Geographic Routing with Constant Stretch in Large Scale Sensor Networks with Holes

Transcription

1 IEEE 7h Inrnaional Confrnc on Wirl an Mobil Compuing, Nworking an Communicaion (WiMob) Gographic Rouing wih Conan Srch in Larg Scal Snor Nwork wih Hol Myounggyu Won, Rau Solru, Haiji Wu Dparmn of Compur Scinc an Enginring, Txa A&M Univriy {mgwon, olru, Abrac Gographic rouing i wll ui for larg cal nor nwork ploymn, bcau h pr no a i mainain i inpnn of h nwork iz. Howvr, u o h local minimum cau by hol/obacl, h pah rch of gographic rouing can grow a O(c ), whr c i h lngh of h opimal pah. Rcnly,, a gographic rouing proocol ba on h viibiliy graph, how ha a conan pah rch can b achiv. Thi, howvr, i poibl wih incra ovrha. To ar hi iu, w propo (Gomric Rouing uing Abrac Hol), a rouing proocol ha provably achiv a conan pah rch, wih lowr mag, pac an compuaional ovrha. W vlop a novl iribu convx hull conrucion (DCC) algorihm ha compacly crib hol. Thi compac rprnaion of a hol i lvrag by no o mak locally opimal rouing ciion. Our horical analyi prov h conan rch propry an avrag rch of. Through xniv imulaion an a harwar implmnaion, w monra h ffcivn of an i faibiliy for larg-cal nor nwork. In our nwork ing, ruc h nrgy conumpion by up o 3%, rouing abl iz by an orr of magniu, whn compar wih. I. INTRODUCTION Gographic rouing ha arac much anion from h WSN rarch communiy, bcau i i impl an calabl. In gographic rouing, a ourc no obain h locaion of a inaion no by uing a locaion rvic [], or hahfuncion in a aa cnric orag chm []. A pack i forwar o h nighbor who i clo o h inaion. Thi gry approach allow nar-opimal pah lngh in uniform an n nwork wihou hol/obacl. Howvr, a Kuhn al [3] prov, h pah lngh can grow a much a O(c ), whr c i h opimal pah lngh, whn h hol/obacl ar prn, bcau of h local minimum phnomnon. In orr o ovrcom h local minimum an ulimaly improv h pah qualiy, om gographic rouing proocol uing h non-local informaion hav bn propo [4][][6][7]. In h proocol, h hol ar inifi bforhan, an h iz an hap of h inifi hol ar propaga o om no o ha forwaring no can u h informaion o avoi h hol. Howvr, for a rourc-conrain WSN, h informaion abou h hol (h iz an hap of h hol) can only b ma known o a limi ub of no,.g., h no loca a h bounary of h hol (rfrr a bounary no), or om of hir nighbor, u o h larg volum of aa o crib h hol. Thu, a racion o h local minimum i only aciva whn a pack rach h no a h local minimum (i.., a uck no), or om nighboring no of h uck no. Thi la racion problm rul in a ubopimal rouing pah. Rcnly, Tan. al. [8] propo, a gographic proocol ha achiv a conan pah rch. In, h hol i rprn a a polygon, which i u o buil h viibiliy graph, a rucur ofn u in compuaional gomry o fin h hor pah, givn obacl. Howvr, uffr from non-ngligibl proocol-rla mag ovrha. Fir, in orr o buil h viibiliy graph, h locaion of all h VON no (h vric of h polygon) n o b floo o all h no in h nwork. Scon, h VON no mu iraivly xchang mag among hmlv unil a compl rouing abl i conruc. Th prformanc riora furhr whn h hol ha a complx hap, ruling in many VON no. In hi papr, w propo a gographic rouing proocol ha achiv a conan pah rch whil incurring l ovrha. In hi proocol, h hol ar compacly crib a a of xrm poin of h convx hull covring h bounary no of h hol. A rou o an from h no ini h convx hull i fficinly hanl for guaran pack livry. A novl iribu convx hull algorihm i inrouc o buil h convx hull of h hol wih low mag complxiy. A iribu xrm poin rucion algorihm furhr ruc h iz of h hol informaion. Conqunly, h locaion of only a fw xrm poin of h convx hull ar locally broaca o no wihin h hop from h hol. Ba on h mall aaba abou narby convx hull, a ourc no inifi h inrfring hol wihin h hop ha block h raigh pah o h inaion wih mall compuaional ovrha. A ourc no hn compu a of inrmia inaion ha gui a pack along h locally opimal pah. Whn a pack rach a no wih br hol informaion, h inrmia inaion ar upa, incrmnally improving h rouing pah. Th conribuion of hi papr ar a follow: W vlop a gographic rouing proocol ha gnra a pah wih conan rch in nwork wih hol. Th proocol ha low mag an compuaional complxiy. W vlop a iribu convx hull algorihm o fficinly ruc h iz of aa ha crib a hol. W prn a horough analyi o prov h corrcn an conan pah rch of our proocol. W prform xniv imulaion an a harwar implmnaion o confirm h ffcivn of our proocol, an i faibiliy for WSN //$6. IEEE 8

