GMP: Distributed Geographic Multicast Routing in Wireless Sensor Networks

Transcription

1 GMP: Ditribte Geographi Mltiat Roting in Wirele Senor Netork Shibo W, K. Selçk Canan Department of Compter Siene, Arizona State Unierity, Tempe, AZ, USA {hibo., Abtrat In thi paper, e propoe a noel Geographi Mltiat roting Protool (GMP) for irele enor netork 1. The propoe protool i flly itribte an tatele. Gien a et of the etination, the tranmitting noe firt ontrt a irtal Eliean Steiner tree roote at itelf an inling the etination, ing a noel an highly effiient retion ratio heriti (alle rrstr). Bae on thi loally ompte tree an the information regaring the loation of it immeiate neighbor, the tranmitting noe then plit the etination into a et of grop an allate a next hop for eah of thee grop. A opy of the paket an the loation of the orreponing grop of etination noe are irete toar the orreponing hop. The imlation relt on NS2 ho that the aerage per-etination hop ont obtaine ing GMP i omparable to the exiting PBM [21] algorithm an ignifiantly le than obtaine by ing LGS [5]. Mot ignifiantly, GMP reqire 25% le hop an energy than alternatie algorithm. 1. Introtion an Relate Work Geographi niat roting protool [4, 13, 31, 17] e loation information to eliminate expenie irele netork operation. In ontrat to ingle-ore ingle-etination niat heme, grop ommniation heme like geoating [15, 2, 28] an mltiating aim at ientifying one-to-many tranmiion path. Mltiating (a oppoe to mltiple niating) preere netork reore by reing renant meaging. Mot exiting mltiat roting protool maintain a itribte trtre for the eliery of mltiat paket. In tree-bae trtre [7, 3, 30] there i only one path for eah etination; mltiple etination may hare part of their path. In meh-bae trtre [18, 10, 19, 8], there may be mltiple path from a gien ore to eah etination. Unfortnately, topology hange, noe failre, an grop memberhip hange an rener the ommniation an reonfigration oerhea of maintaining a itribte tree or meh trtre naeptably high. 1 Thi ork i pporte by NSF grant # , ARIA - Qality- Aaptie Meia-Flo Arhitetre for Senor Data Management In ontrat, in ore-roting bae heme (h a Dynami Sore Mltiat, DSM [6]), the entire mltiat tree i reate by the ore noe in aane an inle in the paket. In DSM, a minimm panning tree bae heriti i e to reate thi roting graph. Eah reeiing noe on thi path eoe the mltiat tree information an rote the paket to the next noe a eie by the ore. Unlike DSM, in Loation-gie Tree (LGT) [5], eah noe only nee to kno it on loation an the loation of it neighbor. The ore noe loally ontrt a mltiat tree oniting of itelf an the etination noe. After reeiing a paket, eah btree root extrat the orreponing et of etination an then repeat the ame proe to partition thi et into bet of etination. [5] preent to tree ontrtion algorithm bae on thi nerlying heme: loation-gie k-ary tree (LGK) an loation-gie Steiner tree (LGS). LGS approximate the Stenier tree ing minimm panning tree (MST) of the etination noe. We note that the LGS heme oerly ontrain the mltiat tree it an generate. In partilar, ine eah noe reate potential mltiat tree ing only the etination noe themele, the tree that it an generate are limite. In ontrat, in thi paper, e propoe a heme hih oe not pt the ame ontraint on the mltiat tree explore. Poition Bae Mltiating (PBM) [21] i another protool hih make foraring eiion bae on loal knolege. Unlike LGS, hoeer, it jointly optimize (a) the progre of the paket toar the etination an (b) the banith age. By oniering all poible bet of it neighbor an aigning eah etination to the loet neighbor in the bet, PBM ientifie a bet hih minimize the optimization riterion. Note that ine eah poible bet of the neighborhoo ha to be oniere, the PBM algorithm an be ery otly hen there are large nmber of neighbor an etination. Frthermore, traeoff beteen remaining oerall itane toar the etination an banith age i not triial Contribtion of thi Paper In thi paper, e propoe a irtal Eliean Steiner tree bae mltiat roting protool here (a) tranmitting noe o not reqire any global knolege to reate a tree hih ill be e for partitioning the etination noe 1

