Hybrid connectivity restoration in wireless sensor and actor networks

Abstract

Wireless sensor and actor networks are becoming more and more popular in the recent years. Each WSAN consists of numerous sensors and a few actors working collaboratively to carry out specific tasks. Unfortunately, actors are prone to failure due to harsh deployment environments and constrained power, which may break network connectivity resulting in disjoint components. Thus, maintaining the connectivity among actors is especially important. This paper proposes hybrid connectivity restoration (HCR), which integrates proactive selection and reactive motion. An actor protectively selects a backup node through its one-hop neighbor table and informs the backup node to supervise its stage. Once it fails, the backup node moves to the best position to restore the connectivity of the failed node’s neighbors reactively. This triggers a local recovery process at the backup node, which is repeated until network connectivity is restored. In order to minimize travel distance, HCR selects the backup node which moves the shortest distance to restore connectivity. Furthermore, HCR opts to reduce the number of messages by just informing the failure to its backup node. The correctness and effectiveness of HCR are validated through both theoretical analysis and simulations.

1 Introduction

Wireless sensor networks (WSNs) are indispensable components of Internet of Things [5–7, 9–11, 15, 19, 23–25, 32, 37, 38]. A wireless sensor and actor network (WSAN) is a special kind of WSN, which has motivated lots of research works [28]. In the corresponding applications such as environmental monitoring, battlefield surveillance, border protection, target searching and tracking, a number of sensors and actors work cooperatively to monitor a specific area and track a target of interest. Sensors are responsible for collecting data, and actors are responsible for processing data and bridging the sensors and the control center. An actor and the sensors connected to it form a self-organized sub-network. All the sub-networks collaborate with each other to carry out tasks. It is desired that all the actors in a WSAN are connected at any time.

Unfortunately, due to harsh deployed environments and limited battery power, actors may deplete energy fast. A sudden loss of a node may break network connectivity resulting in disjoint network components. Therefore, it is important to detect node failures and restore network connectivity as early as possible. Since WSANs are usually deployed far away from the control center and are operated autonomously and unattended, it is difficult and inefficient to control the restoring process in a centralized manner. Connectivity restoration therefore should be a distributed, localized, and self-healing process. In addition, a rapid connectivity restoration is desired in order to reduce the baneful influence of node failures. Moreover, the overhead such as the total travel distance and the total number of messages should be minimized considering the limited energy supply. The average travel distance should be considered as well because one node traveling too far will consume too much energy and may cause another network disconnection. A node failure disrupting network connectivity is called a cut vertex, which is difficult to identify in large-scale WSANs centralized and timely. Though there have been many distributed cut-vertex detection algorithms, they are time-consuming and resource-intensive. As a result, it is very challenging to restore network connectivity in a distributed, localized, and efficient manner.

This paper proposes hybrid connectivity restoration (HCR) considering single-node failures, which integrates proactive selection and reactive motion. The selection of a backup node for a failure node is a proactive process. Each node identifies a backup node and is then monitored by its backup node. Once a node fails, its backup node moves to the best position that connects all the failed node’s one-hop neighbors. It triggers the restoration of the backup node and the restoration is a reactive process. HCR is a distributed and localized scheme, where each node just maintains its one-hop neighborhood information. Since a failed node only affects its direct neighbors’ connectivity, the main idea of the restoration is to move one of the failed node’s neighbors to a new position so that all the failed node’s direct neighbors can be re-connected. The node motion may trigger another disconnection on the moving node, so the restoration is a recursive process, the whole network is connected only when the motion node’s directed neighbors are connected. HCR opts to efficiently restore network connectivity through selecting the most proper backup node and moving it to the best position instead of the failed node’s position. The less distance the node moves, the less influence on the network connection. As aforementioned, only if a node failure breaks its directed neighborhood connectivity, it may further break network connectivity. The node whose failure breaks its directed neighborhood connectivity is called a critical node. On the opposite, the node is called an uncritical node. It should be noted that the uncritical node cannot be a cut vertex while the critical node may be a cut vertex. Though a critical node’s failure may not necessarily break network connectivity, it may bring unnecessary restoration. It is much more efficient and cheaper to identify a critical node and restore its direct neighbors’ connectivity than to identify a cut vertex and restore the network connectivity. To identify a cut vertex requires global information, which is impossible and inefficient in WSANs. At the same time, to identify a critical node just needs one-hop neighborhood information, and the identification is done on the node itself. Moreover, compared with moving a backup node to the failed node’s position in DCR [17], it is better to move the backup node to the position which connects all the failed node’s directed neighbors. Only when all the neighbors are on the boundary of the communication range, the backup node needs to move to the failed node’s position.

Following are our main contributions:

• . We proposed a hybrid connectivity restoration in WSAN, which integrates proactive selection and reactive motion. The proactive selection of a backup node can shorten a recovery process, the HCR offers effectiveness and timeliness.

• . Different from moving a backup node to the position of failed node, HCR moves the backup node to the best position with the shortest travel distance to reconnect the failed node’s one-hop neighbors. This can not only reduce a motion cost but also the total overhead, as the shorter a node travels, the fewer nodes are influenced.

• . HCR opts to reduce the number of messages by just informing the failure to its backup node.

• . The efficiency of HCR is mathematically analyzed and validated through simulations. The bounds on the incurred overhead are derived. HCR outperforms RIM and DCR in terms of the number of relocated nodes, total travel distance, average travel distance, and number of messages for both dense and sparse networks.

