Robot-Assisted Sensor Network Deployment and Data Collection

Transcription

1 -Aited Senor Network Deloyment nd Dt Collection Yu Wng Chnghu Wu Abtrct Wirele enor network he been widely ued in mny liction uch enironment monitoring, ureillnce ytem nd unmnned ce exlortion. Howeer, oor deloyment of enor deice led () bd network connectiity which mke dt communiction or dt collection ery hrd; or () redundncy of coerge which wte energy of enor nd cue redundnt dt in the network. Thu, in thi er, we rooe uing mobile robot to it the enor deloyment nd dt collection for unmnned exlortion or monitoring. We ume tht the robot cn crry nd deloy the enor deice, nd lo he certin communiction ccity to collect the dt from the enor deice. Gien et of interet oint in n re, we tudy the following intereting roblem: () how to decide minimum number of enor deice to coer ll the interet oint; () how to chedule the robot to lce thee enor deice in certin oition o tht the th of the robot i minimum; nd () fter the deloyment of enor, how to chedule the robot to iit nd communicte with thee enor deice to collect dt o tht the th of the robot i minimum. We rooe comlete et of heuritic for ll thee roblem nd erify the erformnce i imultion. I. INTRODUCTION Wirele enor network [] he tremendou roect due to their reltiely lower cot nd cbility of obtining luble informtion from loction tht re beyond humn rech. A enor network conit of et of enor node tht red oer geogrhicl re. Thee enor re ble to erform roceing well ening nd re dditionlly cble of communicting with ech other. Due to it widernge otentil liction uch bttlefield, emergency relief, enironment monitoring, ureillnce ytem, ce exlortion, nd o on, wirele enor network h recently emerged remier reerch toic. Mot of current reerch on wirele enor network ume the cot of ech enor i che thu the number of enor in network could be ufficient lrge (hundred or thound) to coer the trget re nd mintin the network connectiity. Howeer, in mny rel liction (uch ce exlortion), certin kind of enor deice could be ery exenie, nd it i imoible to he thound of them to deloy. In ddition, ince the enor would he reltiely wek rdio, internode ertion i ery common in enor network. On the other hnd, een if the number of enor i ufficient nd the rdio i trong enough, oor deloyment of enor Yu Wng i with Dertment of Comuter Science, Unierity of North Crolin t Chrlotte, USA. Emil: [email protected]. The work of Wng i uorted, in rt, by fund roided by Ok Ridge Aocited Unieritie nd the Unierity of North Crolin t Chrlotte. Chnghu Wu i with Dertment of Science nd Mthemtic, Kettering Unierity, USA. Emil: [email protected] In thi er the term node often rereent ening deice or clled enor. We often interchnge them here. deice could lo led to () bd network connectiity which mke dt communiction or dt collection ery hrd; or () redundncy of coerge which wte energy of enor nd cue redundnt dt in the network. Thu, in thi er, we rooe uing mobile robot to it the enor deloyment nd dt collection for unmnned exlortion or monitoring. We ume tht the robot cn crry nd deloy enor deice, nd lo he certin communiction ccity to collect the dt from thee enor deice. Recent yer he een the growing interet in mobile enor network [] [7] or robot-ited enor network [8] []. In mobile enor network, ll or rtil of the enor node he motion cbility endowed by robotic ltform. Mobile enor network he more flexibility, dtiely nd een intelligence comred with ttionry wirele enor network. Mobile enor cn dynmiclly reoition themele to tify certin requirement on monitoring coerge, network connectiity, or fult tolernce. Howeer, to mke eery enor he motion cbility incree the cot of ech enor nd mybe not feible in mot liction. On the other hnd, robot re lrge comlex ytem with owerful reource nd cn interct with enor node. The new rdigm of robot ited enor network i of ubiquitou enor embedded in the enironment with which the robot interct: to deloy them, to hret dt from them, nd to tk them. In turn, the enor cn roide the robot with model tht re highly dtie to chnge in the enironment nd cn re-tk the robot with feedbck from enor. Therefore, we beliee tht robotic will he rofound effect on enor network. Mot reiou reerch on robot-ited enor network [8] [] tudy uing the robot to chiee coerge, locliztion nd nigtion. In thi er, we focu on coerge nd th lnning. Gien et of interet oint in n re, we tudy the following intereting roblem: () how to decide minimum number of enor deice to coer ll the interet oint; () how to chedule the robot to lce thee enor deice in certin oition o tht the th of the robot i minimum; nd () fter the deloyment of enor, how to chedule the robot to iit nd communiction with thee enor to collect dt o tht the th of the robot i minimum. An illutrtion of thi cenrio i deicted in Figure where the roer uing one th to deloy the enor nd the other th to collect dt from deloyed enor. We rooe comlete et of heuritic for ll thee roblem nd erify the erformnce i imultion. A otentil liction of our rooed robot-ited enor network deign i for unmnned ce exlortion. Unmnned ce exlortion he tremendou roect