2 II. RELATED WORK Rouing proocol for larg cal WSN can b largly ivi ino gographic rouing an hirarchical rouing [9]. Rcnly, S4 [9], a hirarchical rouing proocol, wa hown o achiv h wor ca rch of 3 wih mall pr no a O( N), whr N i h oal numbr of no. Alhough S4 obain a irabl balanc bwn h pah rch an a iz, unlik gographic rouing proocol, h pr no a pn on h nwork iz N. In conra, gographic rouing ui paricularly wll o rourc conrain larg-cal WSN, inc i i al an fully iribu. Howvr, h pah rch of gographic rouing proocol can b a ba a O(c ), whr c i h opimal pah lngh, u o h local minimum cau by opological complx lik hol. Lil wa known abou achiving a conan rch for gographic rouing proocol. Svral gographic proocol hav bn propo o improv h pah qualiy by uing h non-local informaion. In [], a no u h TENT rul o if i i a uck no. If a no fin ilf o b h uck no, BOUNDHOLE algorihm i run o buil a rou aroun h hol by icovring a of h bounary no o gui a pack ou of h local minimum. Th bounary no ar mark an u a lanmark for a fuur pack o bypa h hol. Howvr, h la racion problm (a our i ma a h bounary no) gra h avrag pah lngh. In [7], fining an angl bwn wo ajacn nighbor o b grar han prfin hrhol, h no i lva o ha hi no i avoi by h gry forwaring proc. Howvr, a ciion on whhr a no i a a local minimum or no pn on h locaion of ourc an inaion. Thu, uch huriic approach rul in h frqun failur of h algorihm. To rmy hi problm, in [6], a nwork i ivi ino k rgion. Each no mainain a vcor of iz k, whr ach lmn inica whhr hi no i a local minimum for h i-h rgion or no. To rmin h valu of ach lmn, h local minimum angl b i fin, an if h rgion i covr mor han a crain prcnag by hi angl, h rgion i conir o b h local minimum rgion. Howvr, non of h proocol provi h wor ca pah lngh guaran. Svral rarchr propo o propaga h informaion abou h hol o om no in a limi rgion. In [4], a uck no an om of i nighbor form an unaf ara hap a a rcangl. Th ima hap of h hol i known o h no in hi unaf ara. Thi iribu informaion mol i u o avoi h local minimum. Alhough h pah rch i improv u o h propagaion of h hol informaion o om no, only a fw nighboring no of h uck no ar awar of h informaion, o h la racion problm pri. In [], h hol i rgulariz wih an llip, an h abrac informaion abou a hol i broaca o h no wihin h hop from h bounary of h llip. Howvr, h llip ofn fail o rprn variou kin of hol hap. Mor imporanly, h wor ca pah rch i no guaran. Rcnly, Tan. al. [8] inrouc, a gographic rouing proocol ha achiv a conan pah rch. Fig.. Bounary no p p 3 p pini Convx hull Inrfring hol p4 p An illuraion of noaion an rm. fin a nar opimal rouing pah by xploiing h viibiliy graph. A hol i rprn a a of VON no, an a virual ovrlay nwork coniing of h VON no gui a pack along h clo-o-hor pah ha bypa h hol. On noabl apc of compar wih S4 i ha i pr no a i O(N von ), whr N von (<< N) i h oal numbr of VON no, hrby liminaing h pnncy of h pr no a on h nwork iz. Howvr, hr ar vral chnical iu, ohr han h proocol-rla mag ovrha icu in Scion I, o b olv. Fir, bfor pack ranmiion ar, a ourc no ha o up a rouing pah, by xchanging h viibiliy wih a inaion no. Thi rouing up proc incur aiional ovrha an lay for ach ourc an inaion pair. Scon, h probing mag, u o buil h VON polygon, migh conain a larg amoun of aa, inc h mag i piggyback wih all h locaion of h bounary no ha h mag ha vii unil a nw VON no i foun. III. PRELIMINARIES In hi cion, w fin h noaion an rm u hroughou hi papr. Th noaion ar illura in Figur. W conir a n wirl nor nwork, coniing of N no V = {v,v,..., v N }, uniformly iribu in a wo imnional pac. W aum ha ach no know abou i locaion. Thr ar m hol in h nwork, no by H,H,..., H m. Each hol H i i urroun by a of bounary no, P i = {p,p,..., p n }, p j V. A hol i ihr a clo cycl (p = p n ), or a chain (p p n ) ha form a cycl wih h g() of a nwork. W fin on bounary no in ach P i a h iniiaor a h following: Dfiniion : An iniiaor, no by p ini, i a bounary no in P i wih h high y coorina. If hr ar mor han on bounary no wih h am high y coorina, on wih h low x coorina among hm bcom h iniiaor p ini. A hol H i i rprn a a of xrm poin, no by P i = {p,p,..., p n }, of h convx hull covring h of bounary no P i, whr h xrm poin p j i fin a h following (No ha w will u h rm no an poin inrchangably): Dfiniion : An xrm poin i h cornr poin of a convx hull. A pah bwn ourc an inaion i no by, an h lngh of h pah, a h numbr of hop, bwn 8