The remainder of the paper is organized as follows. Section 2 describes the system model and formulates the problem. Section 3 reviews the related works. Section 4 provides the detailed description of the proposed HCR, and the theoretical analysis is illustrated in Section 5. The simulation results are presented in Section 6. Finally, the conclusions and future work are presented in Section 7.

2 System model and problem formulation

2.1 System model

A WSAN consists of numerous sensors and a few actors (Fig. 1) which are usually randomly deployed in harsh environments far away from the control center. So a WSAN is a self-organized network where sensors are mainly responsible for collecting data and sending data to the nearest actors, and actors are responsible for making decisions. Actors process data, communicate with other actors, and send the results to the sink node. In this paper, it is assumed that actors are movable to connect other actors forming a connected network. In WSANs, actors play the role of gateways to the sink node. Therefore, it is important to maintain actor connectivity. Since sensors are static, they always try to communicate with actors in their communication range without identifying who they are. That means all the actors are the same to sensors. Therefore, actor connectivity decides network connectivity. Here, the coverage of the WSAN is not considered since an actor’s failure will lose all the sensors connected to it. Because the sensors are randomly deployed in the sensing regions, it can be assumed that there is a equal number of sensors within an actor’s transmission range. Therefore, moving actors has a little effect on the total coverage of the WSAN.

Fig. 1

A WSAN with mobile actors

In WSANs, each actor has limited communication radius R _c . Actors can send and receive messages within the communication range to discover other actors. Each actor maintains a one-hop neighborhood table recording its neighbors’ positions and other information. Two-hop neighborhood information or multi-hop neighborhood information can be obtained by exchanging one-hop neighborhood table with neighbors. The more information it gets, the more communication and storage it incurs. However, in most WSAN applications, actors are battery powered and have limited energy. Thus, it is more efficient to maintain less information. In this paper, each actor i just maintains a one-hop neighborhood table denoted as NT(i). NT(i) is a two-dimensional table where each row contains one-hop neighbor information such as unique node ID (ID), local position (POSITION), and critical character (CRITICAL). The critical character is defined as follows.

Critical character: It indicates whether a node failure breaks neighbor connectivity. CRITICAL=1 when the node failure breaks connectivity. Otherwise, CRITICAL=0.

In order to restore connectivity rapidly, each actor also has backup node information. The selection of a backup node is introduced in Section 4.1. The one-hop neighborhood table and backup node information of each actor are maintained and updated during the process of connectivity restoration.

2.2 Problem formulation

Actors are prone to failures due to tough environments or energy depletion. The loss of an actor affects not only the sensors connected to it but also the neighboring actors. The latter case is even worse, so this paper focuses on the latter case. It is assumed that actors are movable, it is possible to restore network connectivity by relocating actors.

Generally, different node failures have different effects on network connectivity. Consider the network shown in Fig. 2. If node A3 fails, the network is divided into two disjoint components. If node A8 fails, the network is still connected and all its neighbors can communicate with each other. A leaf node’s failure does not break connectivity. It is important to note that a node’s failure breaking one-hop neighborhood connectivity does not necessarily mean it breaks network connectivity. For example, node A6 has two neighbors A4 and A7. If A6 fails, A4 and A7 cannot communicate with each other through A6, but A5 still connects A4 and A7. Though only a cut vertex breaks network connectivity, it needs the global network connectivity information to tell whether a node is a cut vertex which is nontrivial and inefficient in large-scale WSANs. It also incurs significant overhead in terms of computation, communication, and storage. As aforementioned, an uncritical node does not break network connectivity, and the identification of an uncritical node just needs one-hop neighborhood information. Moreover, the distance between any two sibling nodes in a WSAN is no more than 2R c, because both of them are in the communication range of the parent node. It takes a node to move at most Rc towards another to reconnect it. Therefore, compared with identifying a cut vertex, it is more efficient and significant to identify a critical node.

Fig. 2

A connected network of mobile actors

Though reconnecting two sibling nodes just needs to move one node for less than Rc, it may trigger another failure, which causes cascading failures. Even if the motion does not trigger other node failures, it may consume a lot of energy. Hence, load balancing should also be taken into consideration. In addition, a node motion may generate some other critical nodes, which increases the risk of node failure in the future. Considering all of these factors, we formulate the following problem.

Given a connected WSAN G=(V,R _c ). V is the set of actors, and R _c is the actors’s communication radius. Each actor V _i has ID and Positions. All the actors are homogeneous with the same R _c . Each node just knows its one-hop neighborhood information. When a single actor i fails, relocate the rest nodes so that (1) network connectivity is restored, (2) the total travel distance is minimized, (3) the average travel distance is minimized, and (4) the total number of messages is minimized.

HCR is proposed in Section 4 to solve this problem efficiently. It is assumed that no two or more actors fail simultaneously, and no node fails during restoration.

3 Related works

Connectivity restoration considering single-node failures has attracted much attention, and there are a lot of surveys [22, 27, 34, 36] focusing on connectivity restoration. The existing schemes can be classified into two categories: proactive restoration and reactive restoration.

