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

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1 IEMS Vol. 10, No. 3, pp , September An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network Namsu Ahn Defense Agency for Technology and Qualty , Boksu-dong, Seo-gu, Daejeon, Tel: , E-mal: namsu.ahn@gmal.com Sungsoo Park Department of Industral and Systems Engneerng KAIST, 373-1, Gusong-dong, Yusong-gu, Daejeon, , Tel: , Fax: , E-mal: sspark@kast.ac.kr Receved, March 12, 2011; Revsed, May 18, 2011; Accepted, May 24, 2011 Abstract. One of the crtcal ssues n wreless sensor network s the desgn of a proper routng protocol. One possble approach s utlzng a vrtual nfrastructure, whch s a subset of sensors to connect all the sensors n the network. Among the many vrtual nfrastructures, the connected domnatng set s wdely used. Snce a small connected domnatng set can help to decrease the protocol overhead and energy consumpton, t s preferable to fnd a small szed connected domnatng set. Although many algorthms have been suggested to construct a mnmum connected domnatng set, there have been few exact approaches. In ths paper, we suggest an mproved optmal algorthm for the mnmum connected domnatng set problem, and extensve computatonal results showed that our algorthm outperformed the prevous exact algorthms. Also, we suggest a new heurstc algorthm to fnd the connected domnatng set and computatonal results show that our algorthm s capable of fndng good qualty solutons qute fast. Keywords: Connected Domnatng Set, Cuttng Plane, Integer Programmng, Wreless Sensor Networks 1. INTRODUCTION Due to the recent rapd advances n dgtal technologes and the mass producton, developments of multfunctonal, low-cost and low-power sensors become possble. Generally, sensor s composed of four major unts: power unt, sensng unt, processng unt and transcever unt, and each unt performs a unque functon. The power unt provdes power to all the other unts, and the sensng unt takes charge of the montorng task and data convertng task. Thus, f a certan phenomenon happens, the sensng unt detects the phenomenon changes, such as temperature, lght, sound and humdty, and then converts the sensed analog data to dgtal data. The processng unt can manpulate the collected data and store the collected data n the storage devce for transmttng. The transcever unt can transmt/receve nformaton over a wreless network. Typcally, a large number of sensors collaborate usng wreless communcaton, and the sensors gather, process and transmt nformaton over a wreless network to a remote runnng applcaton that makes decsons based on ths nformaton. Snce the deployed sensors are clustered and communcate n wreless channels, we call t wreless sensor network (WSN). Nowadays, WSN has many applcatons, such as habtat montorng, forestfre detecton, patent status montorng, home applance, nventory trackng and battlefeld montorng. WSN s featured by no fxed nfrastructure, multhop communcaton and lmted resources (battery capacty and bandwdth). These characterstcs pose new dffcultes n desgnng a routng protocol. Some exstng routng protocols for ad hoc networks (Ng and Lu, 1999; Pe et al., 2000) are based on floodng mechansm (.e., upon recevng a packet, transmt the packet to all of ts neghbors). Therefore f we use these protocols, t : Correspondng Author

2 222 Namsu Ahn Sungsoo Park not only devastates the resources of the sensors, but gves negatve effect on the throughput of the whole network. Furthermore, t probably causes broadcastng storm problem as ndcated n N et al. (2002), whch results n excessve data redundancy, contenton and collson. In Snha et al. (2001), a new routng protocol based on an overlayng vrtual nfrastructure, so called vrtual backbone, s proposed and the authors showed that, usng the backbone can reduce the routng overhead dramatcally. Smply, a vrtual backbone s defned as a subset of sensors, whch can connect all the sensors n the network. Ths concept s frequently used to smplfy the network and mproves the effcency of the routng. In many exstng vrtual backbone schemes whch can be found n Alzoub et al. (2002), Chen et al. (2002), Da and Wu (2004), Stojmenovc et al. (2002) and Wu and Da 2004, a connected domnatng set (CDS) s suggested to use as a backbone. To smplfy our dscusson, we use a connected graph G = (V, E) to represent the network, where