3 an i no by. Givn a pah, h inrfring hol ar formally fin a h following: Dfiniion 3: Givn any wo poin p an q, L(pq) i a of poin on h lf han i of a vcor pq, an R(pq) rprn a of poin on h righ han i of pq. Dfiniion 4: Givn a pah, a hol rprn by a of xrm poin, P, i call h inrfring hol iff hr xi om p k P, k P uch ha p k L() an p k+ R(), orp k R() an p k+ L(). In paricular, wo no ar viibl o ach ohr iff hr i no inrfring hol bwn hm. IV. : GEOMETRIC ROUTING USING ABSTRACTED HOLES A. Proocol Ovrviw Our rouing proocol coni of mainly wo componn: h hol abracion an pack forwaring. Th hol abracion proc i ign o prouc h compac rprnaion of h hol. In hi proc, h hol i rprn a a of xrm poin of a convx hull covring h bounary no of h hol. Th hol abracion proc coni of hr pha. Fir, h bounary no urrouning h hol ar inifi uing ihr aiical [] or opological mho []. Th con pha i h conrucion of h convx hull: a of xrm poin for ach hol i foun by h ingl ravr of a probing pack along h bounary no in a fully iribu mannr. In h la pha, h abrac informaion abou ach hol i broaca o h no wihin h hop from h hol. Whn h hol abracion proc i finih, ach no know abou h locaion of h xrm poin of h hol wihin h hop. Thi informaion i u o mak a rouing ciion. Spcifically, ourc inifi h inrfring hol blocking h raigh pah o inaion, an run our forwaring algorihm o compu a ub of xrm poin among all h xrm poin ha blong o h inrfring hol. Thi ub of xrm poin ar h inrmia inaion u o gui a pack along h pah ha minimiz. A pack carri h locaion of inrmia inaion, an i forwar o ach inrmia inaion by impl gographic forwaring. Whn a pack rach an inrmia inaion, h rouing algorihm i rrun an h prviou inrmia inaion ar upa if ncary. Th ail of h proocol ar prn in h following cion. B. DCC: Diribu Convx Hull Conrucion W aop [] o fin h bounary no. If h bounary no form a clo cycl, p ini i lc uing an xiing lar lcion algorihm on a ring opology, which ha mag complxiy O(n log n), whr n i h numbr of h bounary no [3]. If h bounary no form a chain, on of h wo bounary no a ach n of h chain i lc a p ini. Spcifically, if no p fin ha i i h bounary no a h n of h chain by chcking h numbr of nighboring bounary no, no p n a mag conaining i x an y coorina o h bounary no q a h ohr n of h chain. If no q y coorina i grar han no p y coorina, Algorihm DCC (co for p ini ) : if hop coun = hn : P P {p ini } 3: n a probing pack in counr-clockwi. 4: l : if N pi =or p ini R(p P 6: rmina. 7: n if 8: n if Algorihm DCC (co for p i ) p ) hn : for ach p l P, l P o : if m, l + m P, p m L(p l p i ) hn 3: P P \{p l+,p l+,..., p P } 4: n if : n for p i+ ) hn 7: P P {p i } 8: // EPRA 9: if P > Thrhol hn 6: if p i R(p P : fin p i cr wih minimum i cr : p i p i cr : P P \ p i+ 3: n if 4: // En of EPRA : forwar a probing pack o p i+. 6: l 7: forwar a probing pack o p i+. 8: n if no q bcom p ini. If no q y coorina i h am a no p y coorina, x coorina ar compar, an if no q x coorina i mallr, hn no q lc ilf a p ini. Onc p ini i lc, p ini iniia h DCC algorihm. Algorihm crib h puo co for p ini. p ini a i locaion o h P a h fir xrm poin p, an piggyback h P on a probing pack. Thi pack i n o p ini lf nighboring bounary no, aring o ravr h bounary no of h hol in a counr-clockwi ircion. Upon rciving h pack, h bounary no xamin if i i h xrm poin of h convx hull by xcuing Algorihm. Figur illura h DCC algorihm for p i. For ach bounary no p i,ifp i R(p l p i+ ), for om l, l P, hn p i i h farh (w.r.. h ianc h probing pack ravl) viibl bounary no from p l o far, inc h nx no p i+ i no viibl from p l.sop i i a o h P, an h probing pack conaining h P i forwar o h nx bounary no (Lin 6-8). Howvr, if anohr bounary no p j,j >i+ ha i viibl from p l i foun, p i i l from h P, inc p i i no longr h farh viibl no from p l (Lin -). Th abov proc i rpa unil h probing pack ihr rurn o h iniiaor, or rach h n of h chain if 8