Proactive restoration schemes make use of redundant resources including nodes and paths to increase the robustness of a WSAN. When a node fails, it requires no connectivity restoration because there are redundant resources maintaining connectivity. Since the directional connection between a pair of actors is determined by the communication radius, it can only result in more redundant relay nodes to build K-disjoint paths [30]. In this case, there are K-disjoint paths between any pair of actors in a WSAN. Even K-1 paths fail, there is still a path connecting them. Consider a two-connected WSAN where there are at least two paths between any pair of nodes. CRAFT [18] establishes a bi-connected inter-partition topology while minimizing the longest path length and the number of deployed relay nodes. It strives to form the largest inner simple cycle or Backbone Polygon (BP) around the center of the damaged area where no partition lies inside, and deploys relay nodes to connect each outer partition to the BP through two non-overlapping paths. The advantage of proactive restoration schemes is that it does not disturb a network when node fails, but it requires many redundant resources. The stronger fault tolerance is, the more resources are required. Moreover, it is very difficult to place relay nodes optically, as it needs the global network information. The time complexity is very high in large-scale networks. In [20], it is proven that just listing a set of feasible sites for the relays is already at least APX-hard. Though many heuristic algorithms have been proposed, such as Genetic Algorithm [12, 21], Artificial Bee Colony Algorithm [14], and Concentric Fermat Points [31], it is still extraordinarily time-consuming.

Different from the proactive restoration schemes, the reactive restoration schemes are passive and a recovery process is triggered when a node failure is detected. They do not require reserving redundant resources. The basic idea is to reconnect the failed node’s neighbors. There are two kinds of approaches: cooperative communication and relocating nodes. Cooperative communication is first proposed in [8]. It allows a node to send message beyond its communication radius with the help of its neighbors. Two nodes are able to communicate if and only if the received average signal-to-noise ratio (SNR) is no lees than the fixed threshold. Signal strength diminishes with the increase of transmission distance and overlays at the destination. CSFR [33] adopts cooperative communication to restore connectivity. Taking advantage of neighbors to transport data does not increase the neighbor’s energy consumption a lot. Though it has low overhead from the current perspective, it is still a long-term process which costs a lot of energy in the long run. In addition, it is unacceptable and very time-consuming to select the help nodes.

Currently, most reactive restoration schemes reconnect a network by replacing a failure node with a proper backup node through movement which is a recursive process that may relocate the rest of the nodes. Therefore, which node moves and where to move is nontrivial. Ramezani proposed a distribute method to restore connectivity by using a centralized genetic algorithm [26] at the basic station. It strives to minimize the number of mobile nodes and the average length of all nodes’ paths. It is a heuristic algorithm. As mentioned before, only a cut vertex may break network connectivity. Many approaches decide whether a node is a cut vertex firstly and deal with cut vertex failure only, such as DARA [1], PDARA [2], PCR [16], and NNN [13]. DARA identifies a cut vertex through two-hop neighborhood information. Once a failure happens, the failed node’s neighbors select the most proper backup node considering node degree and distance and inform its sibling nodes. In fact, the process of identifying a cut vertex is not introduced in details in DARA. In a latter improved approach PDARA, it forms a connected dominating set (CDS). PDARA informs a particular node in advance whether a partition occurs in case of failure. They both strive to localize the scope of the recovery process and minimize the movement overhead imposed on the involved actors. In nearest non-critical neighbor (NNN), each actor periodically determines its criticality (i.e., cut vertex or not). In addition, they both maintain two-hop neighborhood information. In order to minimize the message overhead, DCR [17] identifies the critical nodes with one-hop neighborhood information, and the restoration is similar to DARA’s.

Since cut vertex identification incurs significant overhead in terms of messaging and state maintenance, RIM [35] does not distinguish the importance of nodes. All the one-hop neighbors move towards the position of the failed node till the distance is “ R _c /2”. Other nodes perform a cascade inward movement to connect to the connected network when they cannot communicate with the moved nodes. RIM is simple and efficient, especially for sparse networks. But the performance degrades for dense networks. RIM involves too many unnecessary motions, especially the first step. As all the failed node’s neighbors do not know each other, they all move to the position which is “ R _c /2” far from the failed node to maintain neighbor connectivity even when they are within half of the communication radius. These may incur outward motions. In addition, the message overhead is very high.

The above methods mainly focus on the moving distance and message overhead. In fact, network lifetime is the most important factor which depends on energy efficiency and load balancing. Abdelmalek [4] proposed a two-phase restoration algorithm. It searches the redundant nodes using the cluster heads, then restores connectivity, and energy consumption is taken into consideration. CoRF [3] is another connectivity restoration algorithm that strives to increase network lifetime. It selects a backup node according to the fuzzy logic rules. In addition, there are some realistic connectivity restoration methods [29] that take obstacles or terrain elevation into consideration. The direct path movement may be impossible, or is not optimally energy-efficient. In summary, energy efficiency and load balancing are the important evaluation metrics for the connectivity restoration algorithms.

4 Hybrid connectivity restoration algorithm

In this section, a hybrid connectivity restoration (HCR) algorithm is proposed to restore connectivity in WSANs. HCR is a distributed, localized, and efficient approach aiming at minimizing the cost of moving nodes. Instead of identifying a cut vertex, HCR just identifies the critical nodes. Each node maintains a one-hop neighborhood table including unique node ID (ID), local position (POSITION) and critical character (CRITICAL). HCR combines proactive backup node selection and reactive cascade node motion.

4.1 Proactive backup node selection

In order to minimize the number of messages and shorten restoration time, each node will select a backup node from its neighbors before a node failure occurs. During the initialization of a one-hop neighborhood table, each node sends a broadcast message containing its (ID) and (POSITION). All the nodes in its communication range will receive the message.