V and E represent the set of vertces and the set of edges, respectvely. Each vertex v V ndcates a sensor, and there s an edge e(= uv) E whch denotes that sensor u s wthn sensor v's communcaton range and vce versa. A set of sensors s called a domnatng set f each of the sensors n the network s ether n ths set or has a neghbor (sensor u s neghbor of sensor v f there exsts an edge between the two vertces) n the set. Typcally, the sensors n the domnatng set are called domnators, and the other sensors are called domnatees. A domnatng set s called a CDS f the subgraph nduced by the domnatng set s connected, and the connectvty among the domnators s requred for proper routng of sgnals. Therefore f any domnatee want to send a message to domnator(or domnatee), t frst send a message to connected neghbor domnator. If CDS s used as a backbone, we can obtan the followng good characterstcs of the network. Routng overhead can be reduced as shown n Wu and L (1999) because only the sensors n CDS need to mantan the routng nformaton. Thus, f a domnatee wants to send a packet to another domnatee, t sends the packet to ts domnator. Then the domnator wll delver the packet to the destnaton domnatee. Energy effcent area coverage s possble as ndcated n Carle and Smplot (2004) and Chen et al. (2002). Snce CDS s a good approxmaton of an area, domnators n CDS can take over the domnatees sensng task. Thus, f the domnators are actvely performng the sensng task, all of ts domnatees probably enter nto a low-battery sleep state to save energy for future use. Usefulness of the CDS n WSN has been demonstrated n many communcaton protocols such as meda access coordnaton, uncast, multcast/broadcast, locaton-based routng, energy conservaton, resource dscovery and topology control (More comprehensve revew can be found n Blum et al. (2004)). Snce the number of sensors formng the vrtual backbone needs to be as small as possble to decrease the protocol overhead and energy consumpton, t s desrable to form a mnmum szed CDS. Fndng a mnmum szed CDS problem s referred to as the mnmum connected domnatng set (MCDS) problem. For example, vertex set {a, b, c} n Fgure 1 forms a CDS, thus the domnators are {a, b, c} and the domnatees are {d, e, f}. Fgure 1. Example of the CDS. Fndng an MCDS s an NP-Hard problem Garey and Johnson (1979), and many researches have been performed on the MCDS problem and a comprehensve revew on the algorthms can be found n Blum et al. (2004). In ths paper, we revew the centralzed CDS constructon algorthms only (Although CDS can be constructed n a dstrbuted manner (Chen et al., 2009; Wang et al., 2009), t s beyond the scope of ths paper). We revew the performance guaranteed approxmaton algorthms frst. In Guha and Khuller (1998), two algorthms are suggested. One forms a CDS of sze at most 2(1 + H( Δ)) MCDS and the other one constructs a CDS of sze (3 + ln( Δ)) MCDS, where H, MCDS and Δ represent the harmonc functon, sze of the MCDS and the maxmum degree of the gven graph, respectvely. The authors n L et al. (2005), Mn et al. (2006) and Ruan et al. (2004) reported (4.8 + ln5) MCDS, 6.8 MCDS and (3 + ln( Δ)) MCDS approxmaton algorthms, respectvely. There exst many heurstc algorthms for the MCDS problem. Relatvely latest ones can be found n Chen et al. (2010) and Morgan and Grout (2008). However, these algorthms cannot guarantee the qualty of the obtaned solutons because no nformaton on the lower bound can be obtaned. Although solvng the MCDS problem exactly s an mportant research goal, there have been few exact approaches. One smple scheme s enumeratng all subset of the vertces, and the algorthm shown n Fomn et al. (2008) breaks the 2 V barrer by suggestng O (1.9407) V algorthm. However, no mplementaton and performance test of the algorthm can be found. Other two optmal approaches used mathematcal formulatons to obtan MCDS (Morgan and Grout, 2008;