2 well-known grh theoretic roblem, the treling lemn roblem []. We ume tht the D ce doe not he ny obtcle nd the robot cn moe towrd ny direction freely. The objectie of our th lnning i to minimize the totl length of the th which the robot trel. We tudy how to deloy enor nd chedule the robot th uch tht the totl trel ditnce i minimized nd the coerge i gurnteed. Fig.. Illutrtion of the cenrio where robot ( roer) deloy enor node round interet oint nd it to collect dt from the unconnected enor network. Here, the red th i the deloyment th, while the blue one i the dt collection th. due to their reltiely lower cot nd cbility of obtining luble informtion from loction tht re beyond humn rech. The imct of unmnned miion nd the ue of utomted remote monitoring ttion nd robotic ltform in ce he been obered from eerl ucceful enture in the t. Exmle include the NASA Mr roer tht re deigned to negotite unredictble urfce condition nd roide luble dt, ideo mle well hyicl mle through remote control. Our rooed roch cn llow the robot (roer) efficiently deloy nd mintin the enor network which enble dt collection oer lrge re oer extended eriod of time. The rooed coordinted remote dt deloyment nd collection roch cn extend the rech nd lifetime of both ce roer nd mrt enor. II. RELATED WORK Senor Coerge: Since ech enor coer limited re, dequte coerge of lrge re require rorite lcement of enor bed on collectie coerge nd cot contrint. The reiou reerch on enor coerge minly focue on tudying how to determine the minimum et of enor for coering eery loction or certin object (interet oint) in the trget field. Different coerge model nd method re ureyed by Crdei nd Wu []. -Aited Senor Network: Mobile or robotited enor network he been tudied recently. Mot reiou reerch concentrte on uing the robot or mobile enor to hel enor network to chiee coerge [], [], locliztion [6] [9], [], trget detection [], fult-tolernce [], [], nd nigtion [9]. In thi er, we tudy how to ue robot iting the enor deloyment nd dt collection, with focu on efficient th lnning. Pth Plnning: One of the mot imortnt roblem in robotic i th lnning (or clled motion lnning) [], [], which i imed t roiding robot with the cbility of deciding utomticlly which motion to execute in order to chiee certin ecific gol. It rie in riety of form. The common form require finding hort geometric colliion-free th for ingle robot in known ttic workce. In thi er, we do not focu on uch kind of th lnning. The roblem we concentrte on i imilr to A. Model III. MODELS AND PROBLEMS We ume tht et of m interet oint (or clled trget), denoted by P = {,,, m }, re ditributed in -dimenionl lne. The objectie of our miion i to deloy et of enor deice, denoted by S = {,,, n } to form enor network to monitor or trck thee interet oint. Ech enor node i i equied with enor which cn monitor dik region centered t i with rdiu r S, i.e., if the ditnce between l nd i i le thn r S then enor i cn monitor the interet oint l. We ume tht ingle enor cn monitor multile oint inide it ening region. Ech enor node i h n omnidirectionl ntenn o tht it cn tlk to ll enor node or the robot within dik region centered t i with rdiu r T. Herefter, we cll r S nd r T the ening rnge nd the trnmiion rnge reectiely. We ume ll enor node re equied with me hrdwre deice, thu, they he the me fixed r S nd r T. We ume the robot R h lrger trnmiion rnge thn the enor node, i.e., it cn tlk with enor node i if it i inide the trnmiion rnge of i. The robot rk t oint initilly nd need to return fter ll oertion. It cn trel to ny oint in the -dimenionl lne during the oertion. B. The Problem The roblem we tudy i how to efficiently chedule robot to () deloy et of enor node S to gurntee the coerge of ll interet oint P nd () collect dt from thee enor node. Here, the efficiency of the th cheduling men the cheduled th for the robot to trel i hortet. We tret thi roblem two ub-roblem ertely: deloyment roblem nd dt collection roblem. For the deloyment roblem, gien the et of interet oint P, we tudy how to find the oition V = {,,, n } of enor node S where they will be deloyed by the robot, uch tht () the enor network gurntee the full coerge of ll interet oint P nd ue the minimum number n of enor node S; nd () the th Π D = n which the robot will trel to deloy enor t thoe oition h the minimum totl length. For the dt collection roblem, gien the et of deloyed enor V, we tudy how to find the turning oition (or clled ue oint) U = {u, u,, u k } where the robot ue nd collect dt from enor, uch tht () the robot Howeer, our rooed method cn be eily extended to the ce with heterogeneou ening nd trnmiion rnge.

