Optim Lett DOI 10.1007/s11590-009-0168-z ORIGINAL PAPER
Minimum average routing path clustering problem in multi-hop 2-D underwater sensor networks Donghyun Kim · Wei Wang · Ling Ding · Jihwan Lim · Heekuck Oh · Weili Wu
Received: 26 May 2009 / Accepted: 9 December 2009 © Springer-Verlag 2009
Abstract In this paper, we introduce a new clustering problem in underwater sensor networks, namely minimum average routing path clustering problem (MARPCP). To deal with the high complexity of MARPCP, we relax it to a special case of minimum weight dominating set problem (MWDSP). We show an existing algorithm for MWDSP can produce an approximate solution for MARPCP. Also, we design a constant factor approximation algorithm for MARPCP, which is much faster than the first method. Keywords Approximation algorithm · Graph theory · Wireless network clustering · Underwater sensor networks
D. Kim (B) · L. Ding · W. Wu Department of Computer Science, University of Texas at Dallas, Richardson, TX 75083, USA e-mail:
[email protected] L. Ding e-mail:
[email protected] W. Wu e-mail:
[email protected] W. Wang Department of Mathematics, Xi’an Jiaotong University, 710049 Xi’an, People’s Republic of China e-mail:
[email protected] J. Lim · H. Oh Department of Computer Science and Engineering, Hanyang University, Ansan, Gyeonggi 425-791, Republic of Korea e-mail:
[email protected] H. Oh e-mail:
[email protected]
123
D. Kim et al.
1 Introduction For a long time, monitoring oceanic environment has been a dangerous and expensive task. Recently, underwater sensor networks (USNs) are actively studied as a promising solution to relieve the difficulties in the old ways to observe underwater world. USNs are also recognized as an emerging network technology for a wide range of underwater applications for both civilian and military purposes [1]. Unlike conventional terrestrial sensor networks (TSNs), a USN is a network of static sensor nodes and mobile machines such as unmanned underwater vehicles (UUVs) or autonomous underwater vehicles (AUVs). In some applications of USNs such as ocean mineral exploration, a USN is deployed on the floor of the deep ocean. Then, the topology of such USN can be abstracted using a 2-D graph such as unit disk graph (UDG). Interestingly, in the 2-D USN, there is at least one special node called a underwater sink (UW-Sink) which is vertically connected to out-of-water sinks such as buoys or ships. The out-of-water sinks are connected with each other or onshore stations using radio or satellite signal based communication technology. Therefore, we can think the UW-Sink as a gateway between the in-water network (the USN) and the out-of-water network. A UW-Sink has powerful hardware with more energy and is even more expensive than a normal sensor node, whose cost is normally over thousands of dollar. Therefore, such USN is expected to be sparser than TSNs, in which each sensor costs only several dollars. Underwater sensor nodes use acoustic channels for communications instead of radio signals. This is because the radio signal bands used for TSNs cannot travel far inside sea water. While very low frequency radio signals can go further, they consume significant amount of energy and each sensor node has to have a very big antenna to use those signals, which is not practical. Still, the energy consumption of the acoustic channels used for underwater communication is five order of magnitude more than radio signals for TSNs. Replacing the battery of a deployed underwater sensor node is almost impossible or too expensive, and current technology does not allow underwater sensor nodes to harvest recyclable energy sufficiently. Therefore, energy-efficiency is still an inevitably important issue of USNs. In wireless networks having mobility like USNs, clustering guarantees basic level system performance, such as delay and throughput, and helps the networks to be more energy-efficient by reducing collision [2]. In Wireless Sensor Networks (WSNs) such as TSNs or USNs, each clusterhead can fuse messages from its members and send one summary message to a UW-Sink. Since the members of the same cluster are geographically close, the reports generated by them are similar. Therefore, the local data fusion by each clusterhead can reduce the number of messages in the networks significantly while this method does not scarify the accuracy of data seriously. Note that if a data is merged multiple times, its precision can be lowered [3]. In underwater sensor networks, the communication between a clusterhead and a UW-sink can be done via a single-hop path or a multi-hop path. In the first case, each clusterhead transmits the message to a UW-Sink directly [4]. However, this approach is energy consuming especially when a USN is deployed over a vast area since acoustic channels consume energy exponentially proportional to a message transmission range. Therefore, it is desirable to have a maximum transmission range and send a message
