Abstract
We usually use a digraph to represent a wireless network (WN). Correspondingly, a connected dominating set (CDS) of the digraph is usually used to denote a virtual backbone (VB) of the corresponding WN. In this article, focusing on the problem of a minimum strongly connected dominating and absorbing set (MSCDAS) with a bounded diameter (or guaranteed routing cost) for a digraph, which is strongly connected, we introduce two algorithms. One is called the guaranteed routing cost strongly connected dominating and absorbing set (GOC-SCDAS), which can generate a strongly connected dominating and absorbing set (SCDAS) with a performance ratio in respect of the optimal solution. Another is called the guaranteed routing cost strongly connected bidirectional dominating and absorbing set (-GOC-SCBDAS), which can generate a strongly connected bidirectional dominating and absorbing set (SCBDAS) with a performance ratio in respect of the optimal solution and a better routing cost, where and is the transmission range of nodes in the network. Through the simulation experiments, we obtain the conclusion that in terms of the diameter and average routing path length (ARPL) of CDS, the outputs of our algorithms are better than those of the algorithm in (Du et al. 2006).
1. Introduction
Owing to the development of wireless radio communication and very-large-scale technology, WNs such as wireless sensor networks (WSNs) or ad hoc WNs have begun to be widely applied in a lot of fields. For example, in WSNs, since the sensors can be randomly deployed to the expected destination area, WSNs have been successfully applied to numerous fields such as disaster rescue, sea surveillance, climate prediction, bridge health detection, and traffic control [1–7]. However, since there is no predefined infrastructure for facilities with a fixed setup, it is necessary to design a VB for the renewal of network topology and the performance of routing-related tasks [8]. The advantages of a VB established in a network are as follows: when the routing-related tasks are performed to find routing paths, it is enough to search the space of the VB rather than the whole network, which implies that it takes a shorter time for searching routing paths and needs a smaller size of routing table, and then, it causes that the routing maintenance becomes simpler. For constructing VBs in WNs, there are many different methods; particularly, in order to obtain a VB with a better performance, one prefers to find a CDS in a graph, which is modeled a WN containing the VB.
When a VB in a WN is being constructed, the VB size is needed to be considered for the reason that a smaller VB causes less communication overhead. And then, it is a natural idea to construct a minimum VB in a given WN when people hope to reduce the communication overhead of the network. If a connected graph is used to model the WN, then the problem of constructing a minimum VB in the WN is equal to the problem of finding an MCDS in the corresponding connected graph $G$. However, the MCDS problem for a connected graph has been proved to be an NP-hard problem [9]. Therefore, most researchers in this area concentrate on how to find smaller CDSs.
It is worth mentioning that [10] is the first paper to introduce the approximation algorithms computing an MCDS in a unit disk graph (UDG), which is utilized to model a WN with the same transmission radius (or range) for each node. Most of previous researches on the MCDS problem have focused on UDGs [11–14]. These studies all aimed to obtain a smaller CDS to make the best of the existence of a minimum VB.
However, in some WNs, say a WSN, since there are differences of the functionalities and control technologies for connectivity, the powers of these sensors may differ. According to the different requirements on different measured frequencies in collision, a node may be required to change its transmission range. Therefore, for such situations, it is more significant to study a WN with multiple heterogeneous transmission radiuses (ranges) than the one with coincident transmission radius.
Moreover, in some WNs, the energy of the wireless nodes is limited and thus will affect the network lifetime. In other words, the question of how to efficiently use the energy of the wireless nodes is an important issue that the designers of such WNs must consider. Some unnecessary information transmissions can be avoided by choosing an efficient routing method, which can save much energy and extend the life of a network. Hence, when we construct an MCDS for a graph, it is necessary to consider the routing cost in the MCDS. Some research results on the MCDS problem considering the routing cost for a UDG have been obtained in [15–19].