3 An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network 223 Yuan, 2005). However, the performances of the algorthms reported by the authors were not mpressve. In ths paper, we propose an mproved optmal algorthm usng the mathematcal formulaton n Yuan (2005) and compare the performances wth the prevous two approaches. The computatonal results showed that, our algorthm outperforms the prevous optmal approaches n terms of the runnng tme to obtan an optmal soluton. Also, we propose a new heurstc whch use the formulaton and the dea of our mproved optmal algorthm. Ths paper s organzed n the followng way. In Secton 2, we revew notaton and defnton. In Secton 3, the exstng mathematcal formulatons and algorthms for the MCDS problem wll be dscussed, and an mproved optmal approach wll be ntroduced n the same secton. Secton 4.1 shows the performance of the mproved optmal approach suggested n ths research. Secton 4.2 ncludes descrpton of the heurstc whch s suggested n ths research for computng the connected domnatng set effcently, and shows the performance of the heurstc. Fnally, n Secton 5, we conclude ths paper. 2. PRELIMINARIES Ths secton provdes some background nformaton to understand the rest part of the paper. As noted before, WSN can be represented by a smple graph G = (V, E) and the followng defnton and notaton from graph theory wll be used throughout the paper. Open neghbor set, = { } N( u) v ( uv) E, s the set of vertces adjacent to vertex u. Closed neghbor set, N[ u] = N() u { u }, s the set of vertces adjacent to vertex u and u tself. Independent set s a subset of V such that no two vertces are adjacent n G. For example, {d}, {d, e} and {d, e, f} are ndependent sets n Fgure 1. Maxmal ndependent set s an ndependent set such that addng any vertex not n the set breaks the property of the ndependent set. For example, n Fgure 1, the ndependent set {d, e, f} s a maxmal ndependent set snce addton of any other vertces (a or b or c) makes some vertces n the set are connected. Domnatng set s a subset of V such that each vertex s ether n the set or has a neghbor n the set. Note that every maxmal ndependent set s a domnatng set, but the converse s not true. Connected domnatg set s a domnatng set whose nduced subgraph s connected. For example, {a, b, c} s a CDS n Fgure 1. Stener tree s a tree whch, for a gven subset T of V, connects the vertces of T possbly usng the vertces n V\T. For example, n Fgure 1, f T s gven as {d, e, f}, a Stener tree can be constructed by addng the remanng vertces {a, b, c} and the edges ab, ac, ad, be and cf. Vertex cut s a subset of V whose removal dsconnects the graph. If we use maxmal ndependent set and Stener tree, the followng smple procedure can construct a CDS. We frst form a maxmal ndependent set to dentfy a domnatng set, and let the generated domnatng set from the frst step be the vertex set T n Stener tree. Then CDS can be constructed by fndng a Stener tree. Many heurstc algorthms and approxmaton algorthms have been developed usng ths dea, and the latest one can be found n Mn et al. (2006). 3. MATHEMATICAL FORMULATIONS AND OPTIMAL ALGORITHMS To the best of our knowledge, two dfferent mathematcal formulatons exst for the MCDS problem, frst one s shown n Morgan and Grout (2008) and the second one s suggested n Yuan (2005). Two formulatons used dfferent modelng technques. The frst one used a 2V + 2 E number of varables wth 3V + 2 E number of constrants, approxmately. The second one used a V number of varables, but t requred an exponental number of constrants to represent the connectvty of the nduced subgraph. At frst sght, the frst formulaton seems to be better. However, after performng extensve computatonal experments, we concluded that, f an mproved optmal algorthm developed n ths paper s used, the second formulaton may be used to fnd an optmal soluton much faster than the frst formulaton. Generally, representng the connectvty requrement for the graph problems asks for an exponental number of nequaltes. To avod ths, the frst formulaton used the flow varables n the formulaton. The man dea of the formulaton s, when the vertces are selected to form a CDS, any vertex ncluded n the CDS can send a flow to each of the other vertces n the CDS (usng only the selected vertces). Each vertex ncluded n the CDS requres at least one unt of flow, and the root vertex (vertex 1) contans enough flows to supply the requred flows n each vertex n the CDS. Thus, f the root vertex s not ncluded n the CDS, t transfers all of ts flows to one of ts neghborng vertces whch s selected to be ncluded n the CDS. We frst construct a drected graph G = (V, A) by replacng each edge e(= j) E by two arcs j and j, and defne the decson varables as the followng. 1, f vertex s ncluded n the MCDS, x = 0, otherwse fj = amount of flow from vertex to j. 1, f flow s permtted from root vertex to vertex x = 0, otherwse

