This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Secon 2010 proceedings.
Two-Tiered Constrained Relay Node Placement in Wireless Sensor Networks: Efficient Approximations Dejun Yang, Satyajayant Misra, Xi Fang, Guoliang Xue, and Junshan Zhang Abstract— In a wireless sensor network, short range multihop transmissions are preferred to prolong the network lifetime due to super-linear nature of energy consumption with communication distance. It has been proposed to deploy some relay nodes such that the sensors can transmit the sensed data to a nearby relay node, which in turn delivers the data to the base stations. In general, the relay node placement problems aim to meet certain connectivity and/or survivability requirements of the network by deploying a minimum number of relay nodes. In this paper, we study twotiered constrained relay node placement problems, where the relay nodes can only be placed at some pre-specified candidate locations. To meet the connectivity requirement, we study the connected single-cover problem where each sensor node is covered by a relay node (to whom the sensor node can transmit data), and the relay nodes form a connected network with the base stations. To meet the survivability requirement, we study the 2-connected double-cover problem where each sensor node is covered by at least two relay nodes, and the relay nodes form a 2-connected network with the base stations. We focus on the computational complexities of the problems, and propose novel polynomial time approximation algorithms for these problems. For the connected single-cover problem, our algorithms have O(1) approximation ratios. For the 2-connected double-cover problem, our algorithms have O(1) approximation ratios for practical settings and O(ln n) approximation ratios for arbitrary settings. Experimental results show that the number of relay nodes used by our algorithms is no more than twice of the number of relay nodes used in an optimal solution. Keywords: Relay node placement, wireless sensor networks, connectivity and survivability.
1. Introduction and Related Work A wireless sensor network (WSN) consists of many lowcost and low-power sensor nodes (SNs)[1]. These sensor nodes sense and communicate data over multiple hops to the base stations. The fact that the energy requirement for transmission is a super-linear function of the transmission distance [1] necessitates short range communication to improve network lifetime [13]. To prolong network lifetime while meeting certain network specifications, one proposed direction is to deploy a small number of relay nodes (RNs) in the WSN such that they can communicate with the SNs and other RNs [2], [4], [7], [12], [13], [15], [19], [20], [31]. This is studied under the theme of relay node placement. This problem has received considerable attention from the networking community, with related papers published in MobiCom [23], MobiHoc [2], [28], and Infocom [11], [15], [21], [32]. Dejun Yang, Xi Fang, Guoliang Xue and Junshan Zhang are affiliated with Arizona State University, Tempe, AZ 85287. Email: {dejun.yang, xi.fang, xue, junshan.zhang}@asu.edu. Satyajayant Misra is affiliated with New Mexico State University, Las Cruces, NM 88003. Email:
[email protected]. This research was supported in part by NSF grant 0901451 and ARO grant W911NF-09-1-0467. The information reported here does not reflect the position or the policy of the federal government.
In general, the relay node placement problem has been studied under two perspectives, namely the routing structure and the connectivity requirements. The study based on the routing structure may be further classified into either singletiered or two-tiered [7], [12], [13], [20], [23]. The study based on connectivity can be classified into either connected or survivable [2], [12], [15], [32]. In single-tiered relay node placement, the SNs may also forward packets. In two-tiered relay node placement, the SNs transmit their sensed data to an RN or a BS, but do not forward packets for other nodes. In connected relay node placement, the placement of RNs ensures connectivity between the SNs and the BSs. In survivable relay node placement, the placement of RNs ensures biconnectivity between the SNs and the BSs. In what follows, we review the related works. Throughout, we will use R and r to denote the communication ranges of RNs and SNs, respectively. We will use k = 1 to denote connectivity requirement and k ≥ 2 to denote survivability requirement. The problems studied in this paper fall under the two-tiered classification. However, for the sake of completeness, we also present a brief review of single-tiered relay node placement. Lin and Xue [18] proved the hardness of the problem for R = r and k = 1, and presented a 5-approximation algorithm. The steinerization scheme proposed by them was used by following works [2], [3], [4], [7], [11], [12], [15], [19], [20], [28], [32]. The works [2], [3], [4], [15] presented constant approximation algorithms, but with the assumption that R = r. In [7], [20], [32], a more general problem with R ≥ r was studied and constant approximation algorithms were proposed. The two-tiered relay node placement problem has been studied by [7], [11], [12], [19], [20], [28], [32]. The two-tiered model applies well to the highly scalable clustered wireless sensor networks [10], wherein the RNs act as the cluster heads [10], [23], [31]. In the two-tiered problems formulated by Hao et al. [12], an SN has to be connected to at least k RNs and the RNs need to form a k-connected network. In [11], Han et al. studied the scenario where the SNs may use different power levels, while the RNs use a single power level. They presented O(1)-approximation algorithms for the case of R ≥ r and k ≥ 2. Liu et al. [19] proposed two constant approximation algorithms for the problem with R = r and k = 2. Lloyd and Xue [20] studied the problem for R ≥ r and k = 1, and proposed a (5 + ) algorithm, where is any positive constant. Recently, Efrat et al. [7] proposed a PTAS [6] for this problem. Srinivas et al. [28] studied the problem of the maintenance of a mobile backbone using the minimum number of backbone nodes given that R ≥ 2r and k = 1. Zhang et al. [32] studied the problem with R ≥ r and k = 2 and presented a (20 + )approximation algorithm when the BSs are considered.