3 cn communicte with eery enor during the round tri nd mke the minimum number k of to; nd () the th Π C = u u u k which the robot will trel to collect dt on thoe ue oint h the minimum totl length. Notice tht the deloyment roblem nd the dt collection re eentilly the me excet the rnge of coerge i different (one ue the ening rnge, the other ue the trnmiion rnge), thu we ue the me et of heuritic to ole thee two roblem Fig.. The et of interet oint P (blck node) nd the initil oition of the robot (red tringle). Here, re cn be defined A by the 9 ening dik nd their interection. 9 exit mny heuritic for it. The imlet nd mot clicl method i greedy method, in which you lwy greedily elect the ubet which cn coer the mximum number of uncoered element. Thi greedy lgorithm cn chiee n roximtion rtio of O(ln ) where i the ize of the lrget ubet. Inroximbility reult [6], [7] how tht the greedy lgorithm i eentilly the bet-oible olynomil time roximtion lgorithm for et coer under luible comlexity umtion. Algorithm how our greedy lgorithm nd Figure illutrte the reult from Algorithm on the exmle hown in Figure. Algorithm Greedy lgorithm to elect the minimum number of re where to deloy enor Inut: A et of re A = {,,, l } nd et of interet oint P = {,,, m }. Outut: A ubet of re A S = {,,, n } to lce the n enor S. : Initilly, et ll interet oint uncoered nd the uncoered counter k = m. Let the otentil coerge c i of ech re i equl to the number of dik Dj S interecting with thi re. Here, ech interet oint j could be coered by enor lced in re i. We cll j cn be coered by i. : while k! = do : Select re j with the lrget otentil coerge c j (uing ID to brek tie) nd dd it into the elected ubet A S ; : Mrk ll interet oint coered by j coered; : k = k c j ; 6: Udte the c k for ll djcent re k. 7: end while Fig.. Grey re re the ubre elected by Algorithm where enor need to be deloyed. IV. ROBOT-ASSISTED SENSOR DEPLOYMENT In thi ection, we decribe our lgorithm for how to deloy the enor with itnce from the mobile robot. A hown in Figure, we firt ue the ening rnge r S to drw dik Di S for ech interet oint i. To gurntee ll interet oint re coered by enor, we need t let one enor node inide ech dik Di S to monitor i. Howeer, one enor cn it in the interection of multile dik to monitor multile trget. Thu, we define the re formed by the dik nd their interection, denoted by A = {,,, l }, utting enor in n re i coer one or multile interet oint. The firt otimiztion roblem i how to elect the minimum number of re to deloy enor node to gurntee the coerge. Thi roblem i ctully the minimum et coer roblem which im to find the minimum number of ubet to coer the whole ce. The minimum et coer roblem i NP-hrd roblem []. Howeer, there After electing the re to lce the enor, we need to decide their exct oition. Since the oition cn ffect the totl length of the th tht the robot need to iit, we conider the oition roblem joint with the th chedule roblem. In other word, we rooe n lgorithm to chedule the robot to deloy the enor in ech elected re i, o tht the totl length of the th trelled by the robot i minimum. Thi roblem i ctully the treling leeron roblem with neighborhood (TSPN) which i lo NP-hrd roblem [8]. The clicl TSP tudie wht i the hortet round-tri route tht iit ech oint exctly once nd then return to the trting oint, gien et of oint in lne. TSPN tudie wht i the hortet roundtri route tht iit ech re exctly once nd then return to the trting re, gien et of re. There re eerl roximtion lgorithm exit for TSPN, howeer mot of them re ery comlex nd not rcticl t ll. Our lgorithm i n itertie lgorithm in which ech te we dd new turn oint inide one of the uniited re uch tht the ditnce dded to the robot th i minimum. Aume, we he n re needed to be iited (deloying the enor) nd initilly ll re re uniited, the lgorithm will terminte fter n round, ince ech round it dd new turn oint