4 224 Namsu Ahn Sungsoo Park Then the formulaton can be gven as the followng. Mnmze x (1) V Subjectto x + x 1, = 1,, { jj A} j V (2) f { } 1 1 { 1 } 1 = 1 j f j j A j j A j V (3) f, 2,, { } j f = j j A { j j A} j x V (4) 1j 1 j { } j { } j { } 1+ { 1 } j 1 { 0, 1 }, = 1,, j { 0, 1 }, { 1 } f V ( x + x ), j 1j A (5) f V x, j A 1and j 2 (6) f1 V y, j 1j A (7) j j A y x V (8) x V (9) y j j A (10) Constrants (1) mnmze the number of vertces whch are selected to be ncluded n the CDS. Constrants (2) guarantee that, f the vertex s not ncluded n the CDS, t has at least one neghborng vertces whch are selected to be ncluded n the CDS to satsfy the domnance requrement. Constrants (3) assure that the root vertex n the CDS may produce suffcent flows to provde at least one unt flow to the other vertces whch are chosen to be ncluded n the CDS. Constrants (4) state that, f any vertex s chosen to be ncluded n the CDS, t consumes at least one unt flow. Constrants (5) and (6) set upper bounds for the flows. Constrants (7) guarantee that flows from the root vertex to the other vertex j can be sent only f y j s one. Whle constrant (8) forces that, f the root vertex s not ncluded n the CDS, at most one of ts neghbor vertces can be selected n the CDS to send the flows. Constrants (9) and (10) state the bnary nteger requrements on the varables. The authors n Morgan Grout (2008) reported that, when ILOG CPLEX (2008) was used as an optmzaton software to solve the formulaton, they could obtan the optmal solutons for the graphs of up to 100 vertces n 1000 seconds. Fgure 2. Example of the mnmal vertex cut. Now we revew the formulaton and the optmal algorthm shown n Yuan (2005). Before gvng a detaled explanaton, we ntroduce notaton whch wll be used n the formulaton. Let C be a mnmal vertex cut (=whose removal dsconnects the graph and removng any vertex n the cut fals to form the vertex cut), and be a collecton of C. For example, n Fgure 2, vertex cut {a, c, d} s not a mnmal vertex cut, but c or d s a mnmal vertex cut. The author n Yuan (2005) proved that, f at least one vertex s selected from C, C, the set of selected vertces form a CDS. Usng ths characterstc, the authors suggested the followng formulaton for the MCDS problem (The decson varable x ndcates that whether the vertex s ncluded n the CDS or not). Mnmze x (11) V Subjectto 1, C j (12) x 0, 1, V. (13) { } Snce the number of constrants (12) can be huge, representng the formulaton n full s mpractcal. Therefore the author n Yuan (2005) suggested the followng constrant generaton scheme to dentfy the volated constrant of (12) only when needed. Optmal approach n Yuan (2005): Step 1: Construct the ntal formulaton usng constrants (11) and (13) only. Step 2: For each vertex v V, form a subset S of V by settng S = [v]. For each S, construct a mnmal vertex cut by fndng the vertces whch has one or several neghbor vertces n V\S. Then add the correspondng nequaltes of (12) to the formulaton. Step 3: Solve the formulaton optmally and let the set of vertces whch are selected as D. If D forms a CDS, stop the procedure and output the obtaned MCDS. Otherwse, goto Step 4. Step 4: For each vertex v V, construct a subset S of V usng the vertces that v can reach va vertces n D, ncludng v tself. For each S, f we can construct a mnmal vertex cut by fndng the vertces whch has one or several neghbor vertces n V\S, store the correspondng nequalty of (12). Step 5: Perform a par-wse check for the stored nequaltes n Step 4 to dentfy the nequaltes whch are not domnated by the other nequaltes (.e., for gven two vertex cuts C 1 and C 2, f C 1 s a subset of C, 2 C 2 s domnated by C 1 ). Only those nequaltes are added to the formulaton, and goto Step 3. The authors n Yuan (2005) reported that, when the tme lmt was set to 1 hour and ILOG CPLEX 7.0 was used as the optmzaton software, they could obtan the optmal solutons for the graph of up to 80 vertces. Up to ths part of the secton, we revewed two dfferent mathematcal formulatons and the optmal approaches to solve the formulatons, but the performances