2 into grop an (b) they e only loal information ring atal rote eletion. Note that the general Eliean Steiner tree problem i NP-har [14]. Hoeer, a peial ae here there are only three noe, the Steiner point an be allate effiiently a in [24, 11]. Or algorithm, Geographi Mltiat roting Protool (GMP), exploit thi to reate heriti Eliean Steiner tree effiiently (in polynomial time). The progreie natre of the roting heme enable ontino refinement of the relting tree, thereby proiing better mltiat tree than imilar heme ie aboe. A ie aboe, hile reating the minimm panning tree for partitioning the etination, LGS [5] oe not onier any geographi point other than the atal etination themele. PBM [21] trie to balane the peretination hop ont ith the total nmber of hop neee to reah all etination, ith the help of a trae-off parameter. The optimal ale of thi parameter, hoeer, hange from tak to tak an epen on the nmber of neighbor an the itribtion of etination. Th, hooing a ingle itable parameter ale i not eay. In thi paper, e propoe GMP, an effiient Eliean Steiner tree bae geographi mltiat roting protool. The nerlying iea of GMP i that eah tranmitting noe ontrt a heriti Eliean Steiner tree, inling the ore an all etination. The tree i irtal in the ene that it may inle interior ertie that o not orrepon to any atal irele enor noe. The etination are iie into grop bae on thi tree. A in LGS a opy of the paket i then forare to a itable next hop an the proe i repeate by the reeiing noe ntil all etination are reahe. Eentially, there are to main ifferene beteen GMP an LGS: (1) GMP e an effiient an effetie heriti to ontrt Eliean Steiner tree (hih allo all poible Eliean point) hile plitting etination into partition an (2) the betination (root of the btree) toar hih the paket i forare i ontraine to be an atal etination in LGS, hile in GMP it an be any Eliean Steiner point. A hon in Setion 5, thee flexibilitie relt in ignifiantly better mltiat tree. 2. Wirele Netork Moel In thi paper, e aopt a ommonly e enor netork moel [31, 13, 32, 27]: A et, S, of noe i loate in a to imenional geographi area, G. Eah noe i S ha oorinate, oor( i )= x i,y i. Eah noe kno it on oorinate. Thi an be ahiee either throgh an internal GPS eie or throgh a eparate alibration proe [13]. The loation of a noe at a it ID an it netork are. Therefore, there i no nee for a eparate ID etablihment protool. Eah paket i marke ith the loation of the Figre 1. A mltiat tree example. Ho the tree i reate i eribe in Setion 4. next hop an the orreponing noe pik p the paket. The ore noe (generally a prime noe) kno the etination prior to the iemination of the ata paket. In literatre, there are ork ranging from tati grop memberhip [12, 6] to highly ynami enario pporte by the ore noe [25, 5] or a eparate grop management erie [20]. In thi paper, e o not fo on the problem of ho to etablih an maintain mltiat grop. 3. rrstr: An Effiient REDUCTION RATIO Heriti for Eliean Steiner Tree The GMP mltiat roting algorithm e introe in thi paper i bae on Eliean Steiner tree. Note that generating optimal Eliean tree i a otly operation [14] hih nee to be aoie. There are many heriti [23, 1, 33] an approximation algorithm [34, 26] to are thi problem. Althogh they are mh heaper then optimal oltion, mot approximation algorithm are till too otly to be eploye at enor noe. Some exiting heriti [22] apply to retilinear Steiner problem an ome are minimm panning tree bae algorithm [23, 26, 33]. Thee general prpoe heriti o not neearily fit ell for geographi mltiating. In fat, e note that (a) e to the lak of p-to-ate global knolege of the tate of the irele netork, the Steiner tree point ompte by any algorithm are not likely to be atal hop that ill be e in the relting rote an (b) eah reeiing noe in the netork ill hae the opportnity to reajt the Steiner tree bae on it on poition. Therefore, intea of relying on expenie approximation or general prpoe heriti, in thi paper, e fo on the folloing oberation: Oberation 1: hen to etination are far aay from the ore bt are loe to eah other, they are likely to hare bpath. For example, in Figre 1, etination {, } are likely to hare bpath. Oberation 2: hen the angle of the line egment onneting the ore noe an the etination are mall, the noe are likely to hare bpath. For example, in Figre 1, etination {,, } are likely to hare bpath ith etination. Sine hen there are only three noe, the exat Steiner point an be allate effiiently [24, 11], the oberation