A digraph is strongly connected if and only if for any two vertexes and in the digraph, there exists a directed path from to in the digraph. For a strongly connected directed graph (SCDG), denoted by , a subset is called a dominating and absorbing set (DAS) if the following two conditions hold: (1) for any node , there exists a node such that ; (2) for any node , there exists a node such that . A DAS is said to be an SCDAS if it induces a strongly connected subgraph. An MSCDAS is an SCDAS that has the minimum number of nodes. Figure 1(a) shows a sample MSCDAS in a directed graph where all red nodes {1, 2, 3} form an MSCDAS and are called the dominators. All the nodes that have a directed edge from some dominators are called dominatees. In Figure 1(b), all red nodes {1, 3, 10, 12} make an SCDAS. The MSCDAS problem of is the problem of how to solve an MSCDAS of .
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In this paper, for an SCDG, denoted by , which models a WN with nodes’ transmission radiuses in the range , we consider the problem of constructing MSCDAS of the SCDG. Our main works are as follows: (1)We propose a centralized algorithm, called by GOC-SCDAS, which produces an SCDAS with such that for any two vertexes and of , the length of the smallest routing path between them in is , where , is an MSCDAS of , and denote the diameter of (2)We propose another algorithm, called by -GOC-SCBDAS, which produces an SCBDAS with such that for any , , where denotes the cardinality of an MSCDAS of ,
The following is the rest of the article. The related work is introduced in Section 2. The problem statement is formulated in Section 3. Section 4 presents two algorithms for constructing a VB from two different perspectives based on a directed graph model. Our simulation results are in Section 5. Our conclusion is in Section 6.
2. Related Work
In the study of a WN, for the sake of convenience, we usually use a graph to denote a WN and a CDS of the graph to model the corresponding VB for that WN. So far, the research on CDSs has received widespread attention. [20] pointed out the computation of the MCDS was an NP-hard problem for a general graph and even for a UDG [9]. Thus, most studies simply find CDSs with a reasonable approximation ratio. Based on the UDG model, [10] firstly introduced an algorithm for computing the MCDS of a UDG. In [10], using two approximation algorithms, which are polynomial-time, Guha and Kuller obtained two CDSs with performance radio and , respectively, where , . During the process of constructing a sufficiently small CDS, the computation of the upper bound on the maximal independent set (MIS) is one uneasy work. In [21], Wan et al. obtained a result that the cardinality of each MIS does not exceed . Later, people further improve this bound [19, 22–25]. [24] presented an upper bound of MIS , which is the best bound on the cardinality of MIS in a UDG. However, most of previous studies have ignored the importance of the routing overhead. We know that when we obtain an MCDS, the shortest paths in such an MCDS-based VB may be unavailable. Currently, there are a few papers that have constructed a CDS with a bounded routing path length, whose size is slightly larger than the size of MCDS [26–31]. In order to obtain a CDS in a UDG, Kim et al. designed an algorithm, called by CDS-BD-D, which generates a CDS with a bounded diameter [29]. [17] presented an algorithm and presented a CDS with the bound by the algorithm. Further, in [26], they proved that for two nodes and in UDG, there is a path with inner nodes in from to such that its length does not exceed . In [27], Ding et al. defined a new concept, called by -minimum routing cost CDS (-MOC-CDS), where . Let be a graph, be a CDS; is called a -MOC-CDS if the CDS has the property that (and are any two nodes in). For the sake of convenience, they turned the -MOC-CDS problem into another equivalent problem, called by -2hop-DS problem. The authors pointed out that -2hop-DS is also NP-hard [27] when . In [30], a centralized algorithm, called by GOC-MCDS-C, was proposed by Du et al. At the same time, they presented the corresponding distributed algorithm and pointed out the GOC-MCDS problem is still NP-hard when . They claimed that these algorithms produced a CDS with and ( and are any two nodes). It should be noted that the CDS’s cardinality is fairly large. For UBGs and quasi-UDGs, Wu et al. considered the similar problem to -MOC-CDS; the details can be found in [31]. However, in the above studies, none of them investigate the problem of SCDAS with a guaranteed routing cost for an SCDG. Possibly because of different amounts of remaining electricity or the implementation of energy-saving devices on some nodes, the different nodes in a network may have different transmission ranges. For the case of nodes with different transmission ranges, there are two papers to discuss the problem of SCDAS for nonuniform networks (see [32, 33]), but they did not concern the routing cost in the corresponding graph. In [34], Liu et al. extended the conclusion in [30] to heterogeneous WSNs, and they formulated the -MOC-SCBDS problem. In order to solve the -MOC-SCBDS problem, they proposed a centralized algorithm and a corresponding distributed algorithm. They claimed that these -MOC-SCBDS algorithms produced a strongly connected bidirectional dominating set (SCBDS) with , where . However, the authors of [34] did not consider the guaranteed routing cost for an SCDG with heterogeneous transmission ranges; in this case, the proof procedure for the upper bound of in [34] may be incorrect.
