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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 Clustering Wireless Sensor Nodes Using Caterpillar Graph Dr H B Walikar Ishwar Baidari Professor Asst. Professor Dept of Computer Science Dept of Computer Science Karnatak University Karnatak University Dharwad, India Dharwad, India e-mail: walikarhb@yahoo.com e-mail: ishwarbaidari@gmail.com Abstract— When sensors nodes are deployed and organized in the related maintenance cost or energy efficient clusters to form of clusters, they could use either single hop or multi hop mode minimize energy consumption suitable for sensor nodes with of communication to send their data to their respective cluster heads. energy constraints or for load balancing to distribute the We implemented algorithm on class of graph called caterpillar workload of a network. The fig1 illustrates the concept of graphs. We also propose, deploying and clustering wireless sensor clusters. nodes in the form of caterpillar graphs. Here our objective is to find Connected Dominating Set (CDS) of a caterpillar graphs. Key words: clustering, cluster head, connected dominating set, caterpillar graphs, tree. 1.Introduction. Clustering analysis is desirable in nearly any field of study where it is beneficial to group data into similar sets depending on one’s objective in analyzing a set of data one might define similarity between elements differently and thus a clustering process could be optimized to provide numerous way of grouping a set of elements. In order to create any sort of clustering algorithm and determine its effectiveness it is Fig1 necessary to find some way to quantity similarity between Wireless sensor networks are networks of wireless nodes that elements. When sensor nodes are organized in clusters they are deployed over an area for the purpose of monitoring could use either single hop or multi hop mode of certain phenomena of interest. The nodes perform certain communication to send their data to their respective cluster measurements process the measured data and transmit the heads. The sensor nodes are randomly and uniformly processed data to a base station over a wireless channels. The distributed[22] over the region and the nodes are organized in base station collects data from all the nodes and analyzes this clusters to take advantage of possible data aggregation at the data to draw conclusion about the activity in the area of cluster head nodes. There are two types of nodes; cluster head interest. These networks are different from the traditional nodes and sensor nodes. The cluster head nodes act as the wireless ad hoc networks. However, when nodes are organized fusion points within the network. During each data gathering in clusters and when they use multi hop communication to cycle the sensor nodes send their sensed data to the closest reach the cluster head the nodes closer to a cluster head have a cluster head node which perform data aggregation. Then the higher load of relaying packets as compared to other nodes. cluster head directly transmits the aggregated data to a base However is most sensor networks nodes are static station. The sensor nodes have simple functionality, since they consequently the nodes closer to the cluster head get perform sensing and relatively short-range communication. overburdened constantly. The cluster heads themselves have However the cluster head nodes are more complex, since they the extra burden of performing long rang transmissions to the coordinate MAC and routing within their cluster perform data distant base station. fusion and perform long range transmissions to the remote We consider a region to be covered by sensor nodes. base station. The overall system design problem involves The number of sensor nodes is determined by the application determining the optimum number of cluster head nodes the requirements. Usually each sensor node has a sensing radius optimum node of communication within a cluster (Single hop and it is required that the sensor nodes provide coverage of the or Multi hop). region with a high probability. The sensing radius of each Various clustering algorithms have been proposed to node depends on the phenomenon that is being sensed as well organize sensor nodes in a wireless sensor network into as the sensing hardware of the node. Thus in general the clusters. [1][2][3][4][5][6]. Each aim to meet certain needs of required number of sensor nodes is dictated by the application the system. This could provide a system having low clustering and hence we assume it to be a constant. 