Look-ahead Geographic Routing for Sensor Networks

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1 1 Look-ahead Geographic Routig for Sesor Networks Jiaxi You 1, Domiik Lieckfeldt 1, Qi Ha, Jakob Salzma 1, ad Dirk Timmerma 1 1 Uiversity of Rostock, Germay 1 {jiaxi.you, domiik.lieckfeldt, jakob.salzma, Colorado School of Mies, US bstract s sesor etworks are deployed over various terrais, the complexity of their topology cotiues to grow. Voids i etworks ofte cause existig geographic routig algorithms to fail. I this paper, we propose a ovel geographic routig algorithm called Forwardig with VIrtual Positio (ViP). We itroduce virtual positio as the middle positio of all direct eighbors of a ode. Istead of comparig odes real geographic positios, ViP employs virtual positios whe selectig the ext hop. Such virtual positio reflects the eighborhood of a sesor ode, as well as the tedecy of further forwardig. For sparsely-deployed etworks, ViP icreases success rate of packet routig without itroducig sigificat overhead. Furthermore, multiple levels of virtual positio ca be obtaied with localized iteratio. We propose the Forwardig with Multi-level Virtual Positio (MVP) algorithm ad the Forwardig with Hierarchical Virtual Positio (HVP) algorithm. Differet levels of virtual positios ad their combiatios are used alteratively to icrease success rate of packet routig i sesor etworks. Key words geographic routig, greedy forwardig, routig hole, wireless sesor etworks. I. INTRODUCTION Wireless Sesor Networks (WSNs) are composed of may sesor odes, which are capable of sesig physical parameters, processig data ad commuicatig wirelessly [1]. Sesor odes are usually battery powered ad left uatteded after deploymet. Sice radio commuicatio amog sesor odes is the mai drai of eergy i WSNs, eergy efficiecy remais a critical desig issue for WSNs. mog various routig algorithms, o-demad routig algorithms such as ODV [] are popular, while their floodig-based route discovery ofte lead to high cotrol overhead [3]. I cotrast, sigle-path geographic routig with Forwardig (GF) is attractive for WSNs [1]. I a basic GF algorithm, a ode commuicates oly with its direct eighbors (1-hop). The eighborig ode that further miimizes the remaiig distace of a packet to its destiatio will be selected as the ext hop. Such localized approach is This work is partially fiaced by the Germa Research Foudatio (DFG) (post graduate program MuSM, GRK 144). Fig. 1. example of ad MFR geographical routig. packet is set from ode S to ode D. The algorithm selects ode as the ext-hop, sice is the eighbor that is the closest to the destiatio. The MFR algorithm selects ode B as the ext hop, sice B makes the greatest progress alog the directio of packet forwardig (S-D). Both ad MFR requires the ext hop to be closer to the destiatio tha the curret ode. effective ad ca be dyamically adapted to chages, which oly require positio iformatio of sesor odes [1]. The algorithm [4] ad the Most Forwardig progress withi Radius (MFR) algorithm [5] are two fudametal approaches of GF based o distace. Fig. 1 illustrates the priciple of the algorithm ad the MFR algorithm. s moder WSNs are becomig popular i various applicatios, their topology is becomig complicated. Due to limited precisio of deploymet, voids ca cause routig holes i the etwork, where ofte lead traditioal GF algorithms to fail [6]. The reaso is the local miimum pheomeo illustrated i Fig.. I sigle-path GF routig algorithm (e.g. or MFR), forwardig of packets towards the sik ca fail at ode, sice there is o direct eighbor closer to the destiatio tha ode itself. Besides, ueve power cosumptio o sesor odes ca also lead to ew routig holes i the later lifetime of WSNs. The objective of our work is to improve the success rate of GF for sparse WSNs or WSNs with small routig holes. To this ed, we preset a ovel geographic routig algorithm sik hole Fig.. example of local miimum.