5 An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network 225 of the algorthms reported by the authors seem to be not mpressve and the approaches have rooms for more mprovements. However, snce the prmary purposes of ther researches were developng good heurstc algorthms, no more efforts n optmal approaches could be found. Comparsons of the algorthm n Morgan Grout (2008), orgnal algorthm n Yuan (2005), and our mproved algorthm wll be gven n Secton 4.1. Note that f t s guaranteed that the number of vertces selected n an MCDS s greater than or equal to two, the followng addtonal nequaltes can be added to the formulaton. j N() j x 1, C V (14) Constrants (14) ndcate that any selected vertex has at least one or more neghborng vertces to establsh the connectvty. Actually, constrants (14) are not requred n a correct formulaton of the MCDS problem, but addng the constrants to the formulaton strengthens the formulaton and t reduces the computaton tme to solve the problem. Now we dscuss our optmal algorthm for the MCDS problem based on the formulaton from Yuan (2005). To solve the MCDS problem for a gven graph, we used a dfferent constrant generaton scheme and ths dfference results n mprovements n the computaton tmes to obtan the optmal solutons. As noted before, snce the number of constrants (12) can be exponental, the constrants (12) need to be dealt mplctly rather than explctly. We frst solve the formulaton optmally wthout constrants (12) and construct the graph usng the vertces whch are selected. When the current soluton s not a CDS, the optmal approach n Yuan (2005) dentfes one vertex cut whch s volated by the current soluton. On the other hand, when there exst several vertex cuts whch are volated by the current soluton, ths approach does not guarantee to dentfy the smallest szed vertex cut. Snce usng a mnmum vertex cut among the volated vertex cuts may tghten the formulaton the most, t probably reduces the computaton tme to solve the problem. In ths paper, when the constructed subgraph from the current soluton conssts of several dsconnected components, we suggest a procedure to fnd a mnmum vertex cut whch separates one component from the other components. We frst explan how to fnd a vertex cut usng an example, and then, a procedure whch dentfes the mnmum vertex cut wll be dscussed. Assume that, when we solved the formulaton wthout constrants (12) usng the example gven n Fgure 3, two vertces a and c are selected (for ths example, suppose that we replaced constrants (14) by x v N[ u] v 1, u V, otherwse a and c cannot be chosen). However, snce the resultng subgraph s not a CDS, we want to dentfy a vertex cut whch s volated by the current soluton a and c. Fgure 3. Example of the MCDS problem. Clearly, snce we want to fnd a vertex cut whch separates the two vertces a and c, vertces b and d form a vertex cut whch s volated by the current soluton. Now we add the correspondng nequalty xb + x d 1 to the formulaton. When we solve the enlarged formulaton, we can obtan the CDS as {a, b} and the procedure stops. However, dentfyng the vertex cut probably be a complcatng step when the sze of the graph s large. Furthermore, t may be harder to fnd the mnmum vertex cut among the vertex cuts. Therefore, n ths paper, we propose an dea whch dentfes the vertex cut usng the modfed graph of G, then a procedure whch can be used to fnd the mnmum vertex cut wll be llustrated (We used the procedure suggested n Thulasramanand Swamy (1992) wth some modfcatons). To dentfy a vertex cut for a gven graph G = (V, E), we frst replace each edge by two antparallel arcs, and then, splt a vertex v nto two vertces v and v, and create an arc drected from v to v. Then, replace an arc that s drected from vertex u to other vertex v by an arc drected from u to v. Lastly, assgn unt capactes to the generated arcs whch are drected from v to v, and allocate very large postve (= M) capactes to the generated arcs drected from u to v. The result of applyng the procedure to the example n Fgure 3 s shown n Fgure 4. Fgure 4. Transformed graph. Therefore, when we send flows from a source vertex to a snk vertex as much as possble, only the arcs whch are drected from v to v lmt the amount of flows. Note that n ths example, we have two source ver-