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All of the above works study unconstrained relay node placement, in the sense that the RNs can be placed anywhere. In practice, however, there may be some physical constraints on the placement of the RNs: there may be a lower bound on the distance between two network nodes to reduce interference; there may be some forbidden regions where relay nodes cannot be placed. For instance, in a WSN application for monitoring an active volcano that is ready to erupt, the inside of the crater (where the temperature could reach many thousand degrees) is a forbidden region for placement of relay nodes (any nodes). There are many such practical applications where the placement in general is constrained. The relay node placement problem, subject to forbidden regions and lower bound on internode distances, is intrinsically harder to solve efficiently, compared to its unconstrained counterpart. As an initial step towards solving this challenging problem, we study constrained relay node placement problems where the RNs can only be placed at a set of candidate locations. Our formulation can be viewed as an approximation to the aforementioned relay node placement problem, subject to the constraints of forbidden regions and lower bound on internode distance in the following sense. Instead of allowing the relay nodes to be placed anywhere outside of the forbidden regions and satisfying the internode distance bound, we further restrict the placement of the relay nodes to certain candidate locations that are outside of the forbidden regions. The use of candidate locations simultaneously approximates the constraint enforced by the forbidden regions and the constraint enforced by the internode distance bound. Misra et al. [21] studied the single-tiered constrained relay node placement problem in WSNs. In this paper, we study the two-tiered constrained relay node placement problem, under both connectivity and survivability requirements. As discussed earlier, the two-tiered problem strongly resembles the real world deployment of WSNs. The problem with connectivity requirement is denoted as the connected singlecover (1CSCP) problem. In this problem, we want to place a minimum number of relay nodes such that (1) each SN is covered by at least one relay node and (2) the relay nodes form a connected network with the base stations. The problem with survivability requirement is denoted as the 2-connected double-cover (2CDCP) problem. In this problem, we intend to place a minimum number of relay nodes such that (1) each SN is covered by at least two relay nodes and (2) the relay nodes form a 2-connected network with the base stations. We formulate the problems, discuss their hardness, and prove that the 1CSCP problem is NP-hard. Further, we present polynomial time constant approximation algorithms. The rest of the paper is organized as follows. In Section 2, we present the preliminary definitions and lemmas. In Section 3, we study the 1CSCP problem. In Section 4, we study the 2CDCP problem. Section 5 presents linear programming formulations of the problems, which we use to evaluate the effectiveness of our proposed algorithms. We present experimental results in Section 6 and conclude the paper in Section 7.
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2. Definitions and Preliminaries In this section, we formally define the problems and present some basic lemmas that will be used in later sections. We consider a heterogeneous wireless sensor network HWSN consisting of three kinds of nodes: sensor nodes (SNs), relay nodes (RNs), and base stations (BSs). We make the following standard assumptions [21], [32]. For given constants R ≥ 2r > 0, the communication range of the SNs is r, of the RNs is R, and of the BSs is much greater than R such that any two BSs can communicate directly with each other (as they are connected via either the wired networks or the satellites). We use d(x, y) to denote the Euclidean distance between two points x and y in the plane. We also use u to denote the location of a node u, where it is unambiguous. We are interested in a two-tiered set-up, where the SNs send the sensed data to an RN or a BS within distance r, while the RNs can forward received packets to an RN or a BS within distance R. We define the hybrid communication graph (HCG) and the relay communication graph (RCG) in the following. Definition 2.1: [Hybrid Communication Graph] Let r > 0 and R ≥ 2r be the communication range for the sensor nodes and that for the relay nodes, respectively. Let S be a set of base stations, X be a set of sensor nodes, Z be a set of candidate locations where relay nodes can be placed. Let Y be a subset of Z. The hybrid communication graph HCG(r, R, S, X , Y) induced by the 5-tuple (r, R, S, X , Y) is a directed graph with vertex set V = S ∪ X ∪ Y and edge set E defined as follows. For any two base stations bi , bj ∈ S, E contains the bi-directed edge (bi , bj ). For a relay node y ∈ Y and a node z ∈ S ∪ Y, E contains the bi-directed edge (y, z) if and only if d(y, z) ≤ R. For a sensor node x ∈ X and a node z ∈ Y ∪ S, E contains the directed edge (x, z) if and only if d(x, z) ≤ r. 2 Definition 2.2: [Relay Communication Graph] Let R > 0 be the communication range for the relay nodes. Let S be a set of base stations, and Z be a set of candidate locations where relay nodes can be placed. Let Y be a subset of Z. The relay communication graph RCG(R, S, Y) induced by the 3-tuple (R, S, Y) is an undirected graph with vertex set V = S ∪ Y and edge set E defined as follows. For any two base stations bi , bj ∈ S, E contains the undirected edge (bi , bj ). For a relay node y ∈ Y and a node z ∈ S ∪ Y, E contains the undirected edge (y, z) = (z, y) if and only if d(y, z) ≤ R. 2 The HCG characterizes all possible pairwise communications between pairs of nodes. The RCG characterizes all possible communications between the relay nodes and/or base stations. We also define in the following the concepts related to the weight and the relay size of an RCG. These concepts will be used later to establish relationships between the weight of a subgraph and the number of corresponding relay nodes. We use the following standard graph theoretic notations: for a graph G, V (G) denotes the vertex set of G and E(G) denotes the edge set of G. Definition 2.3: Let G = RCG(R, S, Z) be a relay communication graph. Let Y ⊆ Z be a set of selected RNs. For each edge e = (u, v) in the RCG, we define its weight induced by
Y (denoted by wY (e)) as
A. Problem Definitions and Notations
wY (e) = |{u, v} ∩ (Z \ Y)|.
(2.1)
Let H be a subgraph of G. The weight of H (denoted by wY (H)) is defined as wY (H) = wY (e). (2.2) e∈E(H)
The relay-size of H (denoted by sY (H)) is defined as sY (H) = |V (H) ∩ (Z \ Y)|.