2 amed Forwardig with VIrtual Positio (ViP)", as well as its improved versios called the Forwardig with Multi-level Virtual Positio (MVP)" algorithm ad the " Forwardig with Hierarchical Virtual Positio (HVP)" algorithm. Each proposed algorithm has two variats usig the ad MFR priciple respectively. The mai advatage of our approaches is that the algorithms simply employ GF throughout the routig process, ad iheretly results i high routig efficiecy as the basic GF algorithms. I the mea time, the amout of cotrol overhead of the proposed algorithms is strictly limited. The basic assumptio of our approaches is that each ode i the etwork is aware of its ow geographic positio as well as the positios of its direct eighbors. II. RELTED WORK Various geographic routig algorithms [6] have bee proposed i recet years to address the local miimum problem of GF. Most of the routig algorithms are composed of two phases. Such algorithms start with a basic GF algorithm, ad recover from local miimum with their complemetary hole-bypassig phases. The Perimeter Stateless Routig (GPSR) is oe of the fudametal algorithms based o plaar graphs [7]. plaar graph is a sub-graph with o-crossig edges, which represets the same coectivity as the origial etwork. GPSR uses GF ad switches to its perimeter mode whe a local miimum is met. The right-had rule is employed i the perimeter mode, where packets are routed couterclockwise alog the edge o the face of a plaar graph. Bose et al. proposed the FCE-1 ad FCE- [8] algorithms that use the perimeter of the plaar graph formed at each ode. Such approaches have high success rate of packet routig, but also high cotrol overhead due to the plaarizatio processes ad the maiteace of the plaar graph iformatio o every sesor ode. I the abovemetioed -phase algorithms, the performace durig the GF phase is much better tha durig their complemetary phases [9]. To icrease the ratio of GF durig routig, Liu et al. preseted the idea of liged Virtual Coordiate System (VCS) [9]. Virtual coordiates are based o iteger umber of hops to the referece odes (achor odes) ad represet a coarse approximatio of ode locatio. The success rate of greedy routig is improved with VCS, while floodig of cotrol message from achor odes to the whole etwork causes large cotrol overhead durig the settig up of VCS. I [10], Stojmeovic et al. proposed to use GF with iformatio of -hop eighbors (-hop GF). The success rate of the GF algorithms is improved with each ode aware of its direct eighbors as well as its -hop eighbors (eighbors of its eighbors). mog all direct ad -hop eighbors, packets are set to the ode which is the closest to the destiatio via 1-hop or -hop forwardig. Sice farther eighbors are available for the selectio of ext-hop, small routig holes ca be avoided efficietly. The authors also proved that such GFbased methods are iheretly loop-free. The method i [10] solely employs the basic idea of GF durig routig. Such a method limits cotrol overhead sice it oly requires local kowledge of sesor odes (iformatio of 1-hop ad -hop eighbors) ad avoids sophisticated complemetary phases to recover from local miimums. III. LGORITHM DESIGN s metioed earlier, -hop GF uses extra kowledge of - hop eighbors i GF. Such iformatio o farther eighbors improves the success rate of GF i a straight-forward maer. However, -hop GF is ot strictly localized to the direct eighbors, therefore, it is ot scalable for further extesio. For example, whe further improvemet of success rate is demaded, odes eed to have iformatio of eighbors withi N hops (N>=3). For a D wireless etwork, the umber of eighbors stored o each sesor odes icreases proportioally to N. Similarly is the cotrol overhead for the maiteace of eighbor iformatio withi N hops.. Forwardig with Virtual Positio (ViP) We itroduce the virtual positio of a ode as the middle poit of all its direct eighbors. If a ode (x,y ) has direct eighbors V,1 (x,1,y,1 ), V, (x,,y, ),, V, (x,,y, ), the virtual positio of ode is: Fig. 3. example of usig virtual positios i GF. The packet is routed from ode 10 to ode 13. Whe usig the geographic positios of odes, ode 1 becomes a local miimum for both ad MFR, as show i (a). With virtual positios, a path ca be foud by MFR-ViP, as show i (b). With d -level virtual positio, both -MVP (-level) ad MFR- MVP (-level) fid paths aroud the hole, as show i (c).