3. Preliminaries
In this article, we focus on the MSCDAS problem with a guaranteed routing cost in a WN. Let () denote the minimum (maximum) transmission range of nodes in the network and denote the Euclidean distance between and . We use a digraph to denote a WN with heterogeneous transmission ranges and to denote the transmission range of node , where is the node set in the network and is an edge set including all directed links in the network such that if and only if . Then, for , if and only if and . Suppose that digraph is strongly connected. We use to represent the number of hops of the shortest directed path from node to node in . For , let denote the shortest directed routing path from to , whose all inner nodes belong to , and denote the subgraph induced by . If for any two nodes , has a directed path from to, then is said to be strongly connected. If , then node is called an in-neighbor node of , at the same time, is called an out-neighbor node of . For and node , let , , and . We use to denote a weight function, where is a node and is the of node . For two given 2-tuple variables and , if and only if one of the following conditions is true: (1) or(2)
Definition 1. Let be an SCDG, , then is an independent set (IS) if and only if for each pair of nodes , , or . is called an MIS if is an IS and for any , is not an IS.
Definition 2. Let be an SCDG, , then is called a bidirectional dominating and absorbing set (BDAS) if for , has at least one node such that is simultaneously dominated and absorbed by . A BDAS of is called an independent bidirectional dominating and absorbing set (IBDAS) if for two nodes , , or . An IBDAS is called a maximal independent bidirectional dominating and absorbing set (MIBDAS) if for any , is not an IBDAS.
Definition 3. Let be an SCDG, . Then, is called an SCBDAS with a guaranteed routing cost () if the following conditions hold: (a) is a BDAS(b) is strongly connected(c), where
Definition 4. Let represent a connected digraph. For , let represent a shortest path from to and represent the length of . Then, is called the diameter of .
Definition 5. Let be an SCDG, , be a breadth-first search (BFS) tree with root node , and . In the BFS , if and , then is called parent and is called child.
Definition 6. Let be an SCDG, , and be a BFS tree with root node . In the BFS , if there exists a node satisfying and , then is called a brother node of .
Lemma 7. Suppose that is an SCDG and is a DAS of . If for two nodes , , where is a constant, then is an SCDAS of .
Proof. Since is an SCDG and , we always have a path from to , and . According to the assumption, it holds that , which means that there exists a path for transmitting a message from node to node via nodes in . Hence, is an SCDAS of .
Lemma 8. Suppose that is an SCDG and is an MIS of , then is also a DAS of .
Proof. According to the assumption that is an MIS of , we have that must be a dominating set. Next, we show that must be an absorbing set of . In contrast, suppose that has a node, say , such that for any node , , which implies that is an IS of , a contradiction.
Let denote a disk with center and radius . Then, the number of independent nodes in does not exceed the maximum number of circles of radius 0.5 that can be packed into the disk with center and radius . Since regular hexagons, each circumscribing a circle of radius 0.5, can densely fill in a given disk, these circles can be replaced with their corresponding circumscribed regular hexagons to compute the bound of the size of MIS in (see [1]).
Lemma 9. Let be an MIS in a UDG , , is a subset of such that each node of it is covered by the disk with center and radius , then .
Proof. Assume that is the disk with center and radius , denotes the area of , and denotes the area of a circle of radius 0.5, and denotes the area of a regular hexagon circumscribing a circle of radius 0.5. Then, According to the above discussion, it holds that . To get a better bound on , we use the area of a regular hexagon circumscribing a circle of radius 0.5 in place of the area of a circle of radius 0.5. Note that for a hexagon circumscribing such a circle near the boundary of , not all of its area may be used. For example, in Figure 2, the part of the hexagon circumscribing circle with center , lies outside of . The area of the part lying outside of is no more than Hence, we have
4. Algorithm Description
It has been proven in [35] that in a disk graph, it is impossible to obtain a polynomial-time -approximation algorithm for -MOC-CDS unless . Note that a directed disk graph is a special disk graph, this implies that finding an -MOC-CDS in a directed disk graph is also NP-hard unless . In this section, we propose two algorithms for constructing a VB with guaranteed routing cost in a WN with heterogeneous transmission ranges.