3 p q r 1 (,r)+(r,p)+(r,q) (,p)+(,q) t 1 (,t)+(t,)+(t,) (,)+(,) 1 (,)+(,) (,)+(,) (a) (b) Figre 2. Retion ratio 1 (,)+(, ) (,)+(, ) enable to eelop a noel algorithm to reate heriti Eliean Steiner tree effiiently (in polynomial time) Retion Ratio Before e introe the propoe heriti to reate Eliean Steiner tree, e firt introe a noel meare, alle retion ratio, hih niformly aptre thee to oberation to gie the ontrtion of Eliean Steiner tree. Gien a etination pair (, ) an a ore noe, retion ratio, RR(,, ), i efine a follo: RR(,, ) =1 (, t)+(t, )+(t, ) (, )+(, ) Here t i the exat Eliean Steiner point of thee three noe {,, } [24, 11]. In the ret of the paper, the ore noe i impliit hen e refer to retion ratio. The retion ratio meare ha the folloing propertie (the proof are omitte e to pae ontraint): The ale of retion ratio i alay le than 1/2. Gien to eqiitant etination, the retion ratio i larger if thee to etination are frther aay from the ore. For example, in Figre 2(a), the retion ratio of (p, q) i larger than the retion ratio of (, ). Gien a pair of etination, the retion ratio i larger if the angle beteen the to line egment onneting the ore noe an the to etination i maller. For example, in Figre 2(b), the retion ratio of (, ) i larger than the retion ratio of (, ) Bai rrstr Algorithm In Figre 3, e preent an iteratie algorithm, rrstr, to ontrt a Eliean Steiner tree, bae on thee propertie of the retion ratio meare. In thi betion, e proie an oerie of thi algorithm. Gien a et of etination point, initially, ore noe mark all it mltiat etination a atie; that i, none of the etination are oere yet. In eah iteration, the algorithm ientifie a etination pair, (, ), ith the larget retion ratio. Gien thi pair, rrstr reate a irtal etination at the loation of the Steiner point of noe {,, }. Note that hen there are only three noe, the Steiner point an be allate effiiently [24, 11]. The rrstr(, et, rr): Thi fntion allate the Steiner tree for noe an etination in et i the rrent noe; et, the lit of all etination; rr, the raio range of ; T i the tree to be retrne; { 1. et all etination an all pair of etination to be atie; 2. allate the retion ratio an Steiner point for all pair of etination ith ; 3. repeat fin an atie pair (, ) ith larget retion ratio an let t be the Steiner point of {,, } if no h pair an be fon, break; if i the ame noe ith a ege to T, eatiate noe ; ele if t i olloate ith a ege an to T, eatiate an ; ele if t i olloate ith / a ege / to T, eatiate /; ele if both (, ) an (, ) le than rr eatiate the pair (, ); ele if (, )/(, ) i le than rr if rr+(t, )+(t, ) i larger than (, )+(, ), eatiate pair (, ); ele a ege / to T, an eatiate /; ele (, t) i le than rr, an rr + (t, )+(t, ) i larger than (, ) +(, ) a ege an to T, eatiate an ; ele /**reate a ne irtal etination**/ reate a ne irtal etination at t, atiate, a ege an to T, allate the retion ratio an Steiner point for all pair (, i), here i i an atie etination 4. retrn T ; } Figre 3. Retion-ratio-bae heriti for Eliean Steiner tree generation Figre 4. An example Eliean Steiner tree generate by rrstr orreponing to ege, an, are then ae to the tree. Sine they are alreay oere, both an are marke a inatie o that any etination pair that ontain either or ill not be oniere in the remaining iteration. The ne irtal etination i ae in the et of etination an marke atie. Th, rrstr ill allate retion ratio for etination pair oniting an all remaining atie etination. Note that, in the extreme ae, the Steiner point an be olloate ith or. If for example the Steiner point i olloate ith, then no ne irtal etination nee to be reate. Intea, the ege i inerte into the tree an i marke inatie. tay atie. Alo note that the Steiner point an be olloate ith the ore noe itelf. In thi ae, ege an are inerte to the tree an noe, an, are marke inatie. 2 1