6 226 Namsu Ahn Sungsoo Park tces a and a (from vertex a) and two snk vertces c and c (from vertex c) nstead of a sngle source vertex and a sngle snk vertex. Generally, when we want to fnd a vertex cut usng the modfed graph, we need to solve the multple sources and multple snks maxmum flow problem nstead of the ordnary maxmum flow problem. However, ths problem can be easly reduced to the ordnary sngle source sngle snk maxmum flow problem. We frst add a vertex (= super-source vertex) and create arcs to the source vertces wth capactes M. Smlarly, we add a vertex (= super-snk vertex) and create arcs from the snk vertces wth capactes M. Then, any flow from the super-source vertex to the super-snk vertex corresponds to a flow n the multple source vertces and multple snk vertces problem as shown n Cormen et al. (1989). If we dentfed the maxmum flow, we also can fnd the mnmum arc cut such that the value of the flow s equal to one (arcs are drected from v to v ). Then we can obtan a vertex cut of the orgnal graph by restorng the vertces correspondng to the cut arcs. Suppose that the soluton from the formulaton wthout constrants (12) s not a CDS, then the selected vertces form several connected components. Now, for each component, we want to dentfy the mnmum szed vertex cut whch separates the current component and the other components, and the correspondng nequaltes (12) are added to the formulaton. Then, the followng procedure can be used to fnd the mnmum szed volated vertex cut. Procedure to fnd the mnmum szed volated vertex cut: Step 1: For a gven undrected graph G = (V, E), construct a drected graph G = (V, A) by replacng each edge e(= j) E by two arcs j and j. Step 2: Replace every vertex v V by two vertces v and v and create an arc drected from v to v. Step 3: Assgn unt capactes to the generated arcs n Step 2. Step 4: Replace an arc that s drected from vertex u V to the another vertex v V by an arc drected from u to v. Step 5: Assgn capactes M to the generated arcs n Step 4. Step 6: Set k as 1. for all k th component do Step 6.1: Create a super-source vertex and create arcs jonng the super-source vertex to the vertces of k th component wth capactes M. Step 6.2: Create a super-snk vertex and create arcs jonng the vertces of the other components to the super-snk vertex wth capactes M. Step 6.3: Fnd a maxmum flow from the super-source vertex to the super-snk vertex. Step 6.4: Identfy a mnmum arc cut, and obtan a vertex cut of the orgnal graph by restorng the vertces correspondng to the arc cut. Step 6.5: Drop the generated vertces and arcs n Step 6.1 and Step 6.2, and ncrease k by one. end for Step 7: Choose the mnmum vertex cut among the vertex cuts dentfed n Step DESCRIPTION OF THE HEURISTIC We suggested an optmal approach and tested the performance of the algorthm n Secton 5.1. However, when the sze of the problem becomes large, solvng the MCDS usng the formulaton faled to provde an optmal soluton wthn an hour. Therefore, we propose a heurstc whch can be used to fnd the CDS. Major characterstcs of the heurstc are lnear programmng relaxaton of the mathematcal formulaton and mnmum vertex cut dentfcaton procedure explaned n Secton 3. The detaled procedure of the heurstc can be gven as follows. Heurstc to compute connected domnatng set: Step 1: Prepare an empty set, and construct the formulaton whch conssts of (11), (13) and (14). Step 2: Drop the ntegralty condtons and let the lower bound for the objectve value as z Step 3: Solve the lnear programmng relaxaton. Step 4: Identfy the most fractonal varable x varable whch s close to one, and assgn the correspondng vertex to the set. Step 5: If the set satsfy the domnance requrement and sze of the set s greater than or equal to z, go to Step 7. Step 6: Fx the dentfed x varables n Step 5 to one n the formulaton and return to Step 3. Step 7: Check the connectvty among the vertces whch are ncluded n the set. If the set s connected, stop the algorthm and output the vertces n the set. Otherwse, go to Step 8. Step 8: Snce the vertces n the set conssts of several dsconnected components, dentfy the mnmum vertex cut whch separates one component from the other components usng the procedure descrbed n Secton 3. Step 9: Add the correspondng nequalty whch s volated by the current soluton, and let z := Set + 1 and return to Step 3. In Step 4, f the several varables are assgned the same values, we choose the varable whose sum of dstances to other vertces ncluded n the set s the mnmum. Note that the number of teratons s fnte, and snce solvng the lnear programmng relaxaton, separatng the mnmum vertex cut and fndng the dstances can be done n polynomal tmes, our heurstc runs n polynomal tme.