50 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 Connected Dominating Set is a subset of nodes in networks itself. The closed neighborhood N[v] of v also includes v, that and it divides node set into two parts. Nodes inside CDS form is, N[v] = N(v) ∪ {v}. With these definitions extended to a connected sub-network. Which is in charge for routing subsets of V, the open neighborhood of S ⊆ V is N(S) = process. Every node out of CDS should have at least one ∪ v ∈ S N(v)-S, and the closed neighborhood of S is N[S] = adjacent node in this CDS. Thus node outside CDS will N(S) ∪ S. The degree δ (v) of v is the size of its open always acquire routing path through this neighbor whenever neighborhood: δ (v) =|N (v)|. The maximum degree of G is its destination is. The performance of a CDS for coverage routing and broadcasting etc., depends on the size of the CDS. ∆ = maxv ∈ V δ (v). For the purposes of analysis of The smaller the size is the less the routing time will be and the overhead, we assume that a local broadcast takes O( ∆ ) time smaller the routing table size is. Thus much work is devoted to (which is true if the MAC layer can schedule local broadcasts reducing the size of CDS. However computing a minimum reliably). Given a subgraph T of G, the T –degree of v is δ T CDS is NP-hard. (v), the number of v’s neighbors that are in T . The maximum In such model there are usually two main types of nodes i.e. degree of T is denoted ∆ (T ). The diameter diam(G) of G is the cluster head which is in charge of the cluster and cluster the maximum number of edges contained in any simple path members which join a cluster and are controlled by the cluster between two nodes in V . The diameter of a subgraph T of G is head. In this paper we consider single – hop (one – hop) denoted diam(T ). cluster using caterpillar graphs. All the members node is such We use an approximation to a minimum connected a cluster are within the range of the cluster head but not dominating set (MCDS). A subset S ⊆ V is a dominating set necessarily within range of each other In this single – hop if N[S] = V. Let G(C) be the subgraph induced by C ⊆ V . C cluster any member node is at most within two hops away from any other member node via the cluster head. This defines is a connected dominating set if, in addition to N[C] = V, G(C) the clusters diameter. The cluster head is in charge of cluster is connected. Since finding an MCDS is an NP-complete maintenance such as resource allocation to member and the problem that is also hard to approximate we present a acceptance of member in to the cluster. Member node can join distributed greedy MCDS approximation algorithm that is a cluster if the cluster head accepts their join request.An similar to the algorithm in. The MCDS nodes are incidentally efficient clustering must elect suitable cluster heads to achieve also the interior nodes of a maximum leaf spanning tree. the clustering schemes main objectives and the cluster heads We use the interior of this tree as the back bone. Thus, each must also accept suitable nodes to become members of their node v in V has a unique dominator in C, denoted dom(v).The clusters. set 〈 v, dom(v) 〉 ∀ v ∈ V is a maximum leaf spanning tree. In this paper we proposed a clustering wireless The nodes of C comprise the interior of this spanning tree, and sensors network using caterpillar graph. Here we using the edges of this spanning tree between nodes in C are called existing liner time algorithm for finding domination number of back bone edges tree, here our objective is to use this algorithm to find connected dominating set (CDS) of caterpillar graph. Wireless sensor networks can be deployed for many 2. Preliminaries application unlike wired networks or cellular networks no Graph terminology physically backbone infrastructure is installed in wireless We use an undirected graph G = (V, E),[20] with m edges and sensor networks. A communication session is achieved either n nodes, to represent a snapshot of the ad hoc network. Each through a single hop if the communication parties are close node in V represents a mobile host, and each edge in E enough or through relating by intermediate nodes otherwise. signifies that two hosts are within transmission range of each The topology of such wireless ad hoc network can be modeled other. The topology of G is the set of edges and nodes. Hence, as a unit disk graph[ ] a geometric graph in which there is an when we say a node movement changes the topology, we mean edge between two nodes if and only if there distance is at one a change in the network that results in a change in either V or unit as show in fig 2. E. Specifically, an edge deletion occurs when two hosts lose communication with each other, and an edge insertion occurs when two hosts move into range of each other. A node deletion in isolation occurs when a host turns off its power, and a node insertion in isolation occurs when a host turns on its power. By “in isolation” we mean that no other change has occurred in the network. Because a node insertion or deletion affects multiple edges, we process these changes to V as multiple changes to E. Finally, the most general node movement models the movement of a host from one part of the network to another; hence, a node movement is a combination of a node deletion from one part of G and a node insertion in another part of G. The open neighborhood N (v) of node v represents all hosts within transmission range of v except for v Fig2 51 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 Although a wireless sensor network