4 (a) (b) Figre 5. A irtal etination (a) may be or (b) may not be benefiial Figre 4 illtrate the proe ith an example. In the firt iteration, pair (, ) i ientifie ine they hae the larget retion ratio, an a irtal etination 1 i reate at the Steiner point of {,, }. Ege 1 an 1 are ae to the tree. Noe an are then eatiate. In the eon iteration, pair ( 1,) i ientifie, an another irtal etination 2 i reate. Ege 2 1 an 2 are ae to the tree. In the thir iteration, pair ( 2,) i ientifie. No irtal etination i reate in thi tep ine the Steiner point of {,, 2 } i at noe itelf. Intea, ege 2 i ae to the tree. At lat, pair (, ) i fon an ege i ae to the tree. Thi algorithm i relate to bt ifferent from onentional ontration bae algorithm, h a [34, 26]. In thi algorithm a fll onnete omponent i ientifie an replae ith a ne point, iteratiely. In rrstr, intea, a etination pair i ientifie an replae ith a irtal etination. One ifferene i that the ore noe i neer ontrate. More importantly, the ontrtion of the Steiner tree i gie ith the retion ratio, hih ientifie thoe pair that are more likely to hare bpath Raio Range aare rrstr The bai rrstr algorithm eribe aboe e the retion ratio meare to gie the ontrtion of Eliean Steiner tree. Hoeer, the fat that it i not alay goo to e extremely hort tep, epeially ithin the raio range of a tranmitting noe, i not properly aptre by thi algorithm. Intitiely, hen the Steiner point t i ithin the raio range, reating a irtal etination at t may not alay be benefiial (Figre 5(b)). Therefore, in ome ae the bai form of rrstr may relt in renant hop. Th, oerly-eager irtal etination aignment hol be aoie. The folloing are the three ae in hih, gien etination an, it i not appropriate to reate a ne irtal etination: When both an are in the range of the rrent noe, a ne irtal etination ol inreae the nmber of hop to an by 1. Therefore, it i not appropriate to reate a irtal etination at their Steiner point. Intea, e mark the pair (, ) a inatie, o that thi pair ill not be oniere in the ftre. Marking a (a) (b) Figre 6. It i not benefiial to reate a irtal etination hen only one of the etination i in raio range pair (, ) a inatie i ifferent from marking to noe an a inatie, in that pair ontaining or other than the pair (, ) an till be atie. If neither nor, bt the orreponing Steiner point t of {,, } i in the raio range of, then a irtal etination may be benefiial in ome ae. Sine roting throgh the irtal etination ill ot one hop on the relting mltiat tree, thi ill be aeptable only if 1+ (t, )+(t, ) rr < (, )+(, ), rr here rr i the raio range of rrent noe. If the lefthan ie i larger, then there i no benefit of ing the irtal etination; therefore, e a ege an an mark pair (, ) a inatie. If the lefthan ie i maller, then e an reate a irtal etination at t. For example in Figre 5(a), it i benefiial to reate the irtal etination at t, bt in Figre 5(b), it i not appropriate to reate the irtal etination. When only i in the raio range, ing a ne irtal etination at the Steiner point may not be efl. If the righthan ie i larger, then intea, ill be e a the Steiner point. In thi ae, ege ill be ae into the tree an ill be marke a inatie. If the left han ie i larger, ill be e a the Steiner point intea of ; ege an ill be ae to the tree an both an ill be marke inatie. For example, in Figre 6(a) the Steiner point beome, an in Figre 6(b) beome the Steiner point intea. The algorithm preente in Figre 3 i raio range aare an implement thee three peial ae to preent renant hop generation an, th, to ae netork reore. 4. GMP Roting bae on rrstr Tree The otline of the GMP roting algorithm bae on the rrstr tree introe in the preio etion i preente in Figre 7. In thi etion, e eribe ho the GMP algorithm operate in etail. Let be a ore noe. Detination groping: firt effiiently ompte a irtal Eliean Steiner tree a eribe in the preio etion. then e thi Steiner tree to plit the etination into