123
Minimum average routing path clustering problem
over a multi-hop shortest path. To find such path, we may be able to use any existing pro-active or re-active routing scheme. Unfortunately, in mobile wireless networks like USNs, both schemes incur burdensome overheads [4]. A clustering-based routing method is a routing scheme over a clustered network, where any routing path contains only clusterheads [5]. It is known that this routing scheme is energy-efficient in mobile wireless networks since it suffers less from collision and reduces the amount of both control messages and routing related information. In addition, this routing scheme is resilient in mobile networks because a routing path should be fixed only when a clusterhead on the path is disappeared. The contributions of this paper can be summarized as follows. First, we propose a new clustering problem in USNs, namely minimum average routing path clustering problem (MARPCP). The goal of this problem is that, given a USN, to find a set of clusterheads for the USN such that the expected hop distance from a node to its nearest UW-Sink in the clustering-based routing scheme is minimized. Second, to overcome the high complexity of MARPCP, we relax this problem to a special form of minimum weighted dominating set problem (MWDSP-R). Third, we show the existing approximation algorithms for MWDSP are also approximation algorithms for MARPCP. Fourth, motivated by the fact that the existing approximation algorithms for MWDSP are very slow, we design a simple algorithm based on maximal independent set (MIS) computation and prove that this is actually a constant factor approximation algorithm for MARPCP. The rest of this paper is organized as follows. In Sect. 2, we introduce related work. In Sect. 3, we formally define MARPCP and relax it to MWDSP-R. Section 4 shows an existing approximation algorithm for MWDSP is also an approximation algorithm for MARPCP. A new constant factor approximation algorithm for MARPCP and corresponding analysis are presented in Sect. 5. At last, we conclude this paper and introduce some future works in Sect. 6.
2 Related work In WSNs such as TSNs or USNs, energy is a very scarce resource. It is well-known that clustering WSNs can bring various benefits such as reducing collision and ensuring minimum system performance [2]. In USNs, each clusterhead can be also used as a local data fusion point, which merges messages from its cluster members and forwards their summary to its nearest UW-Sink. This method reduces the number of messages in the networks and hence helps the WSNs to save more energy. Usually, a clustering scheme can be incorporated with a routing scheme, so called a clustering-based routing scheme [5]. In the scheme, messages move from the head of a cluster to the head of adjacent cluster following the shortest path between the clusterheads. When a message needs to move through multiple clusters, it repeats visiting the clusterhead of each cluster on its way to the destination. In this way, two node can exchange messages through their clusterheads. In this scheme, only clusterheads maintain routing-related information. In addition, routing paths consist of only clusterheads, whose number is significantly smaller than the size of whole network. This kind of clustering-based routing scheme is highly beneficial to WSNs since the
123
D. Kim et al.
overhead caused by message routing can be greatly reduced and it is resilient to node mobility. So far, there have been only a small number of work related to the clustering problem in USNs [4,6]. In [6], they modified the low-energy adaptive clustering hierarchy (LEACH) for WSNs in [7] and proposed LEACH-L. Originally, the LEACH is proposed to make the energy consumption of a network more fair. However, this algorithm updates its global state regularly, which causes a huge overhead and thus this algorithm can be energy consuming. The authors in [6] assumed USNs are less dynamic than normal WSNs such as TSNs and proposed the LEACH-L, which updates its state locally, and reduced the overhead of LEACH. In [4], a clustering scheme is proposed in the context of routing scheme to extend the lifetime of USNs. In their scheme, each clusterhead send a message to it nearest UW-Sink directly. Due to the inefficiency of acoustic communication channels, this scheme cannot be adopted to USNs well which prefer multi-hop communications.