has no physical backbone Lemma 1([16]). If Pk is a chord less path with k vertices, then infrastructure a virtual back bone can be formed by nodes in a m(Pk) = m(Pk-2)+m(Pk-3), k ≥ 4 with m(p1)=1, m(P2)=2 and connected dominating set of the corresponding unit disk graph m(P3)=2, [6][7][8]. Such a virtual backbone plays a very important role Two vertices are twins in a graph if they have the same in routing, broadcasting, and connectivity managements in neighborhood. wireless sensor networks Jou et al [17] proved the following properties. Lemma 2. If H and y are twins in a graph G then m(G) = m(G- 3. Related Work x) = m(G-y) Efficient distributed algorithms for constructing CDS in WSN Lemma 3. If H is an induced subgraph of G, then m(H) < m(g) were studied in [9,6,10,11,12,13,14,15] Wu li of [ 9 ] proposed Lemma 4. ([18]) For any two disjoint graphs U and z m (U their localized connected dominating set method using a ∪ z) = m ( ∪ ). m (z) marking process where a node is marked true if it has two Let V(Pk)={ V1,V2,------ Vk} For each vi E v (Pk) unconnected neighbors It is shown that the set of marked ,H(vi) is the set of its pendent vertices and |H(vi) = ni, I = nodes forms a CDS. In [11] Dai et further extend the pruning 1,2,……k H(vi) is an independent set but it is not maximal in rule to k- hop neighborhood in order to achieve better results. C(Pk). If same vertex of H(vi) belongs to a mis then every Alzobic et a [10,13] proposed a approximation method to vertex of H(vi) must belongs to it otherwise it is not maximal. construct a minimum CDS with performance ratio of 8. In As two vertices of H(vi) are twins in C(Pk), we can construct [15], chen et al also proposed a localized algorithm to build a them in to a single vertex, called hi, that represents the whole CDS for topology maintence where a node become a set H(vi), i= 1,…………..k. Let Gk be the construction group of dominator when two of its neighbors cannot reach each other C(Pk) otherwise that is also a caterpillar graph with at most either directly via one or two dominator. In [14] a distributed one pendent vertex at each vi the contraction graph of a algorithm on CDS was proposed whose performance ratio is complete caterpillar graph is also complete. 172. In [15] another localized algorithm contains three steps. Step 1 constructs a forest in which each tree is rooted at a node 5. Linear Algorithm with the minimum ID among its 1 – hpo away neighbors step Efficient liner algorithm for the domination number of a tree 2 collects neighboring trees. designed by E Cockayne,S Goodman and S Hedetniemi Cock The research work on selecting minimum CDS has never been et al [19] proposed their “a liner algorithm for finding the interrupted work on selecting a minimum CDS has never been domination number of a tree”, Partitioning the tree in to three interrupted because of its dramatic contributions to wireless subsets V1,V2,V3 where V1 consists of free vertices, V2 consists networks. It has been proved that selection of minimum CDS of bound vertices and V3 consists of required vertices. They in a general graph is an NP-hard problem. have coined the one more term called mixed domination(md) set in G is set of vertices M which Contain all required vertices 4. Caterpillar Graphs i.e. V3 ⊆ M and which dominate all bound vertices i.e. every A caterpillar graph C (Pk)[22] is a tree having a chordless path vertex v ∈ v2 is either in M or is adjacent to at least one vertex Pk, called the backbone that contains at least one end point of in M. Free vertices need not be dominated by M but may be every edge. Edges connecting the leaves with the backbone are included in M in order to dominate bound vertices. The mixed called hairs. In a complete caterpillar graph, each vertex of its dominating set in G such a set is called an md set of G. Here backbone has a nonempty set of hairs denoted by CC(Pk) a we are applying this algorithm on caterpillar graphs. Once we complete caterpillar graph with backbone Pk. traced the algorithm on caterpillar graph we get a chord less path which is itself a connected dominating set. Let us consider the algorithm. Let the vertices of network G be partitioned in to three subsets, V1, V2, V3, where V1 consists of free vertices, V2 consists of bound vertices and V3 consist required vertices. A Fig3 mixed dominating set in G is set of vertices M which contains We can use a simple graph G= (V, E) to represent an wireless all required vertices, i.e. V3 ⊆ M and which dominates all sensor network, where V represents a set of wireless mobile bound vertices, i.e. every vertex v ∈ V2 either in M or is hosts and E represents a set of edges. An edge between host adjacent to at least one vertex in M. Free vertices need not be pairs {v, u} indicates that both hosts v and u are within their dominated by M but may be included in M in order to wireless transmitter ranges. To simplify our discussions, we dominate bound vertices. The mixed domination number assume all mobile hosts are homogeneous i.e. their wireless md(G) is the minimum order of a mixed dominating set