5 GMP(, rr, pak) Thi fntion plit the etination into appropriate grop an forar a opy of the paket to a next hop for eah grop of etination i the rrent noe; rr, the raio range of ; pak i the paket { 1. extrat etination from pak to et; 2. et T to be rrstr(, et, rr); 3. et piot to be the hilren of in T 4. for eah p in piot; fin a neighbor n, that minimize (n, p) bjet to (n, (p)) < (, (p)), here (p) i a (p) (p) non-irtal etination in the btree roote at p; if n i fon, lear PERIMODE flag in pak, remoe p from piot, forar a opy of pak to n ith all (p) ae in the opy; ele remoe ege pl an a ege l, here l i the lat hil of p, a l to piot; if p ha only one hil o left, an p i a irtal etination, then remoe ege po an a ege o, a o to piot, remoe p from piot; ele ontine ith the ame p; 5. if piot i empty, retrn; 6. /** all etination in piot are oi no **/ if PERIMODE flag i not et in pak, et the flag on; 7. allate a next hop n by perimeter roting bae on the aerage loation of the etination in piot; 8. forar a opy of pak to n ith etination in piot ae in the opy; } Figre 7. GMP roting algorithm grop. We refer to the iret (terminal or non-terminal) eenant of in the Steiner tree a piot. Note that piot may be atal noe or they may be irtal, in the ene that a piot may not alay orrepon to an atal enor noe. For eah piot p, ientifie all the non-irtal etination in the btree orreponing to thi piot. Thi et i referre to a the grop of thi piot (grop(p)). Next hop eletion: For eah piot p, then ientifie a next hop ithin it on neighborhoo. In mot ae, thi noe (hop(p)) i the neighbor loet to the piot. Hoeer, to preent roting loop, alo reqire that the total itane from the next hop, hop(p), to all etination in grop(p) to be le than the total itane from to all etination in thi grop. For eah piot p, a opy of the paket a ell a the etination in grop(p) are ent to hop(p). When the next hop reeie the paket, it (1) extrat the orreponing etination, (2) remoe itelf from thi lit if it i one of thoe etination, an (3) repeat the aboe proere, ontrting a ne Steiner tree to plit the etination into grop, an eleting next hop. Thi proe i repeate by all hop ntil all the etination are reahe. Figre 8 illtrate the exetion of the GMP algorithm: 1. In Figre 8(a), ontrt a Eliean Steiner tree an hooe a the piot for etination {,,, }. Then, hooe n 1, the neighbor of loet to a next hop. 2. After a imilar proe, noe n 1 forar the meage to noe, hih in thi ae i both the piot an the next hop (thi tep i omitte in the figre). 3. When noe reeie the meage, it extrat the etination lit from the paket, an remoe itelf from the lit. Bae on the Steiner tree it ontrt, a hon in Figre 8(b), it eie not to plit the etination an forar the meage to noe n 2 hih i loet to the piot 2 it ompte for etination {,, }. 4. After a imilar proe, n 2 forar the meage to n 3 ithot altering the etination lit. 5. n 3 ontrt a irtal Eliean Steiner tree (hon by the ahe line in Figre 8()). Note that, in thi ae, 2, hih i at the Steiner point of {n 3, 1,}, i not e a a irtal etination. Thi i beae 2 i in the raio range of n 3 an not loe to the etination { 1,}. Intea, the piot for etination {, } i 1 an the piot for etination i itelf. 6. A opy of the meage i ent to the next hop n 4 for piot 1 ith etination peifie a {, }. Another opy of the meage i ent to n 5 ith the only remaining etination. 7. n 4 en the meage to an repetiely. 8. n 5 en the meage to etination. The relting tree a hon in Figre 1 in Setion Dealing ith Voi Note that the proe e to ontrt the Steiner tree oe not onier the neighbor loation of the rrent noe. Althogh thi i not an ie in mot ae (a an appropriate next hop an be fon for eah elete piot in a ene netork), it i poible that in ome ae there ill be no itable neighbor ith maller total itane to the etination in the piot grop than the rrent noe. In GMP, the ore onier if there exit a neighbor that ha a maller total itane to part of the etination. If there exit h a neighbor, the ore frther plit the grop into to maller part: 1. The ore remoe the lat hil l of the piot p from p hilren lit. (The lat hil of p an eaily be fon if the orer in hih ege are inle to the Steiner tree i ae ith along ith thi ege.) 2. make l a piot by aing it to it on hilren lit. 3. If there i only one hil (ay o) left in p lit of hilren, then if p i a irtal etination, make o a ne piot an remoe p from it hilren lit if p i an atal etination noe, oe not remoe p from it hilren lit. After plitting a grop, next hop are allate for eah nely reate or pate piot. The plitting proe ontine if no ali next hop an be fon for any piot an it grop. Figre 9 ho an example: Noe ontrt the