4 in the th nd coer n uniited re. Algorithm how the detiled lgorithm. Algorithm Pth chedule nd enor lcement lgorithm: to elect the turn oint of the robot to deloy enor Inut: A et of re A S = {,,, n }. Outut: A th Π D = n which the robot ue for enor deloyment. : Initilly, et ll elected re i uniited nd the uniited counter k = n. Let the th Π D =. : while k! = do : For ech edge on i i+ in th Π D nd eery uniited re j, we drw n ellie which ue i nd i+ it foci nd i tngent to j. Let j be the tngent oint. See () for illutrtion. If elect j to iit between i nd i+, the ditnce dded to the th Π D will be i j + j i+ i i+. : It i obiou tht we wnt to elect the uniited re which dd the let ditnce to th Π D. For exmle, in Figure (b),, hence, i better choice thn j. Aume we elect which i the bet for ll edge in Π D nd ll uniited re, we mrk iited, nd inert between i nd i+ in Π D. Thu the number of edge in the th incree by one. k = k. : end while Senor Fig.. Pth Π D (red line) generted by Algorithm. Here, green qure re the oition to lce the enor (lo the turn oint of the robot). Senor Fig. 6. The deloyed enor (green qure) nd their enor rnge (olid circle) fter the enor deloyment he. i j j i+ i j j i+ u u u u Senor Pue Point D! D! Fig. 7. The robot-ited dt collection: the robot trel i the blue th to collect dt from ech enor. Here, the green dh circle i the communiction rnge of the enor. () Fig.. () For ech edge on i i+ in th Π D nd eery uniited re j, we drw the ellie which ue i nd i+ it foci nd i tngent to j. The ditnce dded to the th Π D by iiting j i i j + j i+ i i+. (b) We elect the uniited re which dd the let ditnce to th Π D. In thi exmle, i better choice thn j. Notice tht in Ste of Algorithm we need to drw n ellie which i tngent to j. Thi cn be done by two wy. We cn trt with mll ellie nd incree it ize until it reche j. Howeer, how to decide the initil ize of the ellie nd wht ize to incree t ech te re difficult to nwer. The econd wy to do i uing binry erch. We firt rndomly elect oint b inide j. We ue i b + i+ b the mjor xi to drw the ellie which gurntee to interect with j. Then we reduce the mjor xi by hlf, if the ellie doe not interect with j, we incree the mjor (b) xi, otherwie further reduce it. By recuriely doing thi, we cn find the ellie which i tngent to j efficiently. In rctice, if the ening rnge i mll comred with the ditnce between ll re, we cn jut ue the ellie i b to etimte the otiml ellie. Figure how the th Π D generted by Algorithm. Pth Π D rereented by red line i the th tht the robot will follow to lce the n enor, while the green qure re the oition to lce the enor (lo the turn oint of the robot). Figure 6 how the deloyed enor nd their ening rnge fter the deloyment he. It i cler tht eery interet oint i coered t let by one enor. V. ROBOT-ASSISTED DATA COLLECTION After the robot h deloyed the enor, ll enor begin to collect informtion bout the interet oint. All

5 6 8 Pth by the greedy method. P=.67 Pth by genetic trel leeron method. P=.9 Pth by the rooed method. P= greedy method generic method for TSP rooed method Fig. 8. Smle th found by the three method. informtion will be ent to centrlized control center. Howeer, due to the fct tht the communiction rnge of enor i limited, the enor network my be rtitioned to comonent fr wy from ech other. Adding more enor node cn imroe the connectiity, howeer, it i not feible in mny liction, uch ce exlortion with exenie enor deice. In uch cenrio, the mobile robot cn hel. We ume tht the robot i lo equied with communiction deice nd cn collect dt from the deloyed enor. The th lnning for the robot i gin n otimiztion roblem where we try to minimize the totl ditnce treled by the robot. Here, gien the et of deloyed enor S = {,,, n } nd their oition,,, n, we tudy how to chedule the robot to iit certin ue oint U = {u, u,, u k } where the robot cn collect dt from enor, uch tht () the robot cn communicte with eery enor during the round tri nd mke the minimum number k of to; nd () the th Π C = u u u k which the robot