4 P l P P m 3 Pj Pi+ Pi P cr cr P P S m m P Th largr convx hull P S P m ' P cr P cr P 3 P 4 Fig.. An illuraion of : Sp-4; Sp. Fig.. An illuraion of DCC. Fig. 3. An illuraion of EPRA. h rul whn DCC algorihm i ingra wih EPRA. Y (mr) Fig Bounary poin Exrm poin X (mr) Y (mr) Bounary poin Exrm poin X (mr) Th igh of xrm poin; ruc xrm poin. h cycl i no clo (co for p ini : Lin -7). Bcau h DCC algorihm rquir a ingl ravral of h probing pack, combin wih h p ini lcion procur, h mag complxiy i O(n log n), a br boun han h currnly known on for a iribu convx hull algorihm on a ring opology [4]. Th DCC algorihm gnra a igh convx hull. Thi man ha for a mooh hol, h DCC algorihm migh gnra many xrm poin (i.., if h hol i a prfc circl, all h bounary no will b lc a h xrm poin). A larg numbr of xrm poin will gra h ym prformanc; hu, h numbr of xrm poin mu b conroll in ral im, uring h xcuion of h DCC algorihm. Thu, w vlop h Exrm Poin Rucion Algorihm (EPRA). EPRA limi h probing pack iz by a ur fin hrhol. I o no incur aiional communicaion ovrha, bcau i opra a par of h DCC algorihm. EPRA i mb in h DCC algorihm a hown in Algorihm (Lin 8-4). Figur 3 illura h algorihm. Spcifically, w fin p i cr a h poin a an inrcion bwn h wo lin p i p i an p i+ p i+, whr p = p P. Th Euclian ianc i cr = (p i cr, p i p i+ ), whr (p, uv) i a lin gmn conncing p an p projcion on lin uv, i compu for ach p i cr. Whn h probing pack rach h poin p i R( p i+ ), an h numbr of h xrm poin foun o far i grar han h prfin hrhol (Lin 9), corrponing p cr valu for h xrm poin foun o far ar compu, an p i cr wih h mall p P i cr i aign a a nw p i, an p i+ i rmov (Lin - ). Thi proc i rpa a h probing pack ravr h bounary no. Figur 4 how an xampl of h xrm poin gnra by DCC algorihm, an Figur 4 pic C. Forwaring Algorihm In hi cion, w prn h ail of our forwaring algorihm. Whn h hol abracion proc finih, h no wihin h hop from a hol know abou h locaion of h xrm poin of h convx hull covring h hol. Uing hi informaion, ourc mak a rouing ciion following h p crib blow: Sp: Sourc fir inifi h inrfring hol wihin h hop from i. If hr i no inrfring hol, ourc forwar h pack uing gographic forwaring. Sp: A largr convx hull i conruc from a of poin, ay P, coniing of h xrm poin of h inrfring hol, ourc, an inaion, a hown in Figur. Th largr convx hull can b conruc by applying an xiing cnraliz convx hull algorihm o h P [], bcau ourc ha all h locaion of h xrm poin for h inrfring hol. Th largr convx hull yil wo poibl pah, on along h uppr par of h hull, P, an h ohr on along h lowr par of h hull, P. Sp3: Sourc lc a horr pah bwn P an P. Sourc hn h xrm poin along h lc pah a h inrmia inaion. Thi p i pic in Figur wih h inrmia inaion, m,m, an m 3. Sp4: Sourc hn chck if hr xi any inrfring hol for m. If hr i no inrfring hol, h pack i forwar o m uing impl gographic forwaring. Ohrwi, Sp i xcu, whr nw inrmia inaion ar up for m. Upon rciving h pack, m bcom a nw ourc ; an h algorihm rrun o rflc h nw viion of m. Sp: Sp an Sp ar u o fin h wo poibl pah conncing an m. Figur how h poibl wo pah. An hn, if h lngh of on pah i longr han + m, h ohr on i lc. If boh ar horr han + m, h pah ha i clor o lin m i chon. Hr w xplain ha ha low compuaional ovrha. For Sp, ach of h xrm poin P i i cann o chck if any hol H i wihin h hop inrc wih h lin gmn. Th compuaional complxiy of Sp i hu O(N x ), whr N x i h oal numbr of xrm no in h nwork. Sp can b aily implmn uing an xiing cnraliz convx hull algorihm. W no ha h b prformanc of currnly known cnraliz convx hull algorihm i O(N x log N x ). Th wor ca happn whn 83

5 m'' ' m m H 3 H H v ' ' m H B A v ' ' m H R c u (c) w v a b Fig. 6. An illuraion of rcuriv run of our rouing algorihm. Fig. 7. Symbol for corrcn proof; bouning rgion rprning poibl locaion for inrmia inaion; (c) final bouning rgion R; all h hol inrfr wih h pah, bu uch an xrm ca rarly happn, making h avrag complxiy of much lowr han O(N x log N x ). Howvr, a hown in Figur 6, in Sp4 an Sp, h algorihm migh b iraivly rpa if hr i an inrfring hol, H, for pah m, hn anohr inrfring hol, H 3, for pah m, an o on. In h following cion, howvr, w will how ha h numbr of iraion i boun by a conan C. Conqunly, h oal compuaion complxiy of for ach no i O(N x log N x ). D. Spcial Ca In igning w n o conir