6 n4 1 n3 2 n5 n4 1 n3 2 n5 n3 n4 1 2 n5 (a) (b) () Figre 8. Example of GMP roting. The Steiner tree allate by a rrent noe i iniate by ahe ege n Figre 9. Splitting the et of etination hen there i no ali next hop Steiner tree ith 3 a the piot for the grop of etination {,,, }. Hoeer, neither one of the to poible neighbor, n 1 or n 2, ha a maller total itane to the etination than itelf. Therefore, there i no ali next hop for thi grop. ill plit the etination into to grop an 1 an 2 ill be aigne a ne piot. Sine n 1 an n 2 are no ali next hop for the grop of 1 an 2 repetiely, opie of meage ill be ent to n 1 an n 2. Conier a ae here one or more grop ontain a ingle non-irtal etination, here no neighbor i loer to any of the etination than the rrent noe. In niat heme (here by efinition there i only one etination), a imilar itation i ealt ith by plaing the paket into a perimeter moe[4, 13, 31]. One in perimeter moe, the paket traere the bonarie of the oi area folloing the right han rle ntil a noe that i loer to the etination than the point here the paket enter the perimeter moe i reahe. To orretly perform the right han rle, the graph of the irele noe ha to be planarize firt, bae on Relatie Neighborhoo or Gabriel Graph [29, 9]. Sh planarization an be one by the rrent noe ith only loal information [4, 13, 31]. In GMP, ine there may be mltiple h etination: 1. the ore reate a ingle grop hih ontain all h etination an et the paket for thi grop to be in perimeter moe. 2. allate the aerage of the geographi loation of thee etination an ientifie the next hop in perimeter moe bae on thi aerage loation, a in [21]. Figre 10. GMP allo oi etination to join grop ith other noe. 3. the paket i forare to thi next hop ith all etination in thi grop reore in the paket. 4. hen a noe reeie a paket in perimeter moe, it firt rn GMP to try plitting the etination into grop an fin a ali next hop for eah grop. 5. If ali next hop are fon for all grop, then the paket for eah grop i ot of perimeter moe. 6. If no ali next hop an be fon for any of the grop, the paket remain in perimeter moe an traere the netork ith the ame preio aerage etination. 7. If ali next hop are fon for ome bt not all of the grop, then a ne perimeter grop ill replae noere grop an a ne aerage etination loation i allate for them. The paket tart a freh ron of perimeter roting ith thi ne aerage etination. The perimeter moe operation eribe aboe i imilar to the one e by PBM [21] in the ene that the allation of next hop in perimeter moe i bae on the planarize graph of irele noe. Hoeer, GMP oe allo a oi etination, for hih no itable next hop exit, to join other etination in a grop, o that a ali next hop an be fon for all thee etination a a grop. In ontrat, in PBM [21], one a oi etination i ientifie, the paket for thi etination ill enter perimeter moe. We note that thi i not alay neeary. For example, in Figre 10, noe ha no neighbor loer than itelf to the etination. In PBM, ill not grop an together, an the paket for ill enter perimeter moe at an i forare to n 1. In GMP, an are in one grop an noe n i a ali next hop for thi grop. Therefore the paket i for-

7 SIM.PARAMETER VALUE Simlator n-2.27 Netork ize 1000m X 1000m Nmber of noe 1000 Channel ata rate 1Mbp Ma protool Ma Tranmiion poer 1.3W Reeiing poer 0.9W Meage ize 128B Antenna OmniAntenna Raio Range 150m Table 1. Simlation etp are to n for etination an. Noe n then forar the paket to, hih ill forar the paket to Complexity of the GMP an Comparion ith LGS an PBM The omptational omplexity of GMP ha to part: (a) ontrtion of the Steiner tree an (b) next hop eletion: Let ame there are n etination an m neighbor for the rrent noe,. For ontrting the Steiner tree, e e a priority qee an a 2-D array to maintain the tat of etination pair. In a gien iteration, if there i no irtal etination reate, the omplexity of thi tep i O(log n) for remoing a noe from the priority qee. There an be at mot O(n 2 ) h iteration. If a irtal etination reate, then the omplexity of h a tep i O(n log n) for inerting O(n) pair into the priority qee. There are at mot O(n) irtal etination. Therefore, the omplexity for the ontrtion of the approximate Steiner tree i: O(n 2 )O(log n)+o(n)o(nlog n) =O(n 2 log n). In the next hop eletion tep, the omplexity for allating a next hop for eah piot i O(m), an there are at mot O(n) piot. Hene, the omplexity of eah tep of the GMP algorithm i O(n 2 log n+n m). In ontrat, ine PBM [21] onier all bet of neighbor, it i exponential in m. Therefore, GMP i ignifiantly more effiient than PBM, epeially for ene netork. On the other han, the omplexity, O(n 2 + n m), of LGS i lightly le than that of GMP. Thi i expete a LGS limit the tree reation to the geographi loation of the netork noe. A e emontrate in the next etion, thi oer-ontrain the poible tree an relt in le effetie mltiating. 