3 3 ( x ', y ') = (( x, 1 + x, + K+ x, ) /,( y,1 + y, + K+ y, ) / ) Each ode calculates its virtual positio accordig to (1), ad broadcasts its virtual positio to its direct eighbors. The iformatio of virtual positio is stored o odes themselves ad their direct eighbors. I other words, each ode has the kowledge of its ow virtual positio, ad the virtual positios of its direct eighbors. Fig. 3(a) shows the geographic positios of odes. Fig. 3(b) shows the calculated virtual positios of odes. The virtual positio of ode 11 is the same as its real geographic positio, sice its direct eighbors (ode 6, 7, 8, 10, 1, 14, 15, 16) are located uiformly i the trasmissio circle cetered at the geographic positio of ode 11. I cotrast, the virtual positio of ode 1 (Fig. 3(b)) locates to the left of the geographic positio of ode 1 (Fig. 3(a)), because the positios of its direct eighbors (ode 7, 8, 11, 15, 16) are left-biased i the trasmissio circle of ode 1. The virtual positio of a ode provides a idicatio of how the direct eighbors are located aroud the ode o average, hece it is a suitable metric to demostrate the tedecy of further forwardig durig geographic routig. Cosider the example i Fig. 3, for a packet that eeds to be set from ode 10 to ode 13, both ad MFR get stuck at ode 1, sice there is o eighbor that ca make further progress towards the destiatio. I cotrast, the virtual positio of ode 1 is strogly left-biased due to the void o the right side of ode 1. s a result, MFR-ViP fids a path ( ) aroud the hole usig virtual positios of odes. We propose Forwardig with Virtual Positio (ViP), a look-ahead geographic routig algorithm based o the coordiate system of virtual positios. Istead of usig farther eighbors as i -hop GF, ViP uses the virtual positios of odes to ivolve farther eighbors i the lookahead routig process. Our algorithm is strictly localized, where a ode oly iteracts with its direct eighbors. ViP has two variats called ad MFR-ViP, which are based o the priciple of the ad MFR algorithm, respectively. s a example for a D WSN, whe a packet with destiatio D(x D,y D ) arrives at ode (x,y ), - ViP evaluates its ow virtual positio (x,y ) ad the virtual positios of its direct eighbors. The eighbor has the virtual positio that is the closest to ode D is selected as the ext hop. ccordig to the algorithm, the virtual positio of the selected eighbor must be also closer to the destiatio tha the virtual positio of curret ode. Namely, the exthop ode V,i of meets the followig two coditios: arg mi( ( x ' x ) + ( y ' y ) ) () ad: V, i, i D, i D (1) (x,i ' x D ) + (y,i ' y D ) < ( (x ' x D ) + (y ' y D ) (3) Similarly, MFR-ViP selects the eighbor that makes the greatest progress towards the destiatio of a packet. Give the vectors D ad DV,i, the ext-hop ode V,i of MFR-ViP is the oe that has the miimal dot product of the two vectors: arg mi( D DV ) (4) V, i ad also satisfies equatio (3). ad MFR-ViP termiate whe there is o eighbor that has the virtual positio to make further progress towards the destiatio of a packet. With this idea, odes oly eed to iteract with their direct eighbors to obtai iformatio of virtual positio. Therefore, settig up virtual positios is strictly localized. The message overhead i the worst case is O(M), where M is the total umber of odes i a etwork. B. Forwardig with Multi-level Virtual Positio (MVP) To further improve the success rate of GF, we itroduce the cocept of multi-level virtual positio that cosiders farther odes (eighbors of K-Hop, K>=1). If we refer to the virtual positio derived usig Equatio (1) as the 1 st -level virtual positio, the the K th -level virtual positio (K>=1) of ode ca be doe with K iteratios of local broadcast betwee odes ad their direct eighbors V,1 (x,1,y,1 ), V, (x,,y, ),, V, (x,,y, ), as follows. ( x K, y K ) = (( x,1 ( y,1 + x, + y,, i + K + x + K + y,, ) /, ) / ) Such K th -level virtual positios of odes idicate how the K- hop eighbors are located o average, also the tedecy of further forwardig durig geographic routig. Usig this cocept, we itroduce the Forwardig with Multilevel Virtual Positio (MVP)" algorithm based o K th -level virtual positio (K>=1), where each ode kows its ow K th - level virtual positio, ad the K th -level virtual positios of its direct eighbors. MVP equals ViP whe K is 1. The variats -MVP ad MFR-MVP evaluate the K th -level virtual positios accordig to the priciple of the ad MFR algorithm, respectively. Comparig to ViP, MVP employs a higher level of virtual positios to further improve the success rate of GF. Fig. 3(c) is a example of MVP usig d -level virtual positio, which demostrates that usig virtual positio of higher level idicates better forwardig tedecy i geographic routig, sice it takes farther eighbors ito cosideratio. -MVP ad MFR-MVP stop whe there is o eighbor that has the K th -level virtual positio that makes further progress towards the destiatio of a packet. C. Forwardig with Hierarchical Virtual Positio (HVP) Usig K th -level virtual positio (K>=1) ca itroduce ew local miimums durig routig. For istace, i Fig. 4, a packet eeds to be set from ode 5 to ode 4. Whe usig the geographic positios of odes, the packet ca be successfully delivered (path i Fig. 4(a)). While usig 1 st -level virtual positio, ode 5 has o eighbor to make further progress towards ode 4, thus becomes a local miimum of - (5)

