Constructing Minimum Connected Dominating Set: Algorithmic approach

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1 Constructng Mnmum Connected Domnatng Set: Algorthmc approach G.N. Puroht and Usha Sharma Centre for Mathematcal Scences, Banasthal Unversty, Rajasthan Abstract: Connected Domnatng Set s popularly used for constructng vrtual backbones for broadcastng operaton n WSNs. UD Graph s the most sutable model for a wreless sensor network. In ths paper we provde an algorthm to fnd MCDS n UD Graph. It s based on the computaton of convex hulls of sensor nodes or vertces. Constructng a vrtual backbone n WSNs s an mportant ssue because t reduces unnecessary message transmsson or floodng n the network. It helps n reducng nterference and energy consumpton because a lmted number of sensors are engaged n message transmsson and thus t helps n mprovng the Qualty of Servce (QoS) n the network. 1. Introducton Wreless sensor network has wde varety of applcatons such as battle feld survellance, target trackng, securty, envronmental control, habtat montorng, source localzaton, fre detecton, ol and gas pumpng etc. A wreless sensor network s composed of a set of battery powered sensors. These sensors can communcate wth one another through wreless lnks f they are wthn ther transmsson range, otherwse they can communcate va other sensors between them. Unt Dsk Graph [1] s an mportant class of graphs. A wreless sensor network can be modeled as a UD Graph [2] snce the transmsson range of sensors s based on Eucldean dstance. In ths modelng sensors are denoted as vertces. The sensng coverage area of a sensor s represented by a unt dsk centered at correspondng vertex. The connectvty between two sensor nodes s determned f the frst sensor s wthn the sensng coverage range of the second sensor. Thus there s an edge between two vertces u and v ff d(u,v) 1, where d(u,v) s the Eucldean dstance between u and v. In ths way UD Graph s most sutable model for a wreless sensor network. Domnatng set n a graph G = (V, E) s a subset D of V such that every node u V s n D or adjacent to some node v D. A domnatng set D s called Connected Domnatng Set (CDS) f t s a nduced connected subgraph of G. A Mnmum Connected Domnatng Set (MCDS) s a connected domnatng set wth smallest possble cardnalty DOI : /jgraphoc

2 among all the CDSs of G. Connected Domnatng Sets [4] are popularly used for constructng vrtual backbones for broadcastng operaton n WSNs. Establshng a vrtual backbone n WSNs s an mportant ssue because t reduces unnecessary message transmsson or floodng n the network. It helps n reducng nterference and energy consumpton because a lmted number of sensors are engaged n message transmsson and thus t helps n mprovng the Qualty of Servce (QoS) n the network. Vrtual backbone s bascally a subset of sensor nodes whch can transmt and receve messages throughout the network. CDS s one of the earlest approaches to construct a vrtual backbone n WSNs [4]. To fnd the CDS n UD Graph s a well studed problem. The MCDS problem was shown. The MCDS remans NP hard [1] for UD graph. A CDS s also useful for locaton based routng. In locaton based routng messages are forwarded based on the geographcal co-ordnates of the hosts and topologcal connectvty. In ths paper we provde an algorthm to fnd MCDS n UD Graph. It s based on the computaton of convex hulls of sensor nodes or vertces. Ths paper s organzed as follows. In secton 2, related work s gven. Auxlary defnton and notaton are gven n secton 3. Secton 4 contans the man algorthm of the paper and ts verfcaton through an example. In the Secton 5, dscusson and concluson s gven. Last Secton 6 contans references. 2. Related Work Guha and Khuller [6] studed the MCDS problem and showed that ths problem s NP hard n an arbtrary undrected graph. Later t was shown n [1] that computng an MCDS for a Unt Dsk (UD) Graph s also a NP hard problem. Alzoutr et al. [3] proposed the frst dstrbuted algorthm guaranteeng a constant approxmaton factor for CDS constructon based on MIS n UD Graph. A dstrbuted algorthm to construct small szed connected domnatng set for UD Graph was provded n [4]. It s based on the computaton of convex hull of sensor nodes whch are consdered as vertces. An analyss of the sze of MIS and the sze of an MCDS also has been provded. It s also shown that ths algorthm produces an optmal CDS f the graph s a tree. In the case of grd the approxmaton factor s 2. A technque s presented that produces a CDS wth the sze at most 38* MCDS, where MCDS s the sze of a mnmum CDS. In [7] a dstrbuted algorthm was proposed that also utlzed MIS for CDS constructon n UD Graphs. Ths technque requres large amount of message exchanges and transmsson for the constructon of spannng tree for constructng CDS. An algorthm s proposed to fnd MCDS usng domnatng set n UD Graphs n [2]. Ths algorthm s mplemented n three phases. In frst phase, domnatng sets are found. In second phase, connectors are dentfed, connected through Stener tree. In thrd phase, the CDS obtaned a MCDS. Network needs to adapt to the contnuous topologcal changes due to 60