(2.3)
We call e a weight-0 edge if wY (e) = 0, a weight-1 edge if 2 wY (e) = 1, and a weight-2 edge if wY (e) = 2. Lemma 2.1: Let H be a subgraph of RCG(R, S, Z) and Y ⊆ Z be a selected set of RNs. Assume that each RN ∈ Z \Y in H has degree at least 2 (in H), then wY (H) ≥ 2 · sY (H).2 P ROOF. This lemma can be proved by shifting the weight of an edge to its end nodes. We refer the reader to Lemma 2.1 in [21], while noting the following differences. Instead of the HCG we consider an RCG here, and instead of all RNs we consider only those belonging to Z \ Y. We will also need the results stated in Lemma 2.2 and Lemma 2.3. These lemmas have been proved by Misra et al. in [21], we state them here without the proofs. Lemma 2.2: Let G(V, E) be an undirected 2-connected graph where |V | ≥ 3 and each edge e ∈ E has a unit length l(e) = 1. Let H(V, E ) be a minimum length 2-connected 2 subgraph of G. Then |E | ≤ 2|V | − 3. Lemma 2.3: Let G(V, E) be an undirected connected graph where |V | ≥ 3 and each edge e ∈ E has a unit length l(e) = 1. Let H(V, E ) be a minimum length connected subgraph of G such that two vertices u and v are in the same 2-connected component of H if and only if they are in the same 2-connected 2 component of G. Then |E | ≤ 2|V | − 1. As is standard, a polynomial time α-approximation algorithm for a minimization problem is an algorithm A that, for any instance of the problem, computes a solution that is at most α times the optimal solution to the instance, in time bounded by a polynomial in the input size of the instance [6]. Approximation algorithm A is also said to have an approximation ratio of α. For graph theoretic terms not defined in this paper, we refer readers to the standard textbook [30]. The terms nodes and vertices are used interchangeably, as well as terms links and edges. For concepts in algorithms and the theory of computation, we refer readers to the standard textbooks [6], [9]. 3. Connected Single-Cover Let S be a set of base stations (BSs), X be a set of sensor nodes (SNs), and Z be a set of candidate locations where relay nodes (RNs) can be placed. Let r > 0 be the communication range of the SNs, and R ≥ 2r be the communication range of the RNs. Note that the assumption of R ≥ 2r is valid for most scenarios of HWSNs [12], [28]. In this section, we study the connected single-cover problem.
Definition 3.1: [SCP] A subset Y ⊆ Z is called a singlecover (denoted by SC) of X if every SN in X is within distance r of Y ∪ S. The size of the corresponding SC is |Y|. An SC of X is called a minimum single-cover (denoted by MSC) of X if it has the minimum size among all SCs of X . The singlecover relay node placement problem (denoted by SCP) for (r, R, S, X , Z) seeks an MSC of X . 2 The SCP is closely related to the geometric disk hitting problem (GDHP) which is defined formally in the following. Definition 3.2: [GDHP] Given a set P of points and a set D of disks with radius r. A subset P ⊆ P is called a hitting set if any disk in D is hit by at least one point in P , that is, the center of any disk is within distance r of at least one point in P . The geometric disk hitting problem (denoted by GDHP) seeks a hitting set of minimum size. 2 Definition 3.3: [1CSCP] A subset Y ⊆ Z is called a connected single-cover (denoted by 1CSC) of X if (a) Y is a single-cover of X and (b) the relay communication graph RCG(R, S, Y) is connected. The size of the corresponding 1CSC is |Y|. A 1CSC is called a minimum connected singlecover (denoted by M1CSC) of X if it has the minimum size among all 1CSCs of X . The connected single-cover relay node placement problem (denoted by 1CSCP) for (r, R, S, X , Z) seeks an M1CSC of X . 2 Observe that existing studies [2], [4], [13], [15], [20], [28], [32] considered unconstrained relay node placement. In contrast, our study here focuses on the connected single cover problem subject to constraints. The solutions to some of the unconstrained problems may deploy a relay node on top of a sensor node or another relay node. In practice, there are some physical constraints on the deployment of relay nodes. For example, there should be a minimum distance between two relay nodes to reduce interference. There may also be some forbidden regions where relay nodes cannot be deployed. It is clear that our model is more practical than previous models. This work also differs from the work in [21], because we are studying a two-tiered network while [21] studied the singletiered problem. B. Computational Complexity Before designing algorithms for 1CSCP, we first studied its computational complexity. Using a polynomial time reduction from planar 3SAT [17], we have proved that SCP is NP-hard. We then proved that 1CSCP is NP-hard by a polynomial time reduction from SCP using the ideas of Lloyd and Xue [20]. The proofs are omitted due to space limitations. C. A Framework of Efficient Approximation Algorithms In this subsection, we present a framework of polynomial time approximation algorithms for 1CSCP. The framework decomposes the 1CSCP problem into two sub-problems. It first applies an algorithm A for SCP to compute a single-cover YA of X with small cardinality. It then applies another algorithm B for the Steiner tree problem [14] to augment YA by some other relay nodes YB so that the relay communication graph
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RCG(R, S, YA ∪ YB ) is connected. The union of YA and YB is output as a connected single-cover of X . Algorithm 1 Approximation for 1CSCP(r, R, S, X , Z) Input: Set S of BSs, set X of SNs, set Z of candidate locations for RNs, sensor node communication range r > 0, and relay node communication range R ≥ 2r > 0. An approximation algorithm A for SCP, and an approximation algorithm B for the Steiner tree problem. Output: A connected single-cover given by YA ∪ YB ⊆ Z. 1: Apply algorithm A to obtain a single-cover YA of X . Without loss of generality, we assume that YA is minimal, meaning that none of its proper subsets is a single-cover of X . 2: Construct the relay communication graph G = RCG(R, S, Z). 3: if the nodes in S ∪ YA are not in the same connected component of RCG(R, S, Z) then 4: Stop. The given instance of 1CSCP does not have a feasible solution. 5: end if 6: Assign edge weight in G as in (2.1), with Y substituted by YA , i.e., for an edge (x, y) in G, we have that wYA (x, y) = |(Z \ YA ) ∩ {x, y}|. Apply algorithm B to compute a low weight Steiner tree [14] TB of RCG(R, S, Z), spanning the node set S ∪ YA . Let the set of Steiner points in the tree TB be YB . 8: Output YA,B = YA ∪ YB . 7:
01 01 (b) A single-cover 01
(a) HCG of the instance
01
01
01 (c) A Steiner tree
10 01 01
10 01 01 (d) Optimal solution
Fig. 1. (a) The HCG for 2 BSs (hexagons), 9 SNs (circles), and 18 candidate locations for RNs (squares). (b) A feasible solution to SCP. (c) The resulting 1CSC, which uses 8 RNs. (d) The optimal solution, which uses 7 RNs.