pcial iuaion, whr ourc or inaion (or boh) i loca in a convx hull. To hanl h cnario, h cycl of bounary no i u o gui a pack o ihr lav h convx hull, or rach h inaion in h convx hull, wihou rlying on any graph planarizaion chniqu. Thr ar hr ca o b conir: Ca, Sourc i in a convx hull: ourc compu h hor pah ba on our rouing proocol, an i n h pack o h fir inrmia inaion. If ourc ha a clar pah o h fir inrmia no, h pack i rou o h fir inrmia inaion by uing gry rouing. Howvr, if h pack i block by a hol, hn h pack woul rach on of h bounary no. Th pack hn ar a counr clockwi ravral along h bounary no unil gry rouing o h fir inrmia no can b rum. Ca, Dinaion i in a convx hull: Thi ca i imilarly hanl a h Ca. Th pack follow a of h inrmia inaion prviouly rmin by our rouing proocol. Afr paing h la inrmia inaion, h pack i grily forwar o inaion. If h la inrmia inaion ha a clar pah o inaion, hn rouing can proc. Ohrwi, h pack woul rach on of h bounary no. Thn, h pack ar ravring h of bounary no in a counr clockwi ircion unil gry forwaring o inaion can b rum. Ca 3, Boh n rc an n ar in convx hull: Thi ca can b imply hanl a a combinaion of h Ca an Ca. V. PROTOCOL ANALYSIS A. Corrcn of Convx Hull Conrucion W fir prov h corrcn of h DCC algorihm. Spcifically, w will how ha givn a hol, our algorihm fin all h xrm poin of h igh convx hull covring h hol. Lmma : p ini i an xrm poin. Proof: If h bounary no form a chain, h claim rivially hol. So, w conir only h ca whr h bounary no form a cycl. Aum, by conraicion, ha p ini i no an xrm poin. By finiion, h y-coorina of p ini i largr han any xrm poin. Thu, p ini i no covr by h convx hull, which i a conraicion. No ha if hr i an xrm poin wih h am y-coorina a ha of p ini, h x coorina of all h xrm poin ar grar han p ini. Thu, p ini i no again covr by h convx hull, a conraicion. A crib arlir, DCC algorihm arch for h farh viibl no from h la icovr xrm poin. Th following lmma how ha uch farh viibl no i h nx xrm poin. Lmma : Givn an xrm poin p i, h farh viibl bounary no from p i, ay p i+, i h nx xrm poin. Proof: Aum by conraicion ha p i+ i no h nx xrm poin, i.., hr xi an xrm poin (p i+ ) ha i clor o p i han p i+.if(p i+ ) L(p i p i+ ), a hol i rhap a a concav hull. If (p i+ ) R(p i p i+ ),orif(p i+ ) i on h lin p i p i+, hn p i+ i no viibl from p i. Lmma 3: p ini i h farh viibl bounary no of p n whn P = {p,p,..., p n }. Proof: Sinc h y-coorina of p ini i h high among all bounary no, an h x-coorina of p ini i h low among all bounary no, any bounary no on p ini lf han i ar no viibl from p n. Thu, p ini i a viibl no ha i h farh from p n. Thorm : Givn a hol H i, DCC fin all xrm poin, ay P i = {p,p,..., p n }, of h igh convx hull covring h bounary no of H i. Proof: By Lmma, p = p ini, an ubqun xrm poin ar rmin by Lmma. Laly, by Lmma 3, p i h farh viibl no from p n. Thu, conncing all p i, i n, a p p n - p, w g a convx hull. Now w prov ha hr i no mor xrm poin. Aum in conraicion ha hr i on mor xrm poin in P. Wihou lo of gnraliy, aum ha a poin p i bwn wo xrm poin, p i an p i+ for om i, i P.By i h farh viibl no of p i. Conir h ca whr p L(p i p i+ ). In hi ca, h ruling polygon bcom concav. If p R(p i p i+ ) or p i on h lin p i p i+, Lmma, p i+ 84

6 hn, p i+ i no longr viibl from p i. B. Corrcn an Conan Srch of In hi cion w prov h corrcn of, an how ha a pah gnra by ha a conan rch. Conir pah m, whr m i h inrmia inaion on h uppr hull P, an H i h inrfring hol for a pic in Figur 7. W fir xplain om ymbol an hir gomric propri, an inrouc our main proof. L α b h angl bwn h wo lin gmn m an. Th rang of α i <α<π, bcau if α>π, hn m woul hav bn in R(), bing a poin for a pah P. i h Euclian ianc o a hol along a lin gmn. i mallr han r max h, whr r max i h maximum raio rang, an h i h numbr of hop o which h locaion of H i broaca. If >r max h, a no woul no hav icovr h hol H. δ rprn h maximum high of an inrfring hol for m. No ha h high of inrfring hol for m, δ,i mallr han δ, bcau if no, will choo a pah m. δ can b xpr a in α, whr α π. A lin gmn m i h hor among all poibl conncion bwn h wo poin an m ha chang pning on h hap of h hol. m can b a long a + m. Thorm : i corrc. Proof: In orr o prov guaran pack livry, i uffic o how ha a pack i