3 deactvaton of a node due to exhauston of battery. These changes are taken care by a local repar algorthm that reconstructs the MCDS..e. power aware MCDS usng only neghborhood nformaton. 3. Auxlary Defnton In order to develop the algorthm, we state some defnton and ntroduce some termnology relevant to the paper. 3() Domnatng Set Domnatng Set for a graph G = (V,E) s a subset D of the Vertex Set V such that each vertex u V s ether n D or adjacent to some vertex v n D. The elements of domnatng set are called domnators. Examples of domnatng set n a graph G are gven below: Fgure 1: {1, 3}, {2, 3, 5} and {1, 2, 3, 4} Fgure 2: {4, 6}, {1, 5, 7} and {4, 5, 6} are Domnatng Sets are Domnatng Sets 3() Connected Domnatng Set A Connected Domnatng Set (CDS) of a graph G = (V,E) s a set of vertces wth two propertes: 1. D s a domnatng set n G. 2. D nduces a connected subgraph of G. In Fg.1, {2, 3, 5} and {1, 2, 3, 4} are Connected Domnatng Sets. Smlarly n Fg. 2, {4, 5, 6} s a Connected Domnatng Set. 3() Mnmum Connected Domnatng Set A mnmum Connected Domnatng Set (MCDS) s a connected domnatng set wth smallest possble cardnalty among all the CDSs of G. As n Fgs. 1 and 2, {2, 3, 5} and {4, 5, 6} are Mnmum Connected Domnatng Sets respectvely. 3(v) Independent Set Independent Set of a graph G s a subset of the set of vertces such that no two vertces are adjacent n the subset. For example n Fg.1 {1, 3}, {1, 4, 5}, {2, 4, 5} are ndependent sets. 3(v) Maxmal Independent Set Maxmal ndependent set (MIS) s an ndependent set, whch s not a subset of any other ndependent set..e. t s a set S such that every edge of the graph has atleast one end pont not n S and every vertex not n S has atleast one neghbor n S. An MIS s also a domnatng set. Sx dfferent Maxmal Independent Set of followng cubc graph are {1,6}, {2,5}, {3, 8},{4, 7}, {1, 5, 7} and {4, 6, 8, 2}. 61

4 Fgure 3: MIS n Cubc Graph 3(v) Convex hull the convex hull for a set of ponts X n real vector space s the mnmum convex set contanng X. t s also called convex envelop and denoted by CH(X). It s represented by a sequence of the vertces of the lne segment formng the boundary of the convex polygon. As n the followng example convex hull of the set {1, 2, 3, 4, 5, 6, 7} of ponts s shown. Fgure 4: CH ({1, 2, 3, 4, 5, 6, 7}) 3(v) Unt Dsk Graph A graph G s a Unt Dsk graph f there s an assgnment of unt dsks centered at ts vertces such two vertces are adjacent f and only f one vertex s wthn the unt dsk centered at the other vertex. 3(v) Neghborhood of a vertex Neghborhood of a vertex u n a graph G = (V,E) s a set of vertces whch are adjacent to u n G. It s denoted by N(u) or N[u]. If neghborhood does not nclude u tself, then t s called open neghborhood of u and denoted by N(u). As n fgure 1, N(1) s {2, 6}, N(2) s {1, 3, 6}, N(3) s {2, 4, 5} and so on. If neghborhood ncludes u tself, then t s called closed neghborhood of u and denoted by N[u]. For example n Fg. 2, N[1] s {1, 2, 4} and N[2] s {1, 2, 3, 4}and so on. 4. Algorthm Now we descrbe an algorthm to fnd mnmum connected domnatng set from a connected domnatng set. Ths CDS s found by algorthm descrbed n [4]. We have to do followng steps: 62

5 Step 1: Select a mnmum degree vertex u from the CDS. Step 2: Calculate CH(N[u]). Step 3: Calculate CH(N[]) N(u). Step 4: Check f CH (N[u]) s contaned n U ]) where N(u). Step 5: If step 2 returns true then remove vertex u and go to step 1. Step 6: Otherwse do not remove vertex u and go to step 1. Step 7: Algorthm termnates when all the nodes n C are consdered and the node remans n C construct the MCDS. The above algorthm can be understood wth the help of followng examples: Fgure 5: G = (V, E) and CDS = {c, e, d, f, g} Connected Domnatng Set (CDS) n the above graph found by the algorthm descrbed n [4] s {c, d, e, f, g} say t C. Now we apply our gven algorthm to fnd Mnmum Connected Domnatng set (MCDS). Step1: Select the mnmum degree vertex n C.e. c. Step2: Calculate CH(N[c]).e. convex hull ecd. Fgure 6: CH(N[c]) = ecd Step3: Calculate CH(N[]) N(c) = {e, b, d}. 63

CH(N[e]) = acf CH(N[b]) = gcd CH(N[d]) = cfj Step

Step1. Step 6: Select the next mnmum degree vertex.

also remove by above process and go to step 1.