We illustrate the major steps of Algorithm 1 using the example shown in Fig. 1. The instance has 2 base stations (hexagons), 9 sensor nodes (circles), and 18 candidate locations for the relay nodes (squares). We also have R = 2r. The nodes,
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as well as the hybrid communication graph HCG(r, R, S, X , Z) are shown in Fig. 1(a), where we have omitted the directions of the SN-RN edges for clarity. Algorithm 1 performs the following steps. In Line 1, we obtain a single-cover YA of X , which contains 6 relay nodes, shown in Fig. 1(b). In Line 2, we construct the RCG, which is obtained by deleting from the HCG in Fig. 1(a) all edges incident from a sensor node. In Lines 3-5, we find that the nodes in YA ∪ S are in the same connected component. In Line 6, we assign each edge a weight following equation (2.1). In Line 7, we obtain a Steiner tree spanning the node set YA ∪ S, shown in Fig. 1(c). Here we used the MST-based approximation algorithm [16] to obtain the Steiner tree. The Steiner tree has two Steiner nodes, leading to the use of two more relay nodes: |YB | = 2. Fig. 1(d) shows an M1CSC, which uses 7 relay nodes. The solution produced by Algorithm 1 uses 8 relay nodes, which is not optimal, but close to optimal. Theorem 3.1: The worst case running time of Algorithm 1 is O(TA + TB + |S ∪ Z|2 ), where TA and TB are the time complexities of the approximation algorithms A and B, respectively. Furthermore, we have: (a) 1CSCP(r, R, S, X , Z) has a feasible solution if and only if X has a single-cover Y ⊆ Z such that all nodes in S ∪Y are in the same connected component of RCG(R, S, Z). (b) When 1CSCP(r, R, S, X , Z) has a feasible solution, Algorithm 1 guarantees computing a connected single-cover YA,B of X , which uses no more than (α + 3.5β) times the number of RNs required in an optimal solution Yopt for 1CSCP(r, R, S, X , Z), where α is the approximation ratio of A and β is the approximation ratio of B. 2 P ROOF. Line 1 of Algorithm 1 computes a single-cover YA of X in TA time. Line 2 constructs the RCG, which requires O(|S ∪ Z|2 ) time. Lines 3-5 can be accomplished using depth first search, which also requires O(|S ∪ Z|2 ) time. Line 6 assigns weights to the edges in the RCG, which requires O(|S ∪ Z|2 ) time. Line 7 requires TB time. This proves the time complexity of the algorithm. The correctness of (a) follows from the definition of 1CSCP. Therefore we only need to prove the correctness of (b). We first show that Algorithm 1 is guaranteed to find a connected single-cover when the instance of 1CSCP(r, R, S, X , Z) is feasible. The feasibility of the instance ensures that Line 1 of Algorithm 1 is guaranteed to find a single-cover YA of X . Let Yopt be an optimal solution to 1CSCP(r, R, S, X , Z). Then Yopt is a single-cover of X , and the relay communication graph RCG(R, S, Yopt ) is connected. Since YA is a minimal singlecover of X , each node y ∈ YA is within distance r of a sensor node x ∈ X . Note that node x must be within distance r of a node y ∈ Yopt , as Yopt is a single-cover of X . Therefore y is either in Yopt or within distance R ≥ 2r of a node (y for example) in Yopt . Therefore RCG(R, S, Yopt ∪ YA ) is also connected. Hence Line 7 of Algorithm 1 is guaranteed to find a Steiner tree, thereby producing a connected single-cover YA,B for the instance of 1CSCP(r, R, S, X , Z). In the following, we will prove that |YA,B | is no more than
(α + 3.5β)|Yopt |. Let Ymsc be a minimum single-cover for the given instance. Then we must have |Ymsc | ≤ |Yopt |, as every connected single-cover is a single-cover. Since A has approximation ratio α, we have |YA | ≤ α|Ymsc | ≤ α|Yopt |.
(3.1)
Note that RCG(R, S, YA ∪ Yopt ) is connected. Let Topt be a minimum spanning tree of RCG(R, S, YA ∪ Yopt ). Let Tmin be a minimum Steiner tree in RCG(R, S, Z) which connects all nodes in S ∪ YA . Then we have wYA (Tmin ) ≤ wYA (Topt ).
(3.2)
Since B is a β-approximation algorithm, we have wYA (TB ) ≤ βwYA (Tmin ) ≤ βwYA (Topt ).