3 Notation, assumptions, and problem definition Here are some notations frequently used in the rest of this paper. G = (V, E) is an UDG. For s, d ∈ V, Hopdist(s, d) is the number of hops between s and d over a shortest path between them in G, which is denoted by Minpath(s, d). Eucdist(s, d) is the Euclidean distance between two nodes s and d. We denote the clusterhead of a node s by CH(s). At last, N (s) is the set of nodes which are adjacent to s in G. Later, we will introduce additional notations if necessary. In this paper, we assume all nodes in an USN have the same maximum transmission range. We also assume following data-centric routing model which is very common in WSNs. Each clusterhead (called initial clusterhead) is used as a local data fusion point, and collects reports from all of its clustermembers, merges them, and sends one summary message to its nearest UW-Sink. It is well-known that since all the members in a cluster are geographically close and data generated by them are very similar, the local data fusion can reduce the amount of redundant message significantly without scarifying the accuracy of data. However, in this paper, we assume USNs need to provide accurate data and, thus we will not allow intermediate nodes on the path from each initial clusterhead to a UW-Sink to further merge the pre-fused messages. We assume a scheduling scheme (i.e. TDMA1 ) is used to handle all inter-cluster or intra-cluster collisions and interferences. In addition, we assume following popular clustering-based routing scheme which is also used in many previous literatures such as [5]. That is, in casual clustered multi-hop wireless networks having a hierarchy, Minpath(s, d) is not used as an actual routing path between two nodes s, d ∈ V . Instead, Routepath(s, d) is used, which is defined as follow. Imagine an edge weighted graph G = (V , E ) such that each vertex v ∈ V corresponds to a cluster Cv ⊆ V and there is an edge (v, u) ∈ E if Cv and Cu are adjacent in G. We put a weight w(v, u) = Hopdist(CH(v), CH(u)) on each edge 1 Time-Division Multiple Access.
123
Minimum average routing path clustering problem
(v, u) ∈ E , where CH(v) is the clusterhead of Cv . Then, the number of hops that a message travels from s to d by the clustering-based routing scheme is
Routedist(s, d) = Hopdist(s, CH(s)) +
w( p)
p∈MWP(CH(s),CH(d))
+ Hopdist(d, CH(d)), where MWP(CH(s), CH(d)) is the minimum weight path p between the corresponding nodes of CH(s) and CH(d) in G and w( p) is the average edge weight of the path p. Note that Hopdist(s, CH(s)) = 1 for any s and CH(s), and Routepath(s, d) is the path with length Routedist (s, d) in G. It is well-known that the energy consumption for sending a message in multi-hop wireless networks is closely related to the hop length of a path over which the message travels [8]. Therefore, we use hop distance as a metric instead of Euclidean distance in our cost function. Now, we formally define the cost function in our problem. Let {s1 , . . . , sn } and {U1 , . . . , Um } be the set of sensor nodes and UW-Sinks, respectively. Also, suppose CH1 , . . . , CHc are selected as clusterheads among n sensor nodes. Then, given that every node has the same probability to generate a report and thus each of them generated one report at the same time, the expected hop distance from a node to its nearest UW-Sink in the clustering-based routing scheme is ⎛ 1⎝ n
1≤ j≤c
⎞ min Routedist(CH j , Uk ) + n − c⎠
1≤k≤m
⎛
=
1⎝ n+ n
1≤ j≤c
⎞ min (Routedist(CH j , Uk ) − 1)⎠ .
1≤k≤m
Note that n − c is the total number of messages sent from each clustermember to its clusterhead. Once they are arrived at their clusterheads, they will be merged and sent to their nearest UW-Sinks. Since n is a constant for any fixed input graph, we are going to ignore it in our cost function. Then, our problem can be defined as follow. Definition 1 (Minimum Average Routing Path Clustering Problem) Given a set of sensor nodes {s1 , . . . , sn } and UW-Sinks {U1 , . . . , Um } on the Euclidean plane, minimum average routing path clustering problem (MARPCP) is to find a set of clusterheads such that each sensor node is adjacent to at least one clusterhead, and the expected hop distance from a node to its nearest UW-Sink in the clustering-based routing scheme is minimized. In other words, we want to find a set of clusterheads {CH1 , . . . , CHc } to minimize min (Routedist(CH j , Uk ) − 1) such that ∀i, ∃ j, Hopdist(si , CH j ) ≤ 1. 1≤ j≤c
1≤k≤m
Note that Routedist(CH j , Uk ) cannot be determined before clusterheads are selected. Therefore, clustering a given network such that this objective function of