6 CH(N[e]) = acf CH(N[b]) = gcd CH(N[d]) = cfj Step 4: CH (N[c]) s contaned n U Fgure 7: CH(N[]) ]) where N(c) = {e, b, d}. Fgure 8: U ]) Fgure 9: c]) UCH ( N[ ]) Step 5: Step 4 returns true. Therefore we remove the vertex c from CDS and go to Step1. Step 6: Select the next mnmum degree vertex.e. g and proceed as prevously and we fnd that g wll also remove by above process and go to step 1. Step 7: Select the next mnmum degree vertex.e. d. Step 8: Calculate CH(N[d]).e. convex hull cfj. Fgure 14: CH(N[d]) = cfj 64

Fgure 16: U ]) Fgure 17: d]) UCH ( N[ ]) Step 11: Step 10 returns false. Therefore we do not remove the vertex d from CDS and go to Step1.

7 Step 9: Calculate CH(N[]) N(d) = {b, f, c, j}. CH(N[b]) = gcd CH(N[f]) = ehd CH(N[c]) = ecd CH(N[j]) =dj Fgure 15: CH(N[]) Step 10: CH (N[d]) s not contaned n U ]) where N(d) = {b, f, c, j}. Fgure 16: U ]) Fgure 17: d]) UCH ( N[ ]) Step 11: Step 10 returns false. Therefore we do not remove the vertex d from CDS and go to Step1. Step 12: Select the next mnmum degree vertces e and f, and perform the steps 6 to 11. Lkewse d these two vertces are not deleted from CDS. Fnally CDS left wth the vertces e, d and f. Thus we get the MCDS = {e, d, f}. 5. Concluson UD graph s the most sutable model for WSNs. Vrtual backbone n a WSNs s a subset of sensor nodes whch can transmt and receve the messages through out the network. 65

8 Constructon of a CDS n UD graph s a common way to generate vrtual backbone n WSNs. We ntroduce an algorthm to fnd Mnmum Connected Domnatng Set (MCDS) for UD Graph based on computaton of convex hull of sensor nodes. Ths s an mportant ssue due to many causes such as t precludes unnecessary message transmsson of floodng n the network. It reduces nterference between nodes and energy consumpton because only the nodes of CDS are engage n message transmsson. These all above reasons help to mprove Qualty of Servce n the network. We also calculated the complexty of our algorthm. It turns out to be O(n 2 logn). Though ths complexty s large, however t s better than exstng algorthms whch are NP hard ([1] and [6]). So our algorthm appears to be superor as compare to avalable algorthms. 6. References [1] B. N. Clark, C. J. Colbourn and D. S. Johnson, Unt dsk graphs, Dscrete Math, Vol. 86, pages , 1990 [2] G. N. Puroht, S. Verma and U. Sharma, Powers of a Graph and Assocated Graph Labelng, (IJCNS) Internatonal Journal of Computer and Network Securty, Vol. 2, No. 4, pages 45-49, Aprl 2010 [3] K. Alzoub, P.-J. Wan and O. Freder, New dstrbuted algorthm for connected domnatng set n wreless ad hoc networks n Proc. IEEE HICSS35, January 2002 [4] K. Islam, S. G. Akl and H. Mejer, A Constant Factor Dstrbuted Algorthm for computng Connected Domnatng Sets n Wreless sensor Networks [5] M. Ra, S. Verma and S. Tapasw, A Power Aware Mnmum Connected Domnatng Set for Wreless Sensor Networks, Journal of Networks, Vol. 4, No. 6, pages , August 2009 [6] S. Guha and S. Khuller, Approxmaton algorthms for connected domnatng sets Algorthmca, pages , 1998 [7] Y. L, S. Zhu, M.Tha and D. Du, Localzed Constructon of Connected Domnatng Set n Wreless Networks n Proc. TAWN, June

An Optimization Algorithm for Minimum Connected Dominating Set Problem in Wireless Sensor Network

An Optimization Algorithm for Minimum Connected Dominating Set Problem in Wireless Sensor Network IEMS Vol. 10, No. 3, pp. 221-231, September 2011. An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network Namsu Ahn Defense Agency for Technology and Qualty 275-18, Boksu-dong,