4 4 D. lgorithm Properties I a GF algorithm, a ode eeds to evaluate all the etries of eighbors stored o the odes, ad select the ext hop based o a certai criterio. Therefore, the average storage ad computatioal overhead is proportioal to the umber of eighbor etries o odes. This metric also reflects the cotrol Fig. 4. example of routig holes itroduced by usig virtual positios. ViP ad MFR-ViP. To address this problem, we itroduce the Forwardig with Hierarchical Virtual Positio (HVP) algorithm (Table. 1 shows the pseudo code of HVP). HVP uses the combiatio of all K-level virtual positios (K>=1) ad the geographic positios of odes i a dow-hill fashio. Namely, whe a local miimum is met usig K th -level virtual positio, HVP will lower the level of virtual positio from K th - level to (K-1) th -level (K>=), or from 1 st -level virtual positio to real geographic positio. HVP requires odes to store the geographic positios, as well as all the K-level virtual positios of itself ad its direct eighbors. flag flag_level is added to the packets to idicate the curret level of virtual positio (usig geographic positio whe K=0). I our example (Fig. 4(b)), the packet stuck (usig 1 st -level virtual positio) at ode 5 ca set its flag_level to be 0, ad reach its destiatio via the path by usig the geographic positios of odes. The two variats -HVP ad MFR- HVP start with K th -level virtual positio (K>=1) accordig to the priciple of the ad MFR algorithm, respectively. To esure that HVP is loop-free, such dow-hill process is uidirectioal. Namely, for a packet iitialized with K th -level virtual positio ad adjusted to (K-1) th -level virtual positio after meetig a local miimum, the packet stays o (K-1) th - level ad ca oly be further adjusted to the lower levels (or from 1 st -level virtual positio to geographic positio). PROCEDURE hadle_packet TBLE 1 PSEUDO CODE OF GF-HVP ON NODE N WHILE there is local miimum ND packet.flag_level>=0 { packet.flag_level--; } IF packet.flag_level>=0 { forward packet usig virtual positio of level packet.flag_level idicated by the packet (usig geographic positio whe packet.flag_level is 0); } ELSE { routig failed; } lgorithm GF, GF-ViP, GF-MVP TBLE LGORITHM OVERHED Overhead GF-HVP starts with K th -level virtual positio N-hop GF R: trasmissio rage of odes. D: average umber of odes i a uit area O(π R D) O(π R D (K+1)) O(π (N R) D) message overhead to maitai the iformatio of virtual positios. Table shows cotrol overhead of related algorithms. Here we assume that odes are deployed with the same desity i a D etwork. The proposed ViP, MVP ad HVP algorithms are strictly localized, ad ca termiate based o local kowledge, i.e., they are loop-free. Lemma 3.1. For ay static etwork, ViP is loop-free. Proof: s proved i [10], sigle-path routig algorithms based o the algorithm or the MFR algorithm is iheretly loop-free. ViP employs the virtual positio coordiatio system which is the 1-to-1 mappig of the origial geographic coordiate system of a etwork. Istead of the geographic positios, ViP evaluates the correspodig virtual positios of odes based o the coordiate system of virtual positio. Thus, ad MFR-ViP are also iheretly loop-free. Namly, ViP is loop-free. Lemma 3.. For ay static etwork, MVP is loop-free. Proof: ssume that MVP of K th -level virtual positio (K>=1) is loop free. Sice MVP of (K+1) th -level virtual positio employs the 1-to-1 mappig of the K th -level virtual positio coordiate system. Therefore, MVP of (K+1) th -level virtual positio is loop-free. s proved i Lemma3.1, MVP of 1 st -level virtual positio is loop free. Thus, MVP of K th -level virtual positio is loop free for ay K>=1. Lemma 3.3. For ay static etwork, HVP is loop-free. Proof: HVP starts with K th -level virtual positio (K>=1) ad adjust the flag_level of packets uidirectioally util 0, which refers to the coordiate system of geographic positio. s proved i Lemma3., MVP of K th -level virtual positio is loop free. Therefore, each step i the dow-hill scalig of HVP is loop-free. Thus, HVP starts with K th -level virtual positio is loop free for ay K>=1.