4.1. Centralized Algorithm GOC-SCDAS
In this section, we present an SCDAS construction algorithm (called by GOC-SCDAS) for an SCDG . Traditional CDS construction algorithms often consist of two steps. The first step is to find an MIS D. The second step is to add some nodes to to form a CDS. Our basic idea on the centralized algorithm GOC-SCDAS can be summarized into three steps. During the first step, we choose a node as a root node by using a leader selection strategy [2]. In the second step, we build a BFS tree and then construct a dominating set of such that for every node , there is one path from to . In the last step, we construct another digraph , where . Using a method similar to that applied in the second step, we can obtain a dominating set of , denoted by , which is an absorbing set of . Then, the union of and , denoted by , is an SCDAS of . In order to understand the concept of BFS tree in the algorithm, we give an example about the resulting BFS tree for a directed graph as follows: Figure 3 is an SCDG with 15 nodes, Figure 4(a) shows a BFS tree with root node using a leader selection strategy for , and Figure 4(b) is a BFS tree with root node for the corresponding graph . Figure 5 is the flowchart of GOC-SCDAS.
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Lemma 10. The node set produced by the subroutine is a dominating set (DS) of .
Proof. Note that the input graph is an SCDG. From Line 20, Line 22, Line 26, Line 33, and Line 38 in the subroutine , we conclude that contains all black nodes in at the end of the subroutine . After Line 40, if , then is gray or white. Note that after the loop in Line 21 to Line 25, there are no white nodes in , which implies that is gray. From Line 10, Line 18 and Line 22, we can get that there exists one black node with . Hence, is a dominating set.
Lemma 11. The node set produced by the subroutineis an IS.
Proof. In contrast, suppose that is not an IS; then, there exist two black nodes with and . Suppose that is colored earlier than . Consider the following situations.
Case 1. . This means that and are at the same level in the BFS tree in . According to Line 14~Line 20 in the subroutine , it can be found that is colored black and subsequently node is colored gray; consequently, is gray, which is a contradiction.
Case 2. . This means that node must be parent node and that is child node. According to Line 21 ~ Line 25 in the subroutine Roottree, it can be found that and are subsequently colored black and gray, respectively; consequently, is gray, which is a contradiction.
Theorem 12. The set output by GOC-SCDAS is an SCDAS in .
Proof. According to Lemma 10, and are a dominating set of and a dominating set of , respectively. Since is obtained by reversing all edges in , is an absorbing set of , which implies that is a DAS of . We claim that for any node , there exists one path, denoted by , fromtoin . Suppose that ,. Next, in order to prove above claim, we use the induction on . When or , the result is trivial. Suppose that and that the result is true for . If is even , then according to Lines 29~34, there exists a parent node of , denoted by , such that . According to the hypothesis, we have that there exists one path from to . Hence, is a path from to in , and then, the result is true. If is odd , then according to Line 35~40, there exists a parent node of , denoted by , such that . According to the hypothesis, we have that there exists one path from to . Hence, the result is true.
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Similarly, the following claim is true: for, there exists one path from to in , which is equivalent to the claim that for , there exists one path from to in .
Now, we show that is strongly connected. We need only to prove that for any , there exist two paths between and ; one of them is from to , denoted by , and another is from to , denoted by . Consider the following situations:
Case 1. . From the above discussion, we have that there is a path from to and a path from to . Then, is a path from to . Similarly, there exists a path from to .
Case 2. . Therefore, there is a path from to . Since is an absorbing set, there is a node with . On the other hand, according to the above discussion, there exists a path from to . Hence, is a path from to . A similar argument proves there exists a path from to .
Case 3. . An argument similar to that for Case 2 can be used here.
Case 4. . According to the above discussion, there exists a path from to and another path from to , and thus, is a path from to . Since and is an absorbing set of , there is node with . Therefore, there exists a path from to . On the other hand, since , there exists a path from to . Hence, is a path from to .
Case 5. . An argument similar to that for Case 4 can be used here.