After a round of information exchanging, each node will determine whether it is critical through its one-hop neighborhood table NT. A node is an uncritical node if and only if all its one-hop neighbors form a connected network. In Fig. 2, node A3 has three one-hop neighbors A1, A2, and A4. They form two disjoint components {A1, A2} and {A4}. So A3 is a critical node. Similarly, A4, A5, A6, and A7 are critical nodes. While node A8 is an uncritical node for all its one-hop neighbors A5, A7, A9 form a connected network {A5, A7, A9}, so as node A9. Node A10 is a leaf node and just has one one-hop neighbor A7, so it is also an uncritical node. It is worth mentioning that a critical node’s failure will not necessarily divide a network, e.g., {A5, A6, A7}. However, an uncritical node’s failure must not break network connectivity. Therefore, a backup node needs to be selected only for a critical node.

Different from DARA [1] and DCR [17] which select a backup node based on distance and node degree, HCR selects a backup node based on the minimum moving cost. The minimum moving cost of all backup nodes should also be the failure cost of the failure node. For a critical node i, choose any node j in its one-hop neighborhood table NT(i), compute BestPosition j ^′ it should move to so as to connect all the rest nodes in NT(i). As shown in Fig. 3, node A3 has three one-hop neighbors A1, A2, and A4 which form two disjoint components {A1, A2}, and {A4}. The best positions for A1, A2, and A4 are A1^′, A2^′, and A4^′. They are all in the communication range of the rest sibling nodes. A1^′ and A2^′ are on the boundary of the communication range of A4. A4^′ is the intersection of the communication boundary of A1 and A2. The moving cost for node A1, A2, and A4 are A1A1^′, A2A2^′, and A4A4^′. The node with the minimum moving cost will be selected as the backup node for node A3. Here, node A2 is selected as the backup node and BestPosition is A2^′. The best position for relocating the backup node is shown in Fig. 4 and the pseudo-code for backup node selection is detailed in Algorithm 2.

Fig. 3

An example of backup node selection

Fig. 4

Best position for relocating the backup node. a Node C is out of the communication range of A and B. b Node C is within the communication range of A and out of B

In order to find a backup node and BestPosition for node i, compute the best position j ^′ for each node j∈NT(i) so that $\phantom {\dot {i}\!}d_{j,j^{\prime }}$ is minimum and satisfies Eq. 1. $\phantom {\dot {i}\!}d_{j,j^{\prime }}$ is the distance between nodes j and j ^′, it is the moving cost of node j for failure node i. For all j∈NT(i), the minimum $\phantom {\dot {i}\!}d_{j,j^{\prime }}$ should be the failure cost of node i, and node j is elected as a backup node and j ^′ is BestPosition.

$$ \forall k \in NT(i),\, {k \neq j}, \quad d_{k,j'}\leq R_{c} $$

(1)

Theorem 1

BestPosition for node i must be the intersection of communication circle of any two nodes in NT(i) or the intersection of the line ij and communication circle of node j.

Proof

First, the intersection of a pair of nodes’ communication boundary in NT(i) must connect these pair nodes. If this intersection is within the rest sibling nodes’ communication range, this node can replace the failure node to connect all the failure node’s neighbors. But the intersection may not be the optimal solution. Actually, the intersection is optimal only when the moved node is beyond the communication range of the pair of nodes. As shown in Fig. 4 b, node C is out of the communication range of A and B. The intersection C _new should be the optimal position. But in Fig. 4 a, node C is within the communication range of A. Node I is the intersection of A’s and B’s communication boundary. Node I can connect A and B, node C _new can also connect A and B, and $d_{C,C_{new}} < d_{C,I}$. $d_{C,C_{new}} + d_{C_{new},B} < d_{C,I} +$ d _I,B for Line Axiom, where $d_{C_{new},B} = d_{I,B} = R_{c}$. Thus I is not the optimal position, but node C _new is. □

Lemma 1

The node failure cost is no more than the nearest-neighbor distance in HCR.

Proof

By Theorem 1, BestPosition must be the intersection of any pair of nodes’ communication boundary in NT(i) or the intersection of the two nodes connection line and one node’s communication boundary. As shown in Fig. 4, node C _new is BestPosition. In Fig. 4 a, ∠ CC _new F must be an obtuse angle; otherwise, FC _new will be the tangent line of circle B, and node F will be out of the circle. As ∠ CC _new F is an obtuse angle, $d_{C,C_{NEW}} \leq d_{C,F}$, so as Fig. 4 b. Since each node’s best position moving cost is less than its distance to the failure node, the node failure cost is no more than the nearest-neighbor distance. □

The previous approaches such as DARA and DCR all move the backup node to the position of the failure node, then the failure cost is equal to the nearest-neighbor distance in the best case. So HCR outperforms DARA and DCR in terms of travel distance in a motion. The shorter distance it moves, the smaller impacts on its neighbors’ connectivity.

If there are two or more neighbor nodes with the same moving costs to restore the network connectivity, the one with the smallest failure cost will be selected as the backup node for the next restoration loop with the smallest moving cost. After selecting the backup node and obtaining BestPosition, it will send a broadcast message containing the information of the backup node and BestPosition. After a round of broadcasting, each node will update its backup node and BestPosition again and inform the changes to its new backup node only. By now, proactive backup node selection is finished. Then the node starts to send heartbeat messages to the backup node periodically to declare that it is functional. Once the backup node does not receive the heartbeat message within a period, it will start reactive cascade node motion to restore connectivity.