5 5 IV. PERFORMNCE EVLUTION We implemeted the proposed algorithms usig Matlab. The simulated etwork was a 1000 m 1000 m square plae, where 500 sesor odes were radomly deployed. Packets were geerated with radom pairs of source-destiatio addresses. Differet commuicatio rages of sesors are used. We used a simple disc-commuicatio model: sesor odes i the trasmissio rage ca receive sigals from a trasmitter without loss. For the same deploymet desity, smaller commuicatio rage implies sparser layout of a etwork ad bigger chace to ecouter routig holes durig packet forwardig. The objective of our experimets is to demostrate the performace of our algorithm i sparsely-deployed WSNs, or WSNs with small voids due to o-uiform deploymets. We show oly the results of the variat, -MVP ad -HVP. The performace of MFR-ViP, MFR-MVP ad MFR-HVP are similar from our simulatio. The results were compared with the -Hop algorithm [10] ad its extesio, the 3-Hop algorithm. We use the followig metrics to compare the performace of the simulated routig algorithms: Success rate of GF: This is defied as the percetage of packet delivery with radom packet forwardig. 500 packets with radom source-destiatio pairs were geerated ad forwarded i the simulated etwork. Whe the trasmissio rage of odes is small, voids appear frequetly i the etwork. This metric illustrates the ability of avoidig routig hole of the simulated algorithms. Delay of GF: This is defied as the average hop couts of packets from their source to the destiatio. Give the same success rate, small packet delay (i hops) represets better performace i terms of routig efficiecy ad eergy cosumptio. lgorithm Overhead: The umber of eighbor etries stored o sesor odes implies the storage overhead, as well as the computatioal overhead by choosig of exthop. Furthermore, the cotrol message overhead of maitaiig eighbor iformatio is also proportioal to the umber of etries. Experimetal Results: Fig. 5 shows the average percetage of successful packet forwardig from our simulatio. Fig. 5(a) compares the performace of the algorithm, Hop-, - ViP ad -MVP (usig d -level virtual positios). Whe the commuicatio rage of sesor odes is short, odes have relatively few eighbors, which leads to low success rates of the simulated algorithms. The Hop- ad cosider the -hop eighbors of odes. s the commuicatio rage icreases, Hop- ad demostrate similar success rates, which are higher comparig to. -MVP usig d -level virtual positio results i eve better success rate sice it cosiders 3-hop eighbors. ll the simulated algorithms reach full delivery of packets i deselydeployed etworks (whe trasmissio rage reaches 10m i verage Success Rate verage Success Rate 100% 90% 80% 70% 60% 50% 40% 30% 0% 10% Hop % 90% 80% 70% 60% 50% 40% (a) 30% 0% -HVP (1 level) 10% -HVP ( level) 0 (b) Fig. 5. verage success rate of the algorithms with differet commuicatio rages Fig. 5(a)). Fig. 5(b) illustrates the performace of the dowhill scalig i the -HVP algorithms start with 1 st -level ad d -level virtual positios. Comparig to the correspodig -MVP algorithms usig the same level of virtual positios, -HVP has higher success rate due to its additioal search o virtual positios of lower levels ad the geographic positios of odes. Fig. 6 shows the average delay of the successfully delivered packets from our simulatio. Whe the etwork is sparselydeployed (trasmissio rage < 70m i Fig. 6(a)), the legth of routig paths icreases with the icreasig of the commuicatio rage. The reaso is that more packets with distat source ad destiatio ca be delivered whe the coectivity of a etwork icreases. Hop-, - ViP ad -MVP (usig d -level virtual positio) have bigger average delay, sice additioal packet with sourcedestiatio pair of loger distace ca also be delivered with the look-ahead algorithms. s the trasmissio rage reaches 70m (Fig. 6(a)), the hop cout starts to fall due to the high success rates ad the icreasig step-size of forwardig. We observe that the results of the simulated algorithms are similar whe the success rates reach 100%. Therefore, the look-ahead routig algorithms (Hop-, ad - MVP) have similar routig efficiecy as the algorithm. Fig. 7 illustrates the cotrol overhead of the simulated

