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Clustering Wireless Sensor Nodes Using Caterpillar Graph

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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
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                                                                53                                  http://sites.google.com/site/ijcsis/
                                                                                                    ISSN 1947-5500
                                                              (IJCSIS) International Journal of Computer Science and Information Security,
                                                              Vol. 10, No. 4, April 2012




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                                                                                                           AUTHORS PROFILE
Networks,Chicago,june2004.
[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.




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