4.2 Reactive cascade node motion

As mentioned before, only a critical node will break network connectivity, so when a critical node i fails, its backup node j will detect the failure the first time and trigger reactive cascading node motion. Before the backup node j moves to BestPosition j ^′, it will update its one-hop neighborhood table NT(j). It replaces the failure node’s position by j ^′, and checks whether its one-hop neighbors’ connectivity is broken. If broken, node j will be the failure node and the restoration will be triggered. Node j will update its backup node k and BestPosition k ^′. Node k is selected from its rest neighbors except for the failed node i in order to avoid falling into an infinite loop. The backup node k will move to BestPosition k ^′ to connect all its neighbors in NT(j) and j ^′. After selecting the backup node k and BestPosition k ^′, node j will send a message to backup node k about BestPosition k ^′ it should move to and this will trigger a new round motion of node k. Node j will move to j ^′ to replace the failure node i. This process will be repeated recursively until the failure node’s neighbors are connected.

Figure 5 is an example for HCR cascading node motion. In Fig. 5 a, node A3 fails. Its three one-hop neighbors A1,A2, and A4 become three isolated nodes and they are out of each other’s communication range. For proactive backup node selection, node A4 is selected to move towards A4^′ to connect A1 and A2. Due to A4’s movement, its neighbor A6 is out of the communication range, while A5 is still within the communication range in Fig. 5 b. Before moving to A4^′, node A4 selects a backup node to replace it and node A6 is selected. In Fig. 5 c, node A6 moves to A6^′ to connect node A4^′ and node A5. The movement does not break its connection to node A7, and the whole network is connected. It is worth mentioning that A4^′ is the intersection of A1’s and A2’s communication circles, while A6^′ is the intersection of line A4^′ A6 and A5’s communication circle.

Fig. 5

An example HCR cascading node motion. a Node A3 failed and node A4 is selected to move. b Node A4 moves to A4’. c Node A6 moves to A6’

The pseudo-code for HCR is shown in Algorithm 1. HCR is a hybrid method that selects a backup node for each actor before it fails. At the beginning of network construction, each node will broadcast a message to notify its position within its communication range, and records its neighbors’ ID and Position to build one-hop neighborhood table NEIGH_TABLE (line 3). Then each node will identify whether it is critical through its NEIGH_TABLE. For an uncritical node, its failure will not break connectivity, so failed_cost=0 (line 8). While for a critical node, it will select a backup node and BestPosition (line 6). The algorithm for backup selection is detailed in Algorithm 2. When a node just has two neighbors, move any one towards another has the same travel distance to restore connectivity. For this case, the one with smaller failed_cost is chosen (line 11-16), this is to insure the next restoration has a smaller moving cost. By now, the initialization and proactive backup selection are done. Every actor sends a message to notify its backup node to monitor its status and BestPosition. When a node detects a failure of its neighbors, it will trigger the cascading node motion (line 18-25). The backup node will add BestPosition into its NEIGH_TABLE to tell whether it is critical. If so, it will select its new backup node before moving to replace the failed node (line 21). Otherwise, network connectivity is restored.

The backup node selection is the key part of HCR and the pseudo-code is presented in Algorithm 2. It is worth mentioning that all the selection of backup node is done on the failed node or moved node and it just explores NEIGH_TABLE. During this process, it does not need to send message to other nodes. This reduces the message overhead. According to Theorem 1, BestPosition must be the intersection of any pair of nodes’ communication boundary or the intersection of the two nodes connection line and one node’s communication boundary. Firstly, for each node i in NEIGH_TABLE, compute its Candidate_Set (line 6-13). Then remove the position that is out of the rest nodes’ communication range (line 14-21). Afterwards, choose the closest point in Candidate_Set as BestPosition for node i. Finally, return the node’s ID and BestPosition which has the minimum travel_dist in NEIGH_TABLE.

5 Algorithm analysis

HCR combines proactive and reactive methods to handle network connectivity restoration from a single-node failure in WSANs. The selection of a backup node is proactive, while the restoration is reactive. This scheme shortens the restoration process and reduces the overhead including distance cost and message cost. Next, the performance of HCR is analyzed.

First and foremost, network connectivity after a single-node failure is restored. It is assumed that no other node fails during the restoration process and any two nodes can communicate with each other directly if they are R _c apart or closer. Then network connectivity is not weakened and no new critical node is introduced during the restoration. In addition, load balancing is taken into consideration and no node travels too far while others too close. The overhead of communication and the complexity of computing are also analyzed. We introduce the following theorems.

Theorem 2

HCR restores network connectivity after a single-node failure.

Proof

Uncritical node failure will not break network connectivity since all its neighbors are connected when it is removed from the network. HCR identifies a critical node at the initialization time and selects a backup node and decides BestPosition. When a critical node fails, it will trigger the restoration, and its backup node moves to BestPosition to reconnect all its one-hop neighbors. In the following cascading node motion, the moved node will reconnect its sibling nodes until all its siblings are connected. In order to avoid an endless loop, each node can only move once during the restoration, and a moved node will not be selected as a backup node in the future. This can guarantee that HCR terminates in limited number of steps. □

Theorem 3

The total number of critical nodes does not increase.