6 6 verage Delay [hop cout] verage Delay [hop cout] (a) Hop HVP (1 level) -HVP ( level) 1 (b) Fig. 6. verage delay of the algorithms with differet commuicatio rages algorithms idicated by the umber of eighbor etries o odes. With similar performace i terms of success rate, results i lower storage ad computatioal overhead compared to the -Hop algorithm. s a extesio of the -Hop algorithm, the rapidly icreasig overhead of the 3-Hop algorithm shows that the N-Hop idea is ot scalable. I cotrast, the umber of eighbor etries of our ad - MVP algorithms stays low as the algorithm. The storage ad computatioal overhead of -HVP icreases liearly alog with the level of virtual positio it starts with. V. CONCLUSION ND FUTURE WORK I this paper, we preset a ovel geographic routig algorithm amed Forwardig with Virtual Positio (ViP)", as well as its two extesios called the Forwardig with Multi-level Virtual Positio (MVP)" algorithm ad the Forwardig with Hierarchical Virtual Positio (HVP) algorithm. Each of the three algorithms has two variats based o the priciple of the algorithm ad the Most Forwardig progress withi Radius (MFR) algorithm, respectively. Simulatio results demostrate that our proposed algorithms improve the success rate of geographic routig for sparsely-deployed WSNs ad Etries of Neighbors ,, -MVP Hop- 3Hop- -HVP (1 level) -HVP ( level) 0 Fig. 7. Number of eighbor etries with differet commuicatio rages WSNs with small routig holes. The proposed algorithms solely employ Forwardig (GF) throughout the routig processes, ad iheretly results i high routig efficiecy as the basic GF algorithms. Furthermore, the cotrol overhead of our proposed algorithms is strictly limited. For future work, the trade-off betwee the overhead caused by usig higher levels of virtual positios ad the resulted performace will be formally aalyzed. Cotext iformatio such as remaiig eergy will be cosidered as the routig parameter for our algorithms. REFERENCES [1] K. kkaya ad M. Youis, " survey of routig protocols for wireless sesor etworks", Elsevier d Hoc Network Joural, 005, Vol. 3/3, pp [] C.E. Perkis ad E.M. Beldig-Royer, "d hoc o-demad distace vector (ODV) routig", i IEEE Workshop o Mobile Computig Systems ad pplicatios, Feb [3] Pham, N. N., You, J., ad Wo, C. " Compariso of Wireless Sesor Network Routig Protocols o a Experimetal Testbed", I Proceedigs of the IEEE iteratioal Coferece o Sesor Networks, Ubiquitous, ad Trustworthy Computig - Vol - Workshops - Volume 0 (Jue 05-07, 006). SUTC. IEEE Computer Society, Washigto, DC, [4] G. G. Fi. "Routig ad addressig problems i large metropolitascale iteretworks", Techical Report ISI/RR , Iformatio Scieces Istitute, Mars [5] H. Takagi ad L. Kleirock, "Optimal Trasmissio Rages for Radomly Distributed Packet Radio Termials", IEEE Tras. Comm., vol. 3, o. 3, pp , [6] Nadeem hmed, Salil S. Kahere, Sajay Jha, "The holes problem i wireless sesor etworks: a survey", CM SIGMOBILE Mobile Computig ad Commui-catios Review, v.9., pril 005. [7] Brad Karp ad H. T. Kug, "GPSR: perimeter stateless routig for wireless etworks", i Proceedigs of the CM/IEEE MobiCom '000, pages 43-54, 000. [8] P. Bose, P. Mori, I. Stojmeovic, ad J. Urrutia, "Routig with guarateed delivery i ad hoc wireless etworks", Wireless Networks, 7: , 001. [9] Ke Liu, Nael bu-ghazaleh, "liged Virtual Coordiates for Routig i WSNs", Mobile dhoc ad Sesor Systems (MSS), 006 IEEE Iteratioal Coferece o, vol., o., pp , Oct [10] Stojmeovic, I. ad Li, X, "Loop-Free Hybrid Sigle-Path/Floodig Routig lgorithms with Guarateed Delivery for Wireless Networks", IEEE Tras. Parallel Distrib. Syst. 1, 10 (Oct. 001),

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