in G; transmitter ranges are the same. In other word, if there is an such a set is called an md- set of G. edge e = {v, u} in E, it indicates u is within v’s range and v is The construction and correctness of the next within u’s range. Thus the corresponding graph will be an algorithm is based on the following theorem. undirected graph. The graph in fig3 represents the Theorem[19] Let T be a tree having free, bound and required corresponding wireless sensor network vertices V1, V2, and V3 respectively. Let v be an end vertex of T which is adjacent to vertex u. Then 52 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 10, No. 4, April 2012 (i) If v ∈ V1, then md(T) = md(T-v); Step 8.Set DOMSET ← DOMSET U {v} (ii) If v ∈ V2 and T’ is the tree which results from Step 9.If u is bound then label u as free; deleting v and relabeling u as “required”, then Step 10.Set G ← G –v. md(T) = md(T’); od (iii) If v ∈ V3 and u ∈ V3, then Step11. [Process last vertex] If the last vertex v is not free md (T) =1+md(T-v); then DOMSET ← DOMSET U {v} (iv) If v ∈ V3 and u ∉ V3 and if T’ is the tree which results from deleting v and relabeling u as Grouping sensor nodes into clusters in order to achieve the “free”, then md (T) =1 + md (T’). network scalability objective. Every cluster would have a leader often referred to as cluster head(CH). Recently a Proof.(i) If v ∈ V1, then since v is free it need not be number of clustering algorithm have been specifically dominated in mixed dominating set of T. Thus any mixed designed for WSN. These proposed clustering techniques dominating set D of T-v is also a mixed dominating set of widely vary depending on the node deployment. In this T2 i.e. md (T) ≤ md (T-v). Conversely, let D be an md set algorithm we need to deploy sensors in the form of caterpillar of T and let the free end vertex v be a adjacent to vertex u. graphs and tracing the algorithm on caterpillar graphs finally it Now if v ∉ D, the D is also a mixed dominating set of T- left with path which is itself a connected dominating set and v. On the other hand if v ∈ D then D-{v} U {u} is mixed all the nodes in the connected dominating sets are cluster heads (CH).A CH may also be just one of the sensors or a dominating set of T-v Thus in either case. node that is richer in resources. The cluster membership may Md (T-v) < |D| = | D-{v} U {u}| = md (T). be fixed or variable. In addition to supporting network (ii) the proof of this case, where the end vertex v is bound, scalability. Clustering has numerous advantages It can localize is virtually identical to case (i) i.e v must be dominated in the route set up within the cluster and thus reduce the size of any md- set of T. In this case we can show that if D is an the routing table store at the individual node. md set of T then so is D’ = D-{v} U {u}, i.e. there is an md –set of T which contains u. But this md –set D’ must 6. Conclusion also be an md-set of T-v, in which u is considered a We studied the problem of the design of wireless sensor required vertex. networks from the point of view of the caterpillar graphs (iii) The proof of this case is obvious and is omitted. retaining the connected dominating set (CDS) of caterpillar (iv) Let D be an md – set of T’ in which v is deleted and u graphs. The CDS is itself a cluster head of the sensor nodes. And we utilize the exiting linear time algorithm for finding is labeled ‘free’. Then clearly, D U {v} is a mixed domination number of a tree. Applying this algorithm dominating set of T, i.e. md (t) < 1+md (T’). systematically on caterpillar graphs we get a connected Conversely let D be an md- set of T. Since v is required, v dominating set. ∈ D. We need to consider two cases. If u is also in D, REFERENCES then D-{v} is mixed dominating set of T’ similarly if u ∉ D then, since u is free in T’, D-v is also mixed [1] S Guha and S Kuller, “Approximation algorithms for connected dominating set in T’. In either case md (T’) < md (T) – 1 dominating sets’, Proc.of 4th Annual Europen Symposium on Algoriths,(1996). and with the previous inequality we conclude, md (T) = [2] J. Wu and H.L. 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[15] X.Cheng,M.Ding,D.Du and X .Jia,Virtuval Backbone Construction in 1.Dr.H.B.Walikar is currently a Vice –Chancellor of Karnatak Multi Hop Ad Hoc Wireless Network, Wireless Communications and Mobile University,Dharwad and received M.A. in Mathematics from the same Computing, 6(2):183-190,2006 Univesity and he was the first person to inroduce the connected [16] Z.Furedi, The Number of Maximal Independent Sets in Connected domination theory.And don tremonds work in the theory of domination. Graph, Journal of Graph Theory 11(1987)463-470. [17] J.Liu, Maximal Independent Sets in Bipartite Graphs, Journal of Graph 2.Ishwar Baidari currently working as a Ass.professor in Dept. of Computer Theory 17(1993)495-507. Science, Karnatak University,Dharwad obtained his degree I n MCA from Karnatak University,Dharwad. 54 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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