(3.3)
We can write wYA (Topt ) as wYA (Topt ) = w1YA (Topt ) + w2YA (Topt ), where w1YA (Topt ) is the sum of the weights of the weight-1 edges in Topt and w2YA (Topt ) is the sum of the weights of the weight-2 edges in Topt . Let u be any node in Yopt \ YA . It follows from the geometric property of minimum spanning trees that u is incident with at most 5 weight-1 edges in Topt . Suppose, to the contrary, u is incident with 6 weight-1 edges (u, v1 ), (u, v2 ), (u, v3 ), (u, v4 ), (u, v5 ), and (u, v6 ). Then we have {v1 , v2 , . . . , v6 } ⊆ S ∪ YA , and d(u, vi ) ≤ R, 1 ≤ i ≤ 6. Therefore there exist i and j (1 ≤ i < j ≤ 6) such that d(vi , vj ) ≤ R. Replacing the weight-1 edge (u, vi ) by the weight-0 edge (vi , vj ) leads to a spanning tree of RCG(R, S, YA ∪ Yopt ) with a smaller weight than the weight of Topt , which is a contradiction. Therefore we have (3.4) w1YA (Topt ) ≤ 5|Yopt |. Since Topt is a tree, it has at most |Yopt | − 1 weight-2 edges. Hence (3.5) w2YA (Topt ) ≤ 2(|Yopt | − 1). Therefore wYA (Topt ) ≤ 5|Yopt | + 2(|Yopt | − 1) = 7|Yopt | − 2.
(3.6)
Combining (3.3), (3.6), and Lemma 2.1, we have 1 YA β w (TB ) ≤ wYA (Topt ) ≤ 3.5β|Yopt |. 2 2 Combining (3.1) and (3.7), we have |YB | ≤
|YA | + |YB | ≤ (α + 3.5β)|Yopt |.
(3.7)
(3.8)
This proves the theorem. We further remark on the implication of this framework. With different choices of algorithms A and B, we will end up with approximation algorithms for 1CSCP with different approximation ratios and running times. For example, we can use the PTAS for GDHP of [22] as algorithm A, and the (1+ ln23 )-approximation algorithm of [27] as algorithm B. This combination leads to the following. Corollary 3.1: The 1CSCP(r, R, S, X , Z) problem has a polynomial time (6.43 + )-approximation scheme. 2
Though the above combination gives a good approximation ratio, the time complexity is high [5], [27]. In our numerical study, we used the 22-approximation algorithm for GDHP in [5] as algorithm A, and the 2-approximation algorithm of [16] as algorithm B. This combination leads to the following. Corollary 3.2: The 1CSCP(r, R, S, X , Z) problem has a 29-approximation algorithm with a running time of 2 O(|Z|2 |X |4 + |S ∪ Z|2 log |S ∪ Z|). We note that an alternative approach to this problem could be to apply algorithm B on a subgraph of HCG(r, R, S, X , Z) without the SN-SN edges to obtain a connected single cover. However, since the number of SNs that are connected to a RN cannot be bounded by a constant, proving a meaningful approximation ratio for this approach is an open problem. Our best solution is the (6.43 + )-approximation scheme. 4. Two-Connected Double-Cover In this section, we study the 2-connected double-cover problem. As in Section 3, let S be a set of base stations (BSs), X be a set of sensor nodes (SNs), and Z be a set of candidate locations where relay nodes (RNs) can be placed. Let r > 0 be the communication range of the SNs, and R ≥ 2r be the communication range of the RNs. A. Problem Definitions and Notations Definition 4.1: [DCP] A subset Y ⊆ Z is called a doublecover (denoted by DC) of X if every SN is within distance r of at least two nodes in S ∪ Y. The size of the corresponding DC is |Y|. A DC is called a minimum double-cover (denoted by MDC) of X if it has the minimum size among all DCs of X . The double-cover relay node placement problem (denoted by DCP) for (r, R, S, X , Z) seeks an MDC of X .2 Definition 4.2: [2CDCP] A subset Y ⊆ Z is called a 2connected double-cover (denoted by 2CDC) of X if for every sensor node x ∈ X there exists a pair of node-disjoint paths from x to two base stations in the hybrid communication graph HCG(r, R, S, X , Y). The size of Y is |Y|. Y is called a minimum 2-connected double-cover (denoted by M2CDC) of X if it is a 2CDC of X , and has the minimum size among all 2CDCs of X . The 2-connected double-cover relay node placement problem for (r, R, S, X , Z) (denoted by 2CDCP) seeks an M2CDC of X . 2 The problem we are studying here is closely related to the {0, 1, 2}-survivable network design problem (SNDP) defined in the following. Definition 4.3: [{0, 1, 2}-SNDP] Let G = (V, E) be an undirected graph with nonnegative weights on all edges e ∈ E. For each pair of vertices u, v ∈ V , there is a connectivity requirement c(u, v) ∈ {0, 1, 2}. The {0, 1, 2}-survivable network design problem (SNDP) seeks a minimum weight subgraph H of G such that for any two vertices u, v ∈ V , H contains at least c(u, v) node-disjoint paths between u and v. 2 B. A Framework of Efficient Approximation Algorithms We present a general framework to solve 2CDCP which uses an approximation algorithm A for the DCP problem,