uccfully rou from o h nx inrmia inaion, ay m, inc ach im a pack arriv a h nx inrmia inaion m, m bcom, an run h am algorihm from h Sp. Thrfor, onc w prov a uccful livry for m wihou any loop or arbirarily long pah, a guaran livry can b prov by inucion. Th maximum high of an inrfring hol for m mu b mallr han δ, bcau ohrwi uch a hol woul hav bn c a an inrfring hol for (No δ δ). Thrfor, w obain h uppr boun for h poibl poiion of nw inrmia inaion, which i pic a a o lin A in Figur 7. Nx aum ha a poin v i h nw inrmia poin ha blong o h inrfring hol for m. On obrvaion i ha v + vm mu b mallr han + m, bcau if v + vm >+ m, hn woul hav lc a pah m. Thrfor, uch a poin v mu b boun by an llip B having an m a foci an paing hrough a poin. Coniring h bounari w compu an h rang of an angl bwn m an v ( o π), h poibl locaion for any nw inrmia poin ar boun by rgion R a hown in Figur 7(c). Thi rgion canno b arbirarily larg, inc δ i a mo which pn on h conan paramr h. Thorm 3: ha conan rch. Proof: Wihou lo of gnraliy, w rprn our nwork a a Uni Dik Graph (UDG). Mor prcily, w aop h k boun gr uni ik graph whr h gr of ach no i boun by k [6]. Howvr, k boun gr uni ik graph can b conruc from a gnral uni ik graph [3]. A hown in Thorm, for any pair of inrmia poin u an v, incluing an, poibl locaion for nw inrmia inaion u o an inrfring hol for a pah uv ar boun by om rgion R. By Kuhn [3], h oal numbr of no N R in rgion R i givn by: N R (k +) 8 π (A(R)+p(R)+π), whr A(R) i h ara of R, an p(r) i h primr of R. A(R) coni of a rcangl abc in uppr par of R, an a riangl uv in lowr par of R. Th ara of h rcangl i ( + m ) in α ( + uv ) in α, an h ara of h riangl i uv in α. Th primr of R i 6 +in α + uv. Thu, h oal numbr of no in R i givn by: N R (k+) 8 π {(3 in α+) uv + in α (+)+6+π} (k +) 8 π {(3 +) uv π} ( <α<π). By h aumpion of n an uniform iribuion of no an h propry of gry forwaring, a pack i forwar ouwar from a poin a ach p of h algorihm unl i hi h bouning rgion R. Thi impli ha ach no in rgion R i vii a mo onc. Thu, h oal numbr of hop H R in R i boun by: H R N R. Now conir all (u, v) pair bwn, an aum ha h i h maximum hop coun of h nwork. Th oal numbr of hop from o, H i givn by: H (u,v) ((k +) 8 π {(3 +) [(k +) 8 π {(3 +) uv π}] (u,v) uv π}., whr (u,v) uv i h hor pah in h Viibiliy Graph [7]. By [8], h hor pah bwn an in h Viibiliy Graph i boun by om conan facor of Euclian ianc bwn an a h following: (u,v) uv in( π ɛ ). Thrfor, w g: H ((k +) 8 π {(3 +) in( π ɛ ) π}, whr r max h. C. Avrag Srch of W hav hown ha h wor ca pah rch of i conan. Now w horically analyz h avrag pah rch of whn h ym paramr h i givn a an inpu. In hi analyi, w conir a quar rgion in which no ar uniformly an nly ploy (i.., a pah bwn wo no can b hough of a a lin gmn conncing h wo). W aum ha ach no ha circular communicaion rang wih raiu, an h hol ar abrac a convx hull. W fir conir h ca wih a ingl hol in h nwork. A hown in Figur 8, boh h rang of inpu h an h ara of riangl ABC, which rprn h viaion from 8

7 / C - h >> Opimal pah Fig. 8. Pah by B h A hol / C - h >> R B h ingl-hol ca; h muli-hol ca. h opimal pah, ar maximiz whn: i) ourc i loca in h mil of on i of h quar; ii) inaion fac from h mipoin of h oppoi i of h quar; an iii) h hol wih wih i loca along h i having. Th following lmma prov h avrag pah rch of for h ingl hol ca. Lmma 4: Th avrag rch λ of for h ingl hol ca i + +. Proof: Th pah lngh of opimal pah i 4 + +, an h pah lngh of i ( h) h +. Thu, h pah rch i f(h) = ( h)+ 4 +h +, an h avrag pah rch λ for inpu h i givn a h following: h= λ = f(h) + +, inc ( +) << an h= 4 +h Now w inviga h avrag rch for h muli-hol ca. A ky obrvaion i ha h muli-hol ca can b conir a a ri of h ingl-hol ca for ach inrfring hol for u o h following raon. A nw rouing ciion i ma whn h pack rach ach inrmia inaion of h hol, an if hr ar mor han wo inrfring hol wihin h hop, hy ar conir a a ingl convx hull covring all h hol. Conir Figur 8. Whn a pack rach a no a B, h no B know abou h hol an h pack i our o h inrmia inaion a A. On iffrnc from h ingl-hol ca i ha hr migh b anohr hol ha inrfr wih h pah from B o A. Howvr, a provn in Thorm, h viaion of h pah BA i boun by h rgion R. Thu, h avrag rch of pah BA, λ bcom: λ = h= ( h)+ 4 +h h h A hol +.. Uing hi propry, w obain h following rul. Thorm 4: Th avrag pah rch of i +. Proof: Givn ourc an inaion, a of inrfring hol, H,..., H n, ar inifi. Th pah rch from o h inrmia inaion of H, ay H i a mo.. Thu, h pah lngh of H i., auming ha opimal pah lngh i. Now w conir h pah H an a hol H a a ingl-hol ca. Similarly, w obain h pah lngh of H H i., whr i h opimal pah lngh of H H.If w rpa hi proc for all inrfring hol, h oal avrag pah rch for i.