Proof

In HCR, the selection of a backup node and BestPosition guarantees that the total number of critical nodes does not increase. The critical of backup node will change into that of the failed node. And the degree of each node will not decrease. The backup node will move to BestPosition to connect all its siblings. In Fig. 6, node A3 fails, and it selects a neighbor to reconnect all its neighbors. In Fig. 6 a, node A1 is selected and moves to node A1^′ where it is on the line A1A4, and $\phantom {\dot {i}\!}d_{A1^{\prime },A4}=R_{c}$, $\phantom {\dot {i}\!}d_{A1^{\prime },A2}<R_{c}$. Since A1 and A2 can connect before failure, we just need to guarantee that node A1^′ is in the communication range of A2. In Fig. 6 b, node A4 is selected and moves to node A4^′ where it is the intersection of circle A1 and A2, $\phantom {\dot {i}\!}d_{A1,A4^{\prime }}=R_{c}$, and $\phantom {\dot {i}\!}d_{A2,A4^{\prime }}=R_{c}$. Node A4 is out of reach by A1 and A2 before the failure. After restoration, the unmoved critical nodes are not changed, e.g., A2 and A4 in Fig. 6 a, and the moved uncritical node becomes a critical node or not. Node A1^′ is critical, where node A4^′ is uncritical. Before the restoration, the failed node is critical, so the total number of critical nodes does not increase. □

Fig. 6

No new critical node is introduced during the restoration. a Node A1 is selected to move. b Node A4 is selected to move

Theorem 4

The maximum distance a node travels in HCR is the communication range R _c .

Proof

In HCR, a backup node is one of the failed node’s neighbors, and it moves to BestPosition to reconnect all its siblings. In the worst case, a backup node moves to the position of the failed node, and it must connect all its siblings. That is the relocated scheme in DCR [17]. It has been proven that the maximum distance a node travels in DCR is the communication range R _c . Lemma 1 proves that HCR outperforms DCR in terms of travel distance in one motion.

Take Fig. 7 as an example. Node A3 is the failed node, and node A4 moves to A4^′ to connect A1 and A2, and $\phantom {\dot {i}\!}d_{A4,A4^{\prime }}<d_{A4,A3}<R_{c}$. Due to the motion of node A4, node A5, and node A6 will be out of reach by A4^′, so before moving to A4^′, A4 will select a node from its one-hop neighbors {A5,A6}. Suppose A6 is selected and BestPosition is A6^′, then $\phantom {\dot {i}\!}d_{A6,A6^{\prime }}<d_{A6,A4}<R_{c}$. □

Fig. 7

The maximum distance a node travels in HCR

Theorem 5

The shorter distance a node travels, the fewer nodes are affected. The probability of a node affected by its moving neighbor is ($1-\frac {2\theta -\sin {2\theta }}{\pi }$), where θ= arccosd/2R _c , d is the travel distance, and R _c is the communication range. It approximately equals 0.62 times of d/R _c .

Proof

Take Fig. 8 a as an example. Node A moves to A ^′. The nodes in the shaded area like node B are within the communication range of node A, while out of reach by A ^′. The travel distance equals d. It is assumed that the nodes are deployed randomly, so a node has an equal chance to reside at any place. The probability of a node affected by its moving neighbor is the probability that it is located in the shaded area. Therefore, the probability is the shaded part area divided by the whole communication area. According to symmetry and area formula of a sector, the shaded part area equals π R _c ²−(2θ·R _c ²− sin2θ·R _c ²), where θ= arccosd/2R _c . So the probability of a node affected by its moving neighbor is ($1-\frac {2\theta -\sin {2\theta }}{\pi }$). In Fig. 8 b, the solid line reflects the probability against d/R _c , where the dashed line reflects the growth rate of probability. It shows that the growth rate is approximately 0.62. □

Fig. 8

The influence of node moving distance. a An example of the influence of node moving.b The influenced probability of a node against d/Rc

Theorem 6

The time complexity of the backup node and BestPosition selection is O(n ³), where n is the number of failed node’s one-hop neighbors.

Proof

It has been proven that BestPosition must be the intersection of any pair of nodes’ communication boundary in NT(i) or the intersection of two nodes connection line and one node’s communication boundary in Theorem 1. For each node, the intersection of any pair of its sibling nodes’ circle is computed firstly, and the time complexity is O(n ²). Then the intersection of a line that connects this node to its sibling node and the sibling node circle is obtained, and the time complexity is O(n). Finally, check whether the intersection is within the communication range of all the sibling nodes and find the minimum travel distance. The time complexity is also O(n ²). Therefore, the time complexity of the best position selection for each node is O(n ²+n+n ²), that is O(n ²). Then the node with the minimum traveling distance will be selected as the backup node, and its best position will be BestPosition. Hence, the time complexity of the backup node and BestPosition selection is O(n ³). □

Theorem 7

The total message complexity of HCR is O(N), where N is the number of actors.

Proof

The selection of a backup node and BestPosition is done at the failed or moved nodes in HCR. It just maintains one-hop neighborhood table for each node. In addition, the failed node only sends a message to its backup node about its movement. In the worst case, there are N−2 nodes moving. Each moved node sends a movement message to its backup node. Therefore, the total message complexity of HCR is O(N), where N is the number of actors. It is worth noting that the exchange with neighbors at a new position does not count in HCR, and it is considered as a part of status update for maintaining one-hop neighborhood table. □

Theorem 8

The time it takes HCR to restore network connectivity is proportional to N and R _c , where N is the number of actors and R _c is the communication range.

Proof

Firstly, HCR proactively selects a backup node and BestPosition before a node failure, so when a node fails, the backup node will move to BestPosition. This will trigger cascading motion. For each moved node, it will select its new backup node and BestPosition before its moving and send a message to notify its backup node to move to BestPosition. According to Theorem 7, the time complexity is O(n ³) where n is the number of its neighbors. Usually, n is very small compared with the total number of nodes N, so the computing time can be ignored. In the worst case, there are N−2 nodes moving, and each node moves at most R _c , so the total time it takes HCR to restore network connectivity is (N−2)×R _c , which does not exceed (N×R _c ). □

6 Simulation results

Extensive simulations have been conducted to evaluate the performance of the proposed HCR compared with the previous algorithms RIM and DCR. The simulation settings and performance metrics are introduced in Section 6.1, and the detailed results and analysis are presented in Section 6.2.