will trel to collect dt on thoe ue oint h the minimum totl length. For the firt hlf roblem, we firt ue trnmiion rnge r T of ech enor i to drw the re to be coered, nd then run the greedy lgorithm (Algorithm ) to elect minimum number of ue oint to coer ll enor node. The roblem i eentilly the me the one in the deloyment he excet tht the rnge of coerge i trnmiion rnge r T inted of ening rnge r S. For the econd hlf roblem, we need to chedule the robot to iit thee elected re uing hortet round tri. By uing the me heuritic (Algorithm ), we cn find olution Π C nd return the turn oint u u u k of the robot, hown the blue th in Figure 7. Notice tht if ome enor cn communicte nd trnfer dt with ech other, then it will uffice for the robot to iit only one of thee enor to ick u dt. For thi itution, we cn merge thee enor trnmiion rnge to union re nd ue it ingle re in the inut of Algorithm inted of eerl indiidul re. By king the enor to incree their trnmiion rnge, the connectiity of the enor network cn incree, which will led to le re the robot need to iit. Thi i trdeoff between the communiction cot lu ower conumtion t enor nd the ower conumtion t the robot. For exmle, if the trnmiion rnge of ech enor i infinitely lrge, then the robot doe not need to moe to collect the dt. If the trnmiion rnge i infinitely mll, the robot need to iit ech enor t it oition to collect the dt. VI. SIMULATION STUDIES We crried out eerl imultion exeriment to elute the rooed method. A we he dicued erlier, the enor deloyment nd dt collection re ctully one roblem. Therefore, we only imulte in the context of enor deloyment. Concluion mde from the imultion in enor deloyment cn be lied to the dt collection roblem. In the imultion, ll enor he the me ening rnge. For imlifiction, the iiting oint i of ech re i i choen to be the center of i. Howeer, thi imlifiction doe not undermine the irtul of the rooed roch. In the imultion, we comred the trel ditnce by the rooed roch with two trditionl method: greedy method nd ner-otiml olution from treling lemn roblem. In the greedy method, the robot trt from the current interet oint nd goe to the next oint which i cloet to the current one until ll interet oint he been iited nd return to the originl oition. The ner otiml olution from treling lemn roblem i obtined by genetic lgorithm [9]. In the imultion, we wnt to know how the rooed roch erform with regrd to the number of interet oint nd the ening rnge comred with the two trditionl method. Figure 8 how ce of the imultion, in which interet oint, hown blck qure dot, re rndomly generted within field. In thi ce, the ening rnge i 8. The totl trel ditnce found by the greedy method, the generic TSP method, nd the rooed method re.67,.9 nd A we cn ee, when there i oerling between the dik, the trel ditnce found by the rooed roch cn be coniderbly mller thn the ditnce found by the genetic TSP method nd the the greedy method on the interet oint. In the next two imultion exeriment, we will comre the three method with regrd to the number of interet oint nd the ening rnge. Figure 9 how the exeriment in eluting the rooed method with regrd to the number of

6 Fig greedy method rooed method generic t method The length of trel th Number of interet oint Trel ditnce comrion with regrd to number of interet oint. Fig.. greedy method rooed method generic t method The length of trel th Sening rnge Trel ditnce comrion with regrd to ening rnge interet oint nd fixed ening rnge. The ening rnge in thi imultion i 8, nd the number of interet oint rie from to 9. We cn ee tht the totl trel ditnce incree lmot roortionlly to the number of interet oint. Among the three method, our rooed roch cn lwy chiee the mllet trel ditnce. Figure how trel ditnce found by the three method with regrd to fixed number () of rndom interet oint nd riou ening rnge (from to 9). It how tht for the greedy method nd the genetic TSP method, the totl trel ditnce do not ry much with regrd