( + + n )+ n + + n.. VI. SIMULATION RESULTS W xn h original implmnaion of [8] an conuc imulaion o valua h prformanc of our proocol. W ranomly ploy 3, no in a,,m rgion. Th locaion an hap of h hol in a nwork wr prfin, an h no ini h hol ar no conir. Th communicaion rang of h no wa 3m, an h avrag no niy wa 9 pr raio rang. W compar our proocol wih, mor pcifically - R. i alo compar wih h claic an wily u gographic rouing proocol, GPSR, an h cnraliz hor pah rouing. Spcifically,, ourc an inaion pair wr ranomly lc among h no ha ar no in convx hull. Th following mric wr u: avrag hop rch (pah rch i fin a h oal hop coun ivi by h hop coun of h hor pah rouing), maximum hop rch, proocol-rla mag ovrha, an proocol rla mmory ovrha. W vari h following paramr: h, h numbr of hop wihin which no rciv h hol informaion, an b, h inrval of broacaing. A. Hop Srch Two nwork cnario wr conir for hi xprimn: on wih h wo hol wih high concaviy hown in Figur 9, an h ohr on wih many hol wih low concaviy a pic in Figur 9. For hi xprimn, w h o a ufficinly larg numbr. (W will how how h affc h prformanc in h nx ubcion.) For boh cnario, ouprform GPSR. Epcially for h hol wih high concaviy, how ramaic improvmn in hop rch. An obrvaion i ha migh gnra a pah wih arbirarily high pah rch. Thi happn bcau h Dircion Rul 3 in [8] migh rul in a ba pah. Figur 9(c) how an xampl of uch ba pah. A wrong rouing ciion i ma a no p loca a h arrow-mark poiion. Figur pic h chmaic iagram of hi xampl. W borrow om noaion from [8]: u i a ourc; v i a inaion; P CCW an P CW ar h nighboring VON no in a counr clockwi an clockwi ircion rpcivly. Bcau pv inrc wih P CCW P CW, Dircion Rul 3 i appli, an a pack i rou in a clockwi ircion, bcau p i on h righ i of uv. A a rul, h pack i rou back along h long bounary of h lowr zig-zag hol unil i rach no q uch ha qq nx inrc wih pv. Figur how h napho of h hop rch for 3 / pair. Whil achiv a ably low pah rch, 86

8 9 GPSR 9 GPSR 9 VON Y (mr) Y (mr) Y (mr) X (mr) X (mr) X (mr) (c) Fig. 9. Scnario : wo hol wih high concaviy; Scnario : many hol wih low concaviy;(c) an xampl of ba pah gnra by ; v CCW P p CW P u Fig.. Exampl of wrong rouing ciion in. Hop rch Hop rch Exprimn counr Fig.. Snapho of hop rch for iniial 3 - pair. uffr from inrminly ariing ba pah hown a h pik in h lowr graph pi i ignificanly lowr pah rch. Th ba pah affc h avrag prformanc of a hown in Figur. In hi figur, h how h avrag pah rch clo o in boh cnario, a w h h o a ufficinly larg numbr, ouprforming GPSR an. Howvr, w no ha incraing h h valu woul la o an incra in h rouing abl iz. Figur how h maximum rch of iffrn proocol for iffrn cnario. Th prformanc gain of i much highr in h maximum hop rch. B. Impac of h Th ym paramr h rmin how far h informaion on h hol i broaca. Smallr h woul cra h mag ovrha. Howvr, h avrag hop rch woul bcom highr for mallr h valu. Thi i bcau, wih mallr h, mor no ar unawar of h locaion of hol. Thu, h ciion o mak a our aroun a hol i mor likly o b lay, ruling in highr avrag hop rch. Figur 3 pic h rul for Scnario, which confirm our xpcaion. A h incra, avrag hop rch cra a xpc. For vry mall h valu lik or, avrag rch of i wor han, bu from h =4, ar o ouprform. C. Mag Ovrha Similar o h xprimnal ing of [8], w aop h upa inrval b, i.., h xrm poin ar broaca vry b con. W aum ha h bacon inrval i con. W maur h oal numbr of mag n pr con, Avrag hop rch Fig.. GPSR Scnario Scnario Maximum hop rch 3 3 GPSR Scnario Scnario Avrag hop rch. Maximum hop rch. no by M, for, wih h =, an wih h = MAX (MAX i > 6 for Scnario ). Th rul ar pic in Figur 3. Fir, i i obviou ha largr upa inrval b ruc M. Th graph alo how ha mor mag ar n for han ha for rgarl of b or h. Thi i bcau rquir aiional mag ohr han flooing h locaion of VON no o run h ianc vcor algorihm; VON