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and an approximation algorithm B for the {0, 1, 2}-SNDP problem. The framework of polynomial time approximation algorithms has an approximation ratio of α + 5β, where α is the approximation ratio of A, and β is the approximation ratio of B. Our framework for 2CDCP is presented as Algorithm 2. Algorithm 2 Approximation for 2CDCP(r, R, S, X , Z) Input: Set S of BSs, set X of SNs, set Z of candidate locations for RNs, sensor node communication range r > 0, and relay node communication range R ≥ 2r > 0. An approximation algorithm A for DCP, and an approximation algorithm B for the {0, 1, 2}-SNDP problem. Output: A connected double-cover given by YA ∪ YB ⊆ Z. 1: Apply algorithm A to obtain a double-cover YA of X . Without loss of generality, we assume that YA is minimal, meaning that none of its proper subsets is a double-cover of X . 2: Construct the relay communication graph G = RCG(R, S, Z). 3: if the nodes in S ∪ YA are not in the same 2-connected component of RCG(R, S, Z) then 4: Stop. The given instance does not have a 2-connected double-cover. 5: end if 6: Assign edge weight in G as in (2.1), with Y substituted by YA , i.e., for an edge (x, y) in G, we have that wYA (x, y) = |(Z \ YA ) ∩ {x, y}|. For a node x and a node y ((x, y) does not have to be an edge), assign the connectivity requirement to be c(x, y) = 2 − |(Z \ YA ) ∩ {x, y}|. 7: Apply algorithm B to compute a low weight 2-connected subgraph HB of RCG(R, S, Z), spanning the node set S ∪ YA . Let the set of added relay nodes in the solution to B be YB . 8: Output YA,B = YA ∪ YB . The major steps of our scheme are as follows. First, we apply the approximation algorithm A to obtain a double cover YA ⊆ Z of X . This is accomplished in Line 1 of the algorithm. In Line 2, we construct the relay communication graph G = RCG(R, S, Z). The given instance of the problem has a feasible solution if and only if all the nodes in S ∪YA are in the same 2-connected component of G. Lines 3 to 5 check whether there exists a 2-connected component containing all the nodes in S ∪ YA . If there is no such 2-connected component, the algorithm stops. In Line 6, we assign non-negative edge weights to the edges in G according to (2.1). Then, in Line 7, we apply algorithm B to compute a low cost 2-connected subgraph of G spanning all the nodes in S ∪ YA . Finally, Line 8 outputs the locations where to place the RNs. Theorem 4.1: The worst case running time of Algorithm 2 is O(TA + TB + |S ∪ Z|2 ), where TA and TB are the time complexities of the approximation algorithms A and B, respectively. Furthermore, we have:
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(a) 2CDCP(r, R, S, X , Z) has a feasible solution if and only if X has a double-cover Y ⊆ Z such that all nodes in S∪Y are in the same 2-connected component of RCG(R, S, Z). (b) When 2CDCP(r, R, S, X , Z) has a feasible solution, Algorithm 2 guarantees computing a 2-connected doublecover of YA,B of X , which uses no more than (α + 5β) times the number of RNs required in an optimal solution Yopt to 2CDCP(r, R, S, X , Z), where α and β are the approximation ratios of A and B, respectively. 2 Before proceeding to prove Theorem 4.1, we need the following lemma. Lemma 4.1: Let Y be a 2-connected double-cover of X . Let Yc be a subset of Y such that Yc is a double-cover of X . Let Yn = Y \ Yc . Assign edge weight in RCG(R, S, Y) such that the weight of edge (x, y) is wYc (x, y) = |Yn ∩ {x, y}|. Let HYc ,Yn be a minimum 2-connected subgraph connecting all nodes in S ∪ Y. Then, for each node u ∈ Yn , u is connected 2 to at most 5 nodes in Yc , and at most one node in S. P ROOF. For the proof, we refer the reader to the Lemma 4.1 in [21], while noting the following differences. The HCG there corresponds to the RCG here, with X there corresponding to Y here, and Y there corresponding to Yopt \ Y here. P ROOF OF T HEOREM 4.1: The proof of the running time and Part (a) are similar to that in the proof of Theorem 3.1. Here we concentrate on the approximation ratio. Recall that the set of RNs required by A is YA and the set of extra RNs required by the SNDP based algorithm B is YB . Then all the RNs required are given by YA,B = YA ∪ YB . Let Yopt be an optimal solution to the 2CDCP problem. Let cov be an optimal solution to the DCP problem. It is easy to Yopt cov | ≤ |Yopt |. Let Hmin be an optimal solution to see that |Yopt the {0, 1, 2}-SNDP instance that connects all nodes in S ∪ YA , and Hopt be a solution that uses the optimal number of RNs required to solve 2CDCP. Since Hopt is a feasible solution to {0, 1, 2}-SNDP, and B is a β-approximation algorithm for {0, 1, 2}-SNDP, we have wYA (HB ) ≤ β · wYA (Hmin ) ≤ β · wYA (Hopt ).
(4.1)
We need to find an upper bound on wYA (Hopt ) using a function of |Yopt |. Let w2YA (Hopt ) denote the total weights of the weight-2 edges in Hopt , and let w1YA (Hopt ) denote the total weights of the weight-1 edges in Hopt . We have wYA (Hopt ) = w2YA (Hopt ) + w1YA (Hopt ).
(4.2)
Let each RN in Hopt be incident with at most Δ(Hopt ) weight1 edges in Hopt , we have w1YA (Hopt ) ≤ |Yopt | · Δ(Hopt ).
(4.3)
Applying Lemma 2.3 to each of the connected components of the subgraph of Hopt induced by all the 2-weight edges, we have w2YA (Hopt ) ≤ 2 · (2|Yopt | − 1).