to the ening rnge, which i cued by the fct tht they lwy go to the interet oint inted of the oerling re. The trel ditnce found by the rooed roch, howeer, tedily decree with increing ening rnge. Thi demontrte tht when the ening rnge i lrge, there i high robbility of oerling, therefore treling only to the oerling region will e lrge mount of time nd energy cot. Since thi er doe not intend to rooe method for finding ner otiml olution for the treling lemn roblem, we did not comre the th finding lgorithm, decribed in Algorithm, in the rooed method with the greedy method nd the genetic method. VII. CONCLUSION In thi er, we tudied how to ue mobile robot to it the enor deloyment nd dt collection for unmnned exlortion or monitoring uing enor network. Gien et of interet oint, we rooed et of heuritic to () decide minimum number of enor deice to coer ll the interet oint; () chedule the robot to lce the enor deice in certin oition o tht the th of the robot i minimum; nd () chedule the robot to iit nd communicte with thee enor deice to collect dt o tht the th of the robot i minimum. We lo erified the erformnce i imultion, nd the reult demontrted nice erformnce of our method comred with the greedy lgorithm nd genetic lgorithm for TSP. REFERENCES [] I. F. Akyildiz, W. Su, Y. Snkrubrmnim, nd E. Cyirci, Wirele enor network: urey, Comuter Network, ol. 8, no.,. 9,. [] A. Howrd, M. J. Mtric, nd G. S. Sukhtme, Mobile enor network deloyment uing otentil field: A ditributed, clble olution to the re coerge roblem, in Proc. of 6th Interntionl Sym. on Ditributed Autonomou ic Sytem (DARS),. [] T.-L. Chin, P. Rmnthn, K. K. Sluj, nd K.-C. Wng, Exoure for collbortie detection uing mobile enor network, in Proc. of IEEE Interntionl Conf. on Mobile Adhoc nd Senor Sytem,. [] G. Wng, G. Co, T. L. Port, nd W. Zhng, Senor reloction in mobile enor network, in Proceeding of IEEE INFOCOM,. [] Y. C. H. Yongguo Mei, Yung-Hing Lu nd C. S. G. Lee, Reducing the number of mobile enor for coerge tk, in Proceeding of Interntionl Conference on Intelligent nd Sytem,. [6] C. Wu, W. Sheng, nd Y. Zhng, Mobile elf-locliztion uing multidimenionl cling in robotic enor network, Int l J. on Intelligent Control nd Sytem,. 6-7, ol., no., 6. [7] W. Sheng, G. Tewolde, nd Y. Guo, Ditributed robot-ited node locliztion in ctie enor network, in Proc. of the 6 IEEE Interntionl Conference on Mechtronic nd Automtion, 6. [8] H. Lee, H. Dong, nd H. Aghjn, -ited locliztion technique for wirele imge enor network, in Proc. of the Annul IEEE Communiction Society Conference on Senor, Meh nd Ad Hoc Communiction nd Network, 6. [9] P. Corke, R. Peteron, nd D. Ru, Locliztion nd nigtion ited by networked cooerting enor nd robot, Interntionl Journl of ic Reerch, ol., no. 9, ,. [] Y. Mei, C. Xin, S. D, Y. C. Hu, nd Y.-H. Lu, Relcing filed enor node by mobile robot, in Interntionl Conference on Ditributed Comuting Sytem, Workho on Wirele Ad hoc nd Senor Network (WWASN), 6. [] A. Pnouooulou, E. Koly, Y. Koeo, A. Tze, nd J. Lygero, -ited trget locliztion uing cooertie cheme relying on wirele enor network, in Proceeding of IEEE Control nd Deciion Conference, (CDC), 6. [] M. Crdei nd J. Wu, Energy-efficient coerge roblem in wirele d hoc enor network, Comuter Communiction Journl (Eleier), ol. 9, no.,., 6. [] J.-C. Ltombe, Motion Plnning. Sringer, 99. [] P. C.-Y. Sheu nd Q. Xue, Intelligent ic Plnning Sytem. World Scientific Pub Co Inc, 99. [] T. H. Cormen, C. E. Leieron, R. L. Riet, nd C. Stein, Introduction to Algorithm. MIT Pre, Cmbridge, Mchuett,. [6] C. Lund nd M. Ynnkki, On the hrdne of roximting minimiztion roblem, Journl of the ACM (JACM), ol., no., , 99. [7] U. Feige, A threhold of ln n for roximting et coer, J. ACM, ol., no.,. 6 6, 998. [8] A. Dumitrecu nd J.S.B. Mitchell, Aroximtion lgorithm for TSP with neighborhood in the lne, in Proc. of ACM Symoium on Dicrete Algorithm,. [9] J. Kirk, Treling Slemn Problem - Genetic Algorithm, htt://