Case 6. . According to the above discussion, there exist a path from to and another path from to , implying that is a path from to . On the other hand, since is an absorbing set (a dominating set), there is one node with . A similar argument can be used to prove there is one path from to . Hence, it is obtained one path from to :.
The following lemma follows [30].
Lemma 13. The length of the path from node to each black node in is at most hops, where
Lemma 14 [36]. Assume that is a digraph with the transmission range, is an IS, then , where .
Theorem 15. Let be the SCDAS obtained by GOC-SCDAS. Then, . For any two nodes , let represent the shortest routing path between and , which includes only nodes in except for and , be the length of ; then, , where is any one optimal SCDAS of .
Proof. Let () denote the produced by (), and let () denote the produced by (). According to GOC-SCDAS, we need only to add at most additional nodes to such that all nodes in can be connected to form a forward tree with root node . Then, . According to Lemma 11 and Lemma 13, it holds that and . Hence, .
On the other hand, it is obvious that . For any two nodes, let be a path from to , denote the length of . Suppose that is the longest shortest path a node to in , then . Let represent an optimal solution on the SCDAS problem for . Then, there exist two paths as follows: is the shortest path from to that includes only nodes in except for and , and is the shortest path from to that includes only nodes in except for and. Let . Then, the diameter of the subgraph induced by is no less than . It is obvious that. Hence, . According to Lemma 13, for any pair of nodes and of , we have the following inequalities:
4.2. Algorithm for constructing an SCBDAS
Assume that is an SCDG. In this section, one algorithm will be proposed for constructing an SCBDAS of , which is a special case of an SCDAS of . This algorithm includes two stages. Firstly, a BDAS of will be constructed by us. Next, we will add some nodes to to form an SCBDAS. The details can be found in Algorithm 2.
The following lemma follows [35].
Lemma 16. Assume that is an SCDG, is an SCBDAS of . If for any two nodes with , . Then, the following condition holds , where and .
Now, we introduce the -GOC-SCBDAS algorithm. For the convenience of understanding this -GOC-SCBDAS algorithm, we present its flowchart (see Figure 6).
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Algorithm 2: -GOC-SCBDAS. | ||||||||||||||||||||||||||||||||||||||||||||
Lemma 17. For the digraph with transmission ranges , suppose that is the BDAS produced by Algorithm 2. Then, the following conditions are true: (1) is an IS, and(2)for each , , where and
Proof. (1)From Line 8, we know that is an MIBDAS of , which implies that is an IS of (see Line 9). From Line 14 to Line 15, we find that for any , is still an IS of . Hence, after the loop in Lines 13~17, is still an IS(2)For each node , let be the circle of center and radius () and be the circle of center and radius. From Line 7 to Line 17, we find that is an IS, which implies that for any two nodes in , say and , . It is easily seen that for any two nodes , if and are contained in, then and are disjoint and are covered by. Hence, the cardinality of does not exceed the number of circles with radius that are disjoint to each other and are contained in . Since the densest possible packing of disks in a plane is attained with a hexagonal lattice [37], it is expectable that the area of a hexagon circumscribing a circle with radius can be used to replace that of the circle to compute a upper bound of the number of independent nodes covered by . According to Lemma 3.3, we can obtain the process of computing the upper bound on as follows:The area of is The area of a hexagon circumscribing a circle with radius is Note that the part of a hexagon circumscribing a circle near the boundary may lie outside (similar to the circle with center at in Figure 2); however, the area of this part does not exceed Therefore, there is an upper bound of :
Theorem 18. For an SCDG , let be the set produced by Algorithm 2. Then, the following conditions are true: (1) is an SCBDAS in (2)For , it holds that (3),where
Proof. (1)Since and is a BDAS in , is a BDAS in . Now, we will show that is strongly connected. We need only to prove that for , there is a path from to and each node in the path belongs to . Since is an SCDG, there exists the shortest path from to , where . Let be the first node of in the direction from to that does not belong to ; then, there is a node such that and . Let be the shortest path from to . From Line 18 and Line 19 of Algorithm 2, we know that all nodes in are in . Thus, is one path from to . By repeating the above process, one path from to , whose each node belongs to , can ultimately be obtained(2)Suppose that are a pair of nodes such