note that HRSRT is better than ReBAT-EX in recovery time.
As shown in Figure 27, the recovery time decreases for all approaches with more MDCs joining in connectivity restoration. The reason for that is that all MDCs collaborate to restore the connectivity simultaneously so that the recovery time is shortened. Although nonterrain-aware approaches perform better than terrain-aware approaches, which aim at mitigating terrain influences on connectivity restoration, as expected in the recovery time, HRSRT still requires less recovery time than ReBAT-EX.
7. Conclusion and Future Work
Due to the significance of both connectivity maintenance and optimal network topologies discovery for s, in this paper, we have discussed a random terrain based connectivity recovery problem in a disconnected under the constraint that there are only relay nodes (s) and mobile data collectors (s) available. According to different values of , a hybrid connectivity restoration and routing strategy is designed. For , two highly efficient algorithms, the random terrain based path planning () and the Optimized Relay Node Deployment (), constitute . For , is composed of , relay nodes deployment (), and optimal Path Allocation (). All four algorithms collaborate to accomplish the biconnectivity restoration of a disconnected network; meanwhile, the energy cost of data collection and aggregation is minimized. The performance of is analyzed theoretically and validated through simulation. The simulation results show that HRSRT outperforms FeSMoR, MiMSI, and the extended version of ReBAT (namely, ReBAT-EX) in terms of the total/maximum energy cost. Our future work is to investigate the connectivity recovery problem with the consideration of the cost for s and s deployment.
Notations
| : | The Euclidean distance between two segments and |
| : | The Manhattan distance between two segments and |
| : | A path from to , abbreviated as |
| : | The minimum weighted path from to |
| : | The maximum communication range of sensor nodes |
| : | The th segment |
| : | The sensor node that represents the segment |
| : | A minimum spanning tree of graph |
| : | The degree of |
| : | The smallest polygon which includes segments (see the shaded area in Figure 1(a)) |
| : | The convex hulk of (see the perimeter that consists of dashed lines in Figure 1(b)) |
| : | The center of mass of . |
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors wish to thank National Natural Science Foundation of China (nos. 61572010, 61072080, U1405255, 61472083, and 61402110), Fujian Normal University Innovative Research Team (no. IRTL1207), and The Natural Science Foundation of Fujian Province (nos. 2013J01221, 2013J01222, and 2016J01289).