123
D. Kim et al.
MARPCP is minimized seems difficult. The proof of NP-completeness of MARPCP can be found in [9]. Definition 2 (Minimum Weighted Dominating Set Problem) Given a weighted graph G = (V, E) with each vertex v having a non-negative weight w(v), the minimum weighted dominating set problem (MWDSP) is to find a dominating set (DS) D of G such that the total weight v∈D w(v) is minimized. Suppose G is an UDG induced by {s1 , . . . , sn } and {U1 , . . . , Um }. To overcome the complexity of MARPCP, we relax the requirement of MARPCP by transforming a MARPCP instance in G to a MWDSP instance in G as follows: For each node s j , we add a vertex v j to G . We also compute min1≤i≤m (Hopdist(s j , Ui ) − 1) and assign this as the weight w(v j ) of v j in G . In this way, we can create a new graph G with special non-negative node weights from the original input graph G formed by the set of sensor nodes. Note that the MWDSP instance defined over G is a special subset of generic MWDSP. In this paper, we will call the MWDSP in G as MWDSP-R, where R represents “relaxed”. The proof of NP-completeness of MWDSP-R can be found in [9]. It is easy to see that a feasible solution of MWDSP-R is also a feasible solution of MARPCP since both of them are DSs for the same set of nodes. In the following section, we will show an optimal solution of MWDSP-R is an approximate solution of corresponding MARPCP. 4 A simple approach to approximate MARPCP In this section, we study the characteristics of cost functions in MARPCP and MWDSP-R. Based on this result, we show an existing approximation algorithm for MWDSP is an approximation algorithm for MARPCP. Lemma 1 For any clusterhead CH and a UW-Sink U, Routedist(CH, U ) − 1 ≤ 3 (Hopdist(CH, U ) − 1). Proof Suppose Minpath(CH, U ) = CH → v1 → v2 → · · · → vl → U with l = Hopdist(CH, U ) − 1. Now, let’s consider following routing path: basically a message follows Minpath(CH, U ). However, when a message arrives at each vi ∈ Minpath(CH, U ), it visits CHvi (vi ’s clusterhead) and come back to vi and travel to vi+1 and this is repeated until it reaches at U . At the worst case, this path (Fig. 1) will look like CH → v1 → CH1 → v1 → v2 → CH2 → v2 → · · · → vl → CHl → vl → U. Observe that this is a feasible routing path for our clustering-based routing scheme. Furthermore, it is easy to see that its length is at most 3l +1. Since Routepath(CH, U ) is the shortest path among feasible paths, Routedist(CH, U ) − 1 ≤ Routedist(CH, U ) ≤ 3(Hopdist(CH, U ) − 1) has to be true. Corollary 1 Let A and B be a MARPCP and its corresponding (relaxed) MWDSP-R instances, respectively. Denote the cost function of feasible solutions for MARPCP
123
Minimum average routing path clustering problem Fig. 1 In this figure, Minpath(CH, U ) = v1 → v2 → v3 . Then, CH → v1 → CH1 → v1 → v2 → CH2 → v2 → v3 → CH3 → v3 → U is a feasible routing path for our clustering-based routing scheme
CH 3
CH1 v2
CH
v1
v3
U
CH 2
and MWDSP-R by w A (·) and w B (·), respectively. Then, for any feasible solution (DS) F, w A (F) ≤ 3w B (F). Proof By Lemma 1, w A (v, U ) − 1 ≤ 3(w B (v, U ) − 1) for every v ∈ F. Note that F is a DS, which is a set of clusterheads. Therefore, we have (w A (v, U ) − 1) ≤ 3 (w B (v, U ) − 1) = 3w B (F). w A (F) = ∀v∈F
∀v∈F
Theorem 1 An α-approximation algorithm for MWDSP-R is a 3α-approximation algorithm for MARPCP. Proof Denote the optimal solutions of MARPCP and MWDSP-R by OPT A and OPT B , respectively. Then, we have w B (OPT B ) ≤ w B (OPT A ). Furthermore, we have w B (OPT A ) ≤ w A (OPT A ) since in MWDSP-R, a clusterhead can send a message to its nearest UW-Sink through the shortest path between them, but this is not true in MARPCP. Therefore, we have w B (OPT B ) ≤ w A (OPT A ). Now, suppose we have an α-approximation algorithm for MWDSP-R and F is a feasible solution generated by this algorithm. Then, we have w B (F) ≤ αw B (OPT B ). By Corollary 1, we have w A (F) ≤ 3w B (F). Therefore, w A (F) ≤ 3w B (F) ≤ 3αw B (OPT B ) ≤ 3αw A (OPT A ).