7 An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network COMPUTATIONAL RESULTS 5.1 Performance of the optmal algorthms We tested the performances of the optmal approaches for the MCDS problem suggested n Morgan Grout (2008), Yuan (2005) and n ths paper. We mplemented the three approaches and tested them on randomly generated graphs. For each of the approaches, the algorthm run tmes are reported. For the approaches gven n Yuan (2005) and ths research, the number of generated nequaltes (12) to obtan the MCDS are also reported. The purpose of the experment s, to observe the dfferences n the run tmes for the three exact approaches. We used the code obtaned from donparker.net/graph_gen.php to generate the connected random graphs. The code can generate three types of connected graphs: sparse, medum and dense graphs for a gven number of vertces V. The graph classfcaton crteron s the number of edges. The number of edges of the sparse graph s less than V ( V 1)/4, the number of edges of the medum graph s approxmately V ( V 1)/4, and the number of edges of the dense graph s greater than V ( V 1)/4. For each type of the graphs, we generated 10 random graphs wth the number of vertces rangng from 100 to 1000 wth an ncrement of 100. The three approaches were mplemented n C++ and ILOG CPLEX 11.0 was used as optmzaton software. All experments were run on an AMD Athlon TM 64 X2 Dual Core (2.70 GHz) wth 2GB RAM, and runnng tme s gven n seconds. Computatonal results are gven n Table 1, 2 and 3 for each type of the graphs. Note that MCDS denotes the sze of the mnmum connected domnatng set and * ndcates the problem on whch the algorthm faled to obtan the soluton wthn an hour. Also, comparson of the run tme s llustrated n Fgure 5, 6 and 7. V Table 1. Comparson of the optmal approaches for sparse graph problems. Approach n Morgan Grout (2008) Approach n Yuan (2005) Ahn and Park s approach Run tme (sec) Inequaltes Run tme (sec) Inequaltes Run tme (sec) MCDS * * * * * * * * * * * * * * * * * * * V Table 2. Comparson of the optmal approaches for medum graph problems. Approach n Morgan Grout (2008) Approach n Yuan (2005) Ahn and Park s approach Run tme (sec) Inequaltes Run tme (sec) Inequaltes Run tme (sec) MCDS * * * * * * * * * * 700 * * * * * * 800 * * * * * * 900 * * * * * * 1000 * * * * * *

8 228 Namsu Ahn Sungsoo Park When we compared the optmal approaches n terms of the algorthm run tme, each approach showed dfferent performance dependng on the types of the graphs. For sparse graphs, our approach obtaned MCDS n the shortest run tme, then the approach gven n Morgan Grout (2008) follows the next, and lastly, the approach proposed n Yuan (2005) showed the longest run tme. On the other hand, for medum and dense graphs, the approach proposed n Morgan Grout (2008) showed the longest run tme to obtan the MCDS, and the other two approaches showed equvalent performances. Also, for the gven medum and dense graphs, our approach and the approach shown n Yuan (2005) requred no addtonal constrants (12) to obtan the optmal soluton. Ths probably due to the densty of the gven graphs. 5.2 Performance of heurstc In Table 4, we use the same problems from Table 1, Table 2 and Table 3 and test the performance of our heurstc. Run tme column denotes the runnng tme of the heurstc, CDS ndcates the sze of the connected domnatng set obtaned from the heurstc and MCDS means the sze of the mnmum connected domnatng set. We can observe that, our heurstc algorthm showed good performance n medum and dense graph problems, but showed relatvely poor performance n sparse graph problems. Run tme(sec) Fgure 5. Comparson of run tme n medum graph. Table 3. Comparson of the optmal approaches for dense graph problems. V Approach n Morgan Grout (2008) Approach n Yuan (2005) Ahn and Park s approach MCDS Run tme (sec) Inequaltes Run tme (sec) Inequaltes Run tme (sec) * * * * * * * * * * 800 * * * * * * 900 * * * * * * 1000 * * * * * *

9 An Optmzaton Algorthm for Mnmum Connected Domnatng Set Problem n Wreless Sensor Network 229 Run tme(sec) Run tme(sec) Fgure 6. Comparson of run tme n medum graph. Fgure 7. Comparson of run tme n dense graph. 6. CONCLUSIONS Desgnng a routng protocol s one of the mportant ssues n wreless sensor network. Many vrtual nfrastructure based protocols have been suggested, and one of them s utlzng the CDS. Snce formng a small szed CDS s preferable, many researches have been devoted on ths ssue. In ths paper, we proposed an mproved optmal algorthm for the MCDS problem, and extensve computatonal experments showed that our algorthm outperforms the prevous approaches n terms of the runnng tme to obtan an optmal soluton. Also, we proposed the heurstc n ths research whch uses the mathematcal formulaton for the MCDS problem, and the computatonal experments show that our algorthm obtans an qute good soluton n a reasonable amount of tme.snce constructng an MCDS s an mportant ssue n wreless sensor networks, further approach such as branch-and-cut or meta-heurstc methods (genetc algo-

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