5. Simlation Relt In thi etion, e preent the relt of the experiment e arrie to ealate an ompare the performane of the propoe GMP mltiat protool againt exiting mltiat algorithm PBM [21] an LGS [5] a eribe in etail in Setion 1. To ee the impat of raio range aarene, e alo experimente ith GMP nr, the erion of GMP in hih raio range aare eiion hae been trne off. For Figre 11. Total nmber of hop the ake of ompletene, e alo implemente a entralize heriti [16], enote by SMT, to allate the Steiner tree. Thi entralize algorithm ame that the ore noe kno the poition of all enor noe in the netork; th the ore noe an allate a loe to optimal Steiner tree onneting itelf an all etination. The ore noe forar a opy of the ata paket ith the roting information embee in the paket. Natrally, aqiring the p-to-ate global knolege of the netork topology i not pratial for large enor netork. Therefore, e inle thi entralize algorithm only for omparion prpoe. Thirly, e alo experimente ith GRD, hih orrepon to the extreme ae, here paket are inepenently rote for eah etination. Thi algorithm expliitly minimize the per-etination hop ont an ere ell a a loer-bon for the aerage nmber hop for eah etination. The imlation etp i eribe in Table 1. The NS2 imlator ha been moifie to pport mltiating for irele enor netork. In eah experiment, e generate 100 tak. The 1000 noe are niformly itribte in the netork. For eah tak, e ranomly pik a noe a the ore noe an ranomly pik k noe a the etination noe. The ale of k arie from 3 to 25. Eah experiment i rn on 10 ifferent netork an relt are aerage Total Nmber of Hop in the Mltiat Tree The total nmber of hop neee for a ingle mltiating tak i the nmber of tranmiion (foraring) reqire to reah all etination. Figre 11 ho the relt obtaine ing for ifferent mltiat roting protool: PBM, LGS, GMP an GMP nr. In [21], it i note that in PBM the minimm total nmber of hop i ahiee for a λ ale beteen 0 an 0.6. We hae th rn the ame roting tak een time, ith the ale of λ arying from 0 to 0.6. Among the relt orreponing to thee λ ale, only the bet (minimm nmber of hop) one i inle for PBM in Figre 11. In thi figre, e ee that, of the fie protool, GMP relt in the leat nmber of hop. Note that, GMP perform een better than the entralize algorithm, SMT. The retion of GMP ompare to both PBM an LGS i p to 25%. The relt iniate that by ing rrstr etination

8 n4 n3 n5 Figre 12. Per-etination hop ont are iie into grop more effetiely an the ay the next hop i allate bae on the Steiner point of the etination in eah grop i more effetie than the ay PBM an LGS allate the next hop. A ie in Setion 3, hen the raio range i not oniere ring the ontrtion the Steiner tree, GMP nr may generate renant hop. The probability for renany beome larger a the nmber of etination beome larger. Therefore, GMP nr e more hop than GMP. Note, hoeer, that een ithot raio range aarene, GMP nr ork better than PBM an LGS. A ie earlier, in PBM the optimal ale of the trae-off parameter λ epen on the nmber of neighbor an the itribtion of the etination. The ale of λ, hoeer, i fixe ring roting, therefore, the etination are plit into grop by a reeiing noe ometime earlier an ometime later rather than being plit by the noe loet to the Steiner point of the etination. Therefore PBM may generate a larger total nmber of hop. LGS e a MST heriti in hih geographi loation other than loation of noe in the netork are not taken into onieration. Therefore the grop ientifie by LGS may not be a goo a thoe ientifie by GMP. Frthermore, the allation of next hop in LGS i bae on one of the etination in the grop. In GMP, the next hop i allate bae