Algorithm 1 FAST-MARPCP (S = {s1 , . . . , sn }) 1: 2: 3: 4: 5:
Color all nodes in S white. while there is a white node v ∈ S do Color v black and its white neighbors gray. end while Return the set of black nodes as clusterheads.
In [10], the authors introduced a 72-approximation algorithm for MWDSP in UDGs. Also, Dai and Yu proposed a (5+)-approximation algorithm [11]. Combining those with Theorem 1, we have 216 and (15 + )-approximation algorithms for MARPCP. However, the time complexities of both algorithms are very high since they use enumeration techniques and therefore not practical for dynamic USNs. On the other hand, we can solve MWDSP-R using a simple greedy algorithm in [12]. However, the performance ratio of this algorithm is O(ln n), which is not a constant. Therefore, in the following section, we propose a simple and fast heuristic algorithm for MARPCP with a constant performance ratio.
123
D. Kim et al.
5 A faster approximation algorithm for MARPCP with a constant performance ratio In this section, we show an output of Algorithm 1, an MIS of an UDG induced by S, is a constant factor approximate solution of MARPCP. Lemma 2 ([13]) Let M is an MIS included in N (v) of a node v. Then, |M| ≤ 5. Theorem 2 Algorithm 1 is a 30-approximation algorithm for MARPCP. Proof In this proof, we show an MIS produced by Algorithm 1 is a 10-approximation of MWDSP-R. Then, by Theorem 1, this is a 30-approximation algorithm for MARPCP. Consider each subgraph G j formed by sensor nodes with a UW-Sink U j such that a sensor node si is in G j if and only if Hopdist(si , G j ) = min Hopdist(si , Uk ). 1≤k≤m
Let OPT B1 , OPT B2 , . . . , OPT Bm be the restrictions of the optimal solution OPT B to each big partitions G 1 , G 2 , . . . , G m . Also, let MIS1 , MIS2 , . . . , MISm be the restrictions of an MIS to each big partitions G 1 , G 2 , . . . , G m . Let T j be the shortest path tree of G j ∪ {U j } rooted at U j , and l j be the height of T j . Denote the set of nodes in level i in T j by L (i, j) (1 ≤ i ≤ l j ). Here, the level of of the OPT B j the root U j is 0. Let OPT B j ∩ L (i, j) and MIS j ∩ L (i, j) be the restriction and MIS j to the level i nodes of tree T j , respectively. Note that D = mj=1 MIS j is dominated by OPT B . Let M(i, j,k) be the subset of nodes in MISk which are dominated by OPT B j ∩ m l j m L (i, j) . Then, D = . As m j=1 k=1 M(i, j,k) k=1 M(i, j,k) is dominated by i=1 m OPT B j ∩ L (i, j) , it follows from Lemma 2 that k=1 |M(i, j,k) | ≤ 5|OPT B j ∩ L (i, j) |. By Algorithm 1, each of M(i, j,k) must lie in either level i −1, i, or i +1 of its corresponding shortest path tree. Thus, w B (M(i, j,k) ) = (i − 1)|M(i, j,k) ∩ L (i−1, j) | + i|M(i, j,k) ∩ L (i, j) | + (i + 1)|M(i, j,k) ∩ L (i+1, j) | ≤ 2i(|M(i, j,k) ∩ L (i−1, j) | + |M(i, j,k) ∩ L (i, j) | + |M(i, j,k) ∩ L (i+1, j) |) ≤ 2i|M(i, j,k) | since i + 1 ≤ 2i. It follows that m k=1
w B (M(i, j,k) ) ≤ 2i
m
|M(i, j,k) |
k=1
≤ 10i|OPT B j ∩ L (i, j) | = 10w B (OPT B j ∩ L (i, j) ).