6.1 Simulation settings and performance metrics

In order to evaluate the performance of HCR and compare it with the previous algorithms RIM [35] and DCR [17] fairly, we carry out all the simulations on Matlab R2012a with an Intel Core i3-3220 CPU and 8 G RAM computer. In the simulations, numerous mobile actors are deployed randomly in an area of 1000 m ×1000 m. All the actors are homogeneous with the same communication range. The energy cost and lifespan are pursued during the connectivity restoration in varied applications, but the actual energy cost is difficult to model and capture during simulations, so the following four metrics are employed to evaluate the performance of HCR:

• Number of relocated nodes: It reports the average number of relocated nodes during a single-node failure restoration. This metric assesses the scope of connectivity restoration within a network.

• Total distance moved: It reports the total distance that the involved nodes move during the restoration. This metric assesses the efficiency of the restoration methods from the standpoint of a network.

• Average travel distance: It depicts the average distance that the involved nodes travel during the restoration. This metric assesses the efficiency standing in the perspective of a node.

• Number of messages: It captures the total number of messages sent among the nodes during the restoration. This metric assesses the communication overhead of the restoration methods.

In addition, to study the impact of network topology on the performance of HCR, the network topology is varied in different simulations with the following parameters:

• Number of deployed nodes (DN): It reports the number of the deployed nodes in an area. Since the area is 1000 m ×1000 m, this parameter actually represents the density of a WSAN. The larger the DN, the bigger the node density, indicating a stronger network connectivity. In a rather highly connected WSAN, a node has many one-hop neighbors, and it is easier to choose a backup node and BestPosition. It will increase the time of selecting at the same time.

• Communication range (R _c ): It is assumed that all the actors are homogeneous with the same communication range and a pair of nodes can communicate with each other when they are within each other’s communication range. R _c also affects network density DN. Small R _c will generate a rather sparse network, while large R _c will increase network connectivity. This will also increase the travel distance of the involved nodes during the restoration under HCR.

6.2 The overall results

Both sparse and dense networks with different values of DN and R _c are simulated to evaluate the performance of HCR compared with RIM [35] and DCR [17]. The value of DN is chosen from set {10,20,40,60,80,100} and the value of R _c is chosen from set {20,40,60,80,100,120}. For every topology, 10 WSANs are generated, and for every WSAN, 30% of the deployed nodes are randomly selected forming the failed node set. For every WSAN, RIM, DCR, and HCR are run after a node fails and the average values are computed. The overall results are shown in Table 1. We can see that HCR outperforms RIM and DCR in terms of the number of relocated nodes, total travel distance, average travel distance, and number of messages for both high-density and low-density networks. The results and analysis are detailed in the following.

Table 1 Overall simulation results

6.2.1 Number of relocated nodes

Figure 9 a, b reports the number of the relocated nodes during the restoration under various communication range R _c and network size DN, respectively. The plotted results are the average over multiple independent simulations. For any topology, 30% of DN fail randomly. The two figures indicate that DCR and HCR relocate fewer nodes than RIM. This is because RIM requires all the neighbors of the failed node to move inward. In fact, sometimes they may move outward to the position that is R _c /2 away from the failed node. In addition, the number of the relocated nodes grows with the increasing of R _c and DN since the larger R _c and DN, the more neighbors a failed node has. DCR and HCR have the same results when R _c and DN increase, because they both identify critical nodes, and only move one of the neighbors to replace the failed node in a round of restoration. It is worth noting that the candidate selection is processed at the failed node before it fails or moves, so they just maintain one-hop neighborhood table.

Fig. 9

The number of relocated nodes during the restoration while varying a the communication range (with D N=40) and b the network size (with R _c =80)

Moreover, Fig. 9 a shows that HCR outperforms DCR when the communication range is less than 72, and has the similar results when the communication range grows. This is because of the candidate selection. HCR will choose a node to move the least distance to BestPosition and to reconnect the failed node’s neighbors, while DCR chooses the backup node by critical, degree, and distance. Furthermore, DCR moves the backup node to the failed node, and the failed node is not at BestPosition most of the time. When R _c is small, the network is sparse, and the choice of a backup node is too limited, so HCD outperforms DCR. When R _c is large, the network is dense, and it is likely to choose an uncritical node to restore network connectivity, so DCR can get the same result with HCR. Figure 9 b shows that DN has little impact on HCR and DCR in terms of relocated nodes when R _c =80. At the moment, the WSAN is dense for the communication range. Considering Fig. 9 a, b, we can see that R _c is more influential than DN on the number of relocated nodes.

6.2.2 Total moved distance

The total moved distance is an important metric for network connectivity restoration, since the actors are energy-limited while moving is an energy-consuming operation. As shown in Fig. 10 a, b, HCR outperforms RIM and HCR for any R _c and DN. As can be seen in Fig. 10 a, b, RIM grows when R _c or DN increases, and it increases linearly. This is terrible and unacceptable when the network scale is large. While the curve of DCR is unstable, it outperforms RIM when the network is dense. In the simulations, when DN=40 and R _c >62, DCR outperforms RIM. Similarly, when R _c =80 and DN>20, DCR outperforms RIM. At the same time, HCR remains stable when varying R _c or DN.