on the Steiner point of the etination in thi grop relatie to the rrent noe, therefore, thi relt in better path Per-etination Hop Cont Figre 12 ho the aerage per-etination hop ont. A ie in Setion 1, it i not alay poible to minimize both the total nmber of hop an the per-etination hop ont. PBM take into aont thi by ing a trae-off parameter, λ. When λ i mall, PBM relt in a maller aerage nmber of hop an a larger total nmber of hop. When λ i larger, the aerage nmber of hop get larger bt the total nmber of hop beome maller. The ale of λ in Figre 12 i the ame a in Figre 11, i.e. the λ that minimize the total nmber of hop. We ee that PBM, SMT an GMP proie omparable per etination hop ont (loe to the greey oltion, GRD). LGS oe not math the other in thi repet. In LGS the next hop i hoen to be the neighbor loet to a etination that i loet to rrent noe in the grop. Figre 13. An example here LGS relt in a large per-etination hop ont Figre 14. Total energy ot Roting in thi ay ten to reah the etination eqentially an preent the etination from getting iie into grop at intermeiate noe. To ee thi, onier in Figre 13. Let ame that the paket i at noe an the remaining etination are {,, }. The MST reate by LGS ill onit of the folloing three ege: {,, }. Th, noe oe not plit the etination into grop an forar the paket toar ine i loet to. Coneqently, the path taken by the LGS ill eentally be >... > > >... >. Contrat thi ith the tree-trtre mltiat rote ompte by GMP a illtrate in Figre 1. Coneqently, hen LGS i e, etination that are reahe later may take a ignifiantly larger nmber of hop an thi inreae the oerall aerage nmber of hop Energy Conmption Figre 14 ho the energy onmption ring mltiat roting. 2 We ee that ine the nmber of hop e by GMP i le than the nmber for all of PBM, LGS an SMT, the energy onmption of GMP i ignifiantly maller than onmption of PBM, LGS an SMT in all onfigration e tete. In thee experiment, the energy aing ompare to PBM an LGS i p to 25% a hon in Figre Failre De to Perimeter Roting When the netork enity i loer, the probability that there i no neighbor loer to the etination an that the paket enter the perimeter moe beome higher. Beae 2 The energy onmption reporte in thi paper inle the tranmiion poer of ener an the reeiing poer of all litening noe ithin the tranmiion raio range of the ener.

9 the mltiat tree reate by GMP i ignifiantly loer than that of LGS an omparable to that of PBM. Therefore, GMP proie the bet featre of both alternatie. Frthermore, in term of total nmber of hop an energy onmption, GMP i aron 25% better than both of thee alternatie. Referene Figre 15. Nmber of faile tak for ifferent netork enitie traering the oi area follo the right han rle, the length of the path may beome large. Frthermore, thi may relt in faile elierie, epeially hen there i an pper bon aoiate ith the total nmber of hop alloe for eah paket. To obere ho GMP ale againt lo enity netork, e alo i experiment to tet the performane of GMP in netork ith ifferent netork enitie. In thee experiment, the nmber of noe in the netork are 1000, 800, 600 an 400. The enor noe are niformly itribte in the netork. We et the maximm path length for eah etination to be 100, that i a paket i roppe if it hop ont reahe 100. Eah tak in thee experiment rote a meage from a ranom ore to 12 ranomly generate etination. A tak fail if not all etination are reahe. For eah netork ize, e generate 10 netork an ran 100 tak for eah netork. Figre 15 ho the total nmber of faile tak in the total of 1000 tak ing three protool PBM, LGS, GMP. (The other protool o not e perimeter roting) We ee that GMP ha the leat nmber of failre. LGS ha the larget nmber of failre beae it ame a ali next hop an alay be fon an it fail hen a oi etination i ientifie een if the hop ont i le than 100. PBM ill grop all the oi etination an alay mark the paket to be in perimeter moe for thee etination. GMP, on the other han, may grop ome oi etination ith other etination ithot etting the paket to perimeter moe for thi grop aming a ali next hop an be fon for thi grop a eribe in Setion Conlion In thi paper, e preente a noel geographi mltiat roting protool, GMP. 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