(4.4)
It follows from Lemma 2.1 that 1 YA β w (HB ) ≤ wYA (Hopt )(4.5) |YB | = sYA (HB ) ≤ 2 2 β (4 + Δ(Hopt ))|Yopt |. (4.6) ≤ 2 It follows from Lemma 4.1 that β |YB | ≤ (4 + 6)|Yopt | ≤ 5β|Yopt |. (4.7) 2 cov | ≤ α|Yopt |. Therefore Note that we also have |YA | ≤ α|Yopt we have |YA ∪ YB | ≤ (α + 5β)|Yopt |. We can use different combinations of A and B for different purposes. Two examples are summarized in the following two corollaries. Corollary 4.1: The 2CDCP problem has a O(ln n)approximation algorithm with a polynomial running time. 2 P ROOF : This is achieved by choosing A as the O(ln n)approximation algorithm by Rajagopalan et al. [24] and B as the 3-approximation algorithm of Ravi et al. for the {0, 1, 2}SNDP problem [25], [26]. In the study above, we have made no assumption on the number of relay nodes that a sensor node can be connected to. However, in many practical scenarios, the number of relay nodes that a sensor node can be connected to is usually bounded by a constant f . In this case, a simple modification to the constant frequency based algorithm in [29] will result in a constant approximation algorithm as stated in the following corollary. Corollary 4.2: If the number of relay nodes that can be connected to a sensor node is bounded by a constant f , then Algorithm 2 guarantees a feasible solution in polynomial time, with a constant approximation ratio f + 5β, where β is the approximation ratio of B. 2 If f = 4 and β = 2 [8] we obtain a 14-approximation algorithm for solving the 2CDCP problem with bounded relay degree. This approximation ratio is much better than the 20 + approximation ratio even for the unconstrained case [32]. The experimental results illustrate that our algorithms perform pretty close to the optimal. 5. An Efficiently Computable Lower Bound No algorithms have been proposed for the problem of two-tiered constrained relay node placement in the literature. Further, obtaining optimal solution is difficult. Hence for lack of a more suitable comparison, we compare the results from our algorithms with that obtained from a Linear Program (LP) formulation of 1CSCP and 2CDCP. We note that an LP can be solved in polynomial time. Also, the solution to the LP version of the problem denoted by YLP , is a lower bound of the solution to the corresponding Integer Linear Program (ILP) version. That is, YLP ≤ Yopt . The suitability of the use of LP for comparison comes from two facts. Firstly, if A is an α-approximation algorithm for a problem, then YA ≤ α · YLP , consequently, YA ≤ α·Yopt . Secondly, solving the LP may take much less time in comparison to solving the corresponding ILP, which may have an exponential running time.
We formulate the LP for the 1CSCP problem and the 2CDCP problems as a multi-commodity flow problem, with a flow originating from each SN u (the source) and all flows destined to a fictitious sink d, which is connected to the BSs. The value of the flow originating from the source is designated as f. For the 1CSCP problem, f = 1 and for the 2CDCP problem f = 2.
min
|Z|
zj
(5.1)
j=1
subject to, Requirements Constraints: zj − rij ≥ 0, for each j ∈ Z, over all i ∈ X , (5.2) Source Constraints: fiji − fjii = f , (5.3) i∈X (i,j)∈HCG
i∈X (j,i)∈HCG
Sink Constraints: fjdi −
fdji = f ,
(5.4)
Flow satisfaction constraints: fijk − rkj = 0, ∀k ∈ X ,
(5.5)
i∈X ,j∈S
i∈X ,j∈S
j∈Z (i,j)∈HCG
Flow conservation constraints: fijk − fjik = 0, j∈Z,k∈X (i,j)∈HCG
j∈Z,k∈X (j,i)∈HCG
fijk
j∈S,k∈X (i,j)∈HCG∪(d,j)
−
(5.6)
fjik = 0,
(5.7)
j∈S,k∈X (j,i)∈HCG∪(j,d)
Bounds: fijk = 0, i, j ∈ S. 0 ≤ rij , fijk ≤ 1, ∀ rij , remaining fijk s.
(5.8) (5.9)
The variable zj , j = 1, . . . , |Z|, is the flow variable, which represents the maximum flow of any commodity through the RN j for the flow problem to be satisfied. It may be thought of as the fraction of the RN j that is used in solving the relay node placement problem. The variable rij , i = 1, . . . , |X |, j = 1, . . . , |Z|, is the requirement variable. It represents the amount of flow from an SN i required to pass through a RN j. To ensure that the coverage and/or connectivity requirements are satisfied, the flow variables for each RN have to satisfy the requirements constraints given by Equation (5.2). The variable fijk represents the flow of commodity k from node i to j. Equations (5.3) and (5.4) represent the source and the sink constraints respectively. The source flow constraints are easy to understand. The sink constraints are specified with respect to the fictitious sink which is connected to the BSs only. The flow satisfaction constraints, given by Equation (5.5), enforce that the amount of flow of commodity k flowing into an RN j from nodes adjacent to it is exactly equal to the amount of
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flow of k that is required to flow through j. That is, j does not create nor buffer any flow. Equations (5.6) and (5.7) represent the flow conservation at the RNs and the BSs respectively. Corresponding to each BS there is an additional edge connecting it to d for the flows to reach the destination. Equation (5.8) represents the bounds for the flow variables corresponding to the flow between the BSs. These flow variables are set to zero to ensure that each BS can only appear in at most one path from a source to d. 6. Experimental Results To evaluate the effectiveness of the frameworks, we implemented Algorithm 1 and Algorithm 2. For Algorithm 1, we used the 22-approximation algorithm in [5] as A and the 2-approximation algorithm in [16] as B. Our choices were motivated by their faster running times. The numerical results show that the solutions produced are still within twice the optimal for all of our test cases. We use ACS to denote this implementation of Algorithm 1. For Algorithm 2, we used the 3-approximation algorithm in [25] as B. For the unbounded case, we used the O(ln n)-approximation algorithm in [24] as A. We use ALD to denote this combination. For the bounded case, we used a variant of the constant approximation algorithm in [29] as A. We use ACD to denote this combination. To ensure rigorous comparison, for each instance of 1CSCP and 2CDCP problems that are solved by our algorithms, we also solve the corresponding LP instance (for lower bound) and evaluate the proximity of our solutions