Fig. 10

The total moved distance during the restoration while varying a the communication range (with DN=40), and b the network size (with R _c =80)

In Fig. 10 a, the curve of HCR rises slowly when R _c <60, then it remains stable. It decreases with the increase of DN in Fig. 10 b. When the communication range is small, the network is sparse and the failed node has limited neighbors to select, so HCR needs many nodes to move to restore network connectivity. Since each node just needs to move a little, the growth is slow and small. While when the communication range increases, many choices make a rapid convergence. It is the same for DCR. However, DCR moves the backup node to the failed node, while HCR moves the backup node to BestPosition. It has been proven in Lemma 1 and the simulation results also verify that HCR outperforms DCR in terms of total moved distance. Figure 10 b shows a decreasing of the total moved distance for HCR when increasing DN. This is because when R _c is determined, the growing of DN will increase the choices, and moving a shorter distance is enough to restore network connectivity.

6.2.3 Average travel distance

As aforementioned, node motion is a high energy-consuming operation, and nodes are prone to be out of work due to energy depletion. A node failure will incur cascading motion and more energy consumptions creating a vicious spiral. So the average travel distance of the involved nodes is of great importance in assessing the connectivity restoration algorithms.

Figure 11 a shows that the curves of DCR and RIM grow with the increase of R _c while DN=40. HCR also grows until R _c =60, then it maintains stable. RIM outperforms DCR since it moves more nodes, and the biggest distance of RIM is limited to R _c /2, while DCR moves the backup node to the position of the failed node, and each involved node moves more than that in RIM. But they both increase linearly, which is unbearable in large-scale WSANs. Since the backup node moves to BestPosition in HCR, the involved nodes will move less than DCR, and when the network is dense, it is stable to move some distance to restore connectivity. As shown in Fig. 11 a, when R _c >60, the average distance is around 11. RIM obtains the same results as HCR when the communication range is small.

Fig. 11

The average distance a node travels during the restoration while varying a the communication range (with D N=40), and b the network size (with R _c =80)

Figure 11 b shows very different results from Fig. 11 a. The average travel distance for each involved node almost remains unchanged with varied network size when R _c is 80. This indicates that network size does not influence the average travel distance because each node moves at most R _c /2 in RIM, while R _c is fixed to be 80. Both the curves of HCR and DCR decrease when increasing the deployed nodes with R _c =80. The WSAN is dense, thus the failed node has more neighbors which increases the probability of choosing the best replacement position of nodes. Due to moving the backup node to the position of the failed node, HCR moves less than DCR.

6.2.4 Number of messages

Communication cost is another important metric to assess the connectivity restoration methods because communication consumes large bandwidth resource which is very limited in WSANs. Figure 12 a, b reports the total number of messages that need to be sent in restoring connectivity corporately. RIM incurs the highest message overhead since the decision of restoration is made by the failed node’s neighbors in RIM. It needs to send the new position to its sibling nodes, so the communication cost is very lager. It becomes much worse when the density of a network increases because the number of neighbors grows.

Fig. 12

The total number of messages during the restoration while varying a the communication range (with DN=40) and b the network size (with R _c =80)

Figure 12 a, b indicates that HCR and DCR achieve similar message overhead when the network is dense. This is because the decision of restoration is made by the failed node. The communication is maintained just between the failed node and its backup node. Figure 12 b also shows that the number of deployed nodes will influence the messages when the communication range equals 80. It is also true for other communication range, and the detailed results are shown in Table 1. It should be noted that HCR outperforms DCR when the communication range is small. In Fig. 12 a, the curve of HCR is below that of DCR when the communication range is less than 60. That means HCR outperforms DCR when the network is sparse. It contributes to the motion strategy. HCR moves less distance than DCR so that it incurs less motion in cascading moving. In sparse networks, the backup node’s traveling distance has a great impact on the cascading moving. Therefore, HCR outperforms DCR since it moves less during a node’s motion.

In conclusion, HCR outperforms RIM and HCR in terms of the four evaluation metrics on all the aspects whether the network is sparse or dense because of its proactive backup node selection and reactive cascading node motion. Though the proactive backup node selection is time consuming, it is carried out before the node failure. This will improve the response time. During the reactive cascading node motion, the computing of BestPosition is complex and time-consuming. While HCR is a distributed and localized method, the number of each node’s neighbors is very small in all kinds of applications, so the selection of BestPosition will not spend so much time, and the connectivity restoration process will be fast.

7 Conclusions

There is a growing interest in the applications of WSANs in the recent years. Due to the harsh employed environment and limited energy supply, WSAN is prone to be out of work, which may break network connectivity. In this paper, we investigate the problem of restoring network connectivity when a single node fails. A hybrid distributed, localized, and efficient connectivity restoration algorithm HCR is proposed to solve this problem through moving the backup node to BestPosition. Compared with the previous schemes, HCR performs a localized network analysis to identify critical nodes, and only a critical node’s failure triggers the restoration process. It is a compromised proposal between the cut vertex identification and non-identification. It is effective and has low complexity.

The performance of HCR is analyzed mathematically and validated through simulations. The simulation results have confirmed the effectiveness of HCR in terms of all the evaluation metrics. More importantly, HCR is applicable to various network topologies, sparse or dense. The performance of HCR remains stable when varying network topology. Though a comprehensive network will increase the complexity of the selection of BestPosition, it is acceptable.

Though HCR is designed for restoring network connectivity after a single-node failure, that means it can only deal with a single-node failure at a time and handle the sequential node failures. It can be extended to hand multi-node failures at a time by adding one more constraint that no two nodes share the same backup node. In addition, the investigated WSANs are two-dimensional, and we plan to study three-dimensional WSANs in the future. At the same time, coverage is another factor that can be taken into consideration in connectivity restoration, which is also our future research interest.