123
Minimum average routing path clustering problem Fig. 2 In this figure, if all other nodes are selected as MIS nodes, w B (MIS) = 8. On the other hand w B (OPT B ) = 1. Therefore, the example shows that the performance ratio of our algorithm on MWDSP-R is at least 8
Summing up for i = 1, 2, . . . , l j and j = 1, 2, . . . , m, we obtain, w B (D) =
lj m m
w B (M(i, j,k) )
j=1 i=1 k=1
≤ 10
lj m
w B (OPT B j ∩ L (i, j) )
j=1 i=1
= 10w B (OPT B ). This result shows that Algorithm 1 is a 10-approximation for MWDSP-R. Thus, the theorem follows. Note that our analysis on the relaxed MWDSP-R is almost tight since we cannot have better than 8 (see Fig. 2). 6 Conclusion and future work In this paper, we have studied a new clustering problem for an energy-efficient clustering-based routing scheme in multi-hop USNs requiring high data precision. We formally defined this as MARPCP and proposed two ways to get constant factor approximate solutions. In this paper, we particularly focused on USNs. However, we believe our problem model and algorithm can be applied to the other kind of wireless networks. As a future work, we are interested in designing an algorithm for MARPCP with smaller approximation ratio since that of our solution is not small. Also, studying the same question using d-hop clustering might be interesting. At last, we are also interested in an algorithm for this problem, which is still efficient in the presence of frequent clusterhead computation. Acknowledgments This work is supported in part by under grant IIS-0513669, CCF-0750992, and CCF0621829. This work is also supported in part by the Ministry of Knowledge Economy, Korea, under the HNRC (Home Network Research Center) and ITRC (Information Technology Research Center) support program supervised by the Institute of Information Technology Assessment.
References 1. Akyildiz, I.F., Pompili, D., Melodia, T.: Underwater acoustic sensor networks: research challenges. J. Ad Hoc Netw. 3(3), 257–279 (2005)
123
D. Kim et al. 2. Yu, J.Y., Chong, P.H.J.: A survey of clustering schemes for mobile ad hoc networks. IEEE Commun. Surv. Tutor. 7(1), 32–48 (2005) 3. Govindan, R., Hellerstein, J.M., Hong, W., Madden, S., Franklin, M., Shenker, S.: The sensor network as a database. USC Technical Report No. 02-771, September 2002 4. Wang, P., Li, C., Zheng, J.: Distributed minimum-cost clustering protocol for underwater sensor networks (UWSNs). IEEE International Conference on Communications (ICC 2007), pp. 3510–3515, June 2007 5. Belding-Royer, E.M.: Multi-level hierarchies for scalable ad hoc routing. Wireless Netw. 9(5), 461–478 (2003) 6. Li, X., Fang, S.-L., Zhang, Y.-C.: The study on clustering algorithm of the underwater acoustic sensor networks. In: The 14th International Conference on Mechatronics and Machine Vision in Practice (M2VIP 2007), Dec 2007 7. Heinzelman, W.R., Chandrakasan, A., Balakrishnan, H.: Energy-efficient communication protocol for wireless microsensor networks. In: The 33rd Annual Hawaii International Conference on System Sciences, Jan 2000 8. Akyildiz, I.F., Su, W., Sankarasubramaniam, Y., Cayirci, E.: Wireless sensor networks: a survey. Comput. Netw. Int. J. Comput. Telecommun. Netw. 38(4), 393–422 (2002) 9. Wang, W., Kim, D., Willson, J., Thuraisingham, B., Wu, W.: A better approximation for minimum average routing path clustering problem in Multi-hop 2-D underwater sensor networks. Discrete Math. Algorithms Appl. 1(2) June (2009) 10. Ambuhl, C., Erlebach, T., Mihalak, M., Nunkesser, M.: Constant-factor approximation for minimumweight (connected) dominating sets in unit disk graphs. In: International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, (APPROX-RANDOM 2006), pp. 3–14 (2006) 11. Dai, D., Yu, C.: A 5 + -approximation algorithm for minimum weighted dominating set in unit disk graph. Theor. Comput. Sci. 410, 756–765 (2009) 12. Chvatal, V.: A greedy heuristic for the set covering problem. Math. Methods Oper. Res. 4, 233–235 (1979) 13. Wu, W., Du, H., Jia, X., Li, Y., Huang, S.C.-H.: Minimum connected dominating sets and maximal independent sets in unit disk graphs. Theor. Comput. Sci. 352, 1–7 (2006)
123