to the optimal. We denote the solutions for 1CSCP and 2CDCP as LPS and LPD respectively. The tests were run on a 2.4 GHz Linux PC. As in [15], [21] and [32], the SNs X were randomly distributed in a square deployment region. There are two base stations deployed randomly in the region. The value of the transmission ranges were, r = 15 and R = 30. The deployment region consists of K × K small squares each of side 10, with Z, the candidate locations, being the (K + 1)2 grid points. We have also performed simulation studies where the candidate locations are randomly chosen outside of the forbidden regions, and found that the observed performances of our algorithms are independent of the choices of the test cases. Therefore, we only present results where the candidate locations are on a grid. We report two separate deployments of the SNs: the case where the density of the SNs in the region increases and the case where the density is constant. We define the density as the ratio of the number of SNs in the region to the area of the region. For the increasing density case, we chose a constant region of size 100 × 100 sq. units. For the constant density case, with the increase in the number of SNs the size of the region increased. For the increasing density case, the number of candidate locations for RNs was 121. The number of SNs was varied from 20 to 120, in steps of 20. For each setting the results were averaged over 10 test cases. Fig. 2(a) shows the running time of ACS, ALD, and ACD. Since the running time of the algorithms is based on the number of edges (|E|) and the number of vertices (|V |), the X-axis represents the average of |V | + |E|,
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while the Y axis denotes the running time in seconds. The dashed blue line (diamond markers) shows the running time of ACS. The solid red line (star markers) and the dash-dot black line (square markers) show the running time of ALD and ACD respectively. In case of ALD and ACD, the running time decreases at first and then increases. This is because, for the initial data point (20 SNs), the RNs chosen as a cover are not connected, and connecting them with additional RNs increases the running time. However, with an increase in the number of SNs the number of RNs needed for a cover becomes adequate to form a connected network by themselves. With increasing number of SNs the running time increases as it takes more time to find a cover. This also holds true for ACS. However, since ACS is much faster than the other algorithms, it is not obvious in the figure. The bigger dip and faster increase of the running time of ALD and ACD (for 2CDCP case) in comparison to that of ACS (for 1CSCP case) is because of the use of the higher complexity SNDP algorithm, and the higher running time needed to get a double cover respectively. Figs. 2(b) and 2(c) show the average number of RNs required by ACS and LPS, for solving the 1CSCP problem, and by ALD, ACD, and LPD, for solving the 2CDCP problem respectively. Both approximation algorithms perform pretty well in comparison to the lower bound. The number of RNs obtained is never more than twice the cost of the LP solution. Thus the results from our approximation algorithms are never more than twice the optimal value. This indicates that our approximation algorithms perform very well. We note that for 2CDCP, both the approximation algorithms have the same results despite the disparities in their approximation ratios. This is because, the pathological cases where the O(ln n) algorithm performs badly do not occur in the square grid with random SNs deployment. We will study this in the future. For the case of constant density, we studied two sub-cases: density d1 = 0.005 and density d2 = 0.01. For each density value, we used 7 different numbers of SNs. The deployment region sizes were chosen to be 40×40, . . . , 100×100, with the number of SNs ranging from 8 to 50 for d1 , and 16 to 100 for d2 . The result of each configuration was averaged over 10 test cases. Fig. 3(a) shows the running times of ACS, ALD, and ACD for both the densities. For both densities, the algorithms for the 1CSCP problem have low running time, while those for the 2CDCP problem have higher running time, which is expected. The running times for the ALD and ACD, increase with increasing |E|+|V | as more RNs are required for coverage and connectivity, thus taking more time. The running time for d2 = 0.01 is less than that of d1 = 0.005, as with increasing density, there is greater likelihood of the RNs used for coverage to be connected as such. Figs. 3(b) and 3(c) show the number of RNs required by various algorithms. Again, our algorithms perform remarkably well, never requiring more than twice the number of RNs in the optimal solution. 7. Conclusions We have studied the constrained relay node placement in a two-tiered wireless sensor network to meet connectivity
15 10 5 0 1400
1600
1800
2000 2200 |V|+|E|
2400
2600
20 15 10 5 0
2800
(a) running times of ACS, ALD, ACD Fig. 2.
15 10
25
ACS(d=0.005) ACS(d=0.01) ALD(d=0.005) ACD(d=0.005) ALD(d=0.01) ACD(d=0.01)
500
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(a) running times of ACS, ALD, ACD Fig. 3.
40
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LPD ALD ACD
20
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(c) #RNs used by ALD, ACD, LPD
Results with increasing density: 100 × 100 deployment region; |Z| = 121; |X | = 20, 40, 60, 80, 100, 120.
5 0 0
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(b) #RNs used by ACS, LPS
Number of RNs used
Running time (in secs)
20
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LPS ACS
Number of RNs used
25 ACS ALD ACD
20 15
40
LPS(d=0.005) ACS(d=0.005) LPS(d=0.01) ACS(d=0.01)
Number of RNs used
20
Number of RNs used
Running time (in secs)
25
10 5 0
40x40 50x50 60x60 70x70 80x80 90x90100x100 Field Size(units x units)
(b) #RNs used by ACS, LPS
30 20
LPD(d=0.005) ALD(d=0.005) ACD(d=0.005) LPD(d=0.01) ALD(d=0.01) ACD(d=0.01)
10 0
40x40 50x50 60x60 70x70 80x80 90x90100x100 Field Size(units x units)
(c) #RNs used by ALD, ACD, LPD
Results with constant density: seven different deployment regions, from 40 × 40 to 100 × 100; two density values, d1 = 0.005 and d2 = 0.01.
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