no xchang mag wih ach ohr along mulipl hop o mainain rouing abl. A no in Scion VI-B, mallr h prmi mallr mag ovrha. D. Mmory Ovrha In, ach VON no mainain a rouing abl wih h rouing nri for all ohr VON no. Thu, aiional mmory rquirmn for VON no i O(N von ), whr N von i h numbr of VON no. Furhrmor, ach non-von no ha o mainain h viibiliy. In orr o compu h viibiliy, ach non-von no ha o know all h locaion of h VON no; hu h pac complxiy for non- VON no i O(V von ). Compar wih, h mmory ovrha for i horically O(V x ), bcau ach no ha o mainain h locaion of all h xrm poin whn h = MAX. Howvr, V x i ypically mallr han V x u o mor compac rprnaion of a hol. Aiionally, mmory ovrha for i ajuabl by changing h. Tabl I compar h mmory ovrha of an, for h wo cnario. Th iz of rquir mmory for wa mallr han in boh cnario. In paricular, prform br in cnario whr hr ar l hol, bu wih high concaviy. W alo obrv ha mallr h ruc h mmory ovrha. On raon i ha for mallr h, no mainain h a from mallr numbr of hol. 87

9 Avrag hop rch h Numbr of proocol mg pr c (h=) (h=max) b Avrag hop rch Locaion Error (%) Pack rop ra (%) Locaion rror (%) Fig. 3. Impac of h. Mag ovrha. Fig. 4. Effc of locaion rror: avrag pah rch; pack rop ra. TABLE I THE NUMBER OF ROUTING TABLE ENTRIES Cnral rvr (h =) (h = MAX) Scnario 6 74 Scnario (.6) (.4) (, ) (3,4) (4,6) (49,39) (9,) (7,6) (7,4) (, 3) (9,99) E. Impac of Locaion Error In hi cion, w inviga how h locaion rror affc our proocol. W aum ha h x an y coorina of any no can vary a mo ±3m, which i h communicaion rang. W maur h avrag pah rch an pack rop ra for iffrn locaion rror ra, pcifically from % o % of h maximum viaion. Figur 4 pic our fir rul. A hown, h avrag pah rch incra a h locaion rror incra. Nx, Figur 4 how h impac of locaion rror on h pack rop ra. W obrv ha h pack rop ra i narly unil locaion rror i abou 3%, hn ar o rapily incra. VII. SYSTEM IMPLEMENTATION W implmn h DCC an EPRA algorihm on Epic mo running TinyOS... Exprimn wr prform in an inoor b coniing of mo. Each mo i quipp wih a CC4 IEEE 8..4 wirl rancivr an MSP43 procor. Th nir nc implmnaion wa, lin. No locaion ar pr-programm, an a hol i arificially cra by urning off om of h no. Figur pic h rul, whr (x, y) rprn h locaion of ach no. A hown, h bounary no ar inifi an pic a h chain of o lin. Our DCC algorihm icovr h xrm poin ha ar rprn a quar. VIII. CONCLUSIONS AND FUTURE WORK In hi papr, w prn, a gographic rouing proocol ha achiv conan rch wih low ovrha. A hol i compacly rprn by lvraging our iribu convx hull conrucion algorihm. Our xrm poin rucion algorihm furhr ruc h aa iz. Thi aa rprning hol nabl ach ourc no ak an arly our aroun a hol, achiving a conan rch. A fuur work, w will ign a mor fficin algorihm o rou a pack o a no ini a convx hull. W alo plan o vlop a concp of iffrn lvl of convx hull, call iolin, o achiv an vn iribuion of nrgy conumpion. Slcing h propr hrhol for EPRA algorihm alo rmain a our fuur work. Fig.. (.8) (, ) (3,8) (33, 7) (4,) (49,9) (47,) (9,) (7,8) (7,) Sym implmnaion an -b valuaion. REFERENCES [] J. Li, J. Jannoi, D. S. J. D Couo, D. R. Kargr, an R. Morri, A calabl locaion rvic for gographic a hoc rouing, in Proc. of ACM MOBICOM,. [] S. Ranaamy, B. Karp, L. Yin, F. Yu, D. Erin, R. Govinan, an S. Shnkr, Gh: a gographic hah abl for aa-cnric orag, in Proc. of ACM WSNA,. [3] F. Kuhn, R. Wanhofr, Y. Zhang, an A. Zollingr, Gomric a-hoc rouing: of hory an pracic, in Proc. of ACM PODC, 3. [4] Z. Jiang, J. Ma, an W. Lou, An informaion mol for gographic gry forwaring in wirl a-hoc nor nwork, in Proc. of IEEE INFOCOM, 8. [] Q. Fang, J. Gao, an L. J. Guiba, Locaing an bypaing hol in nor nwork, in Proc. of IEEE INFOCOM, 4. [6] C. Liu an J. Wu, Dinaion-rgion-ba local minimum awar gomric rouing, in Proc. of IEEE MASS, 7. [7] N. Ara an Y. Shavi, Minimizing rcovry a in gographic a hoc rouing, IEEE Tranacion on Mobil Compuing, vol. 8, 9. [8] G. Tan, M. Brir, an A.-M. Krmarrc, Viibiliy-graph-ba hor-pah gographic rouing in nor nwork, in Proc. of IEEE INFOCOM, 9. [9] Y. Mao, F. Wang, L. Qiu, S. Lam, an J. Smih, S4: Small a an mall rch compac rouing proocol for larg aic wirl nwork, Nworking, IEEE/ACM Tranacion on, pp , jun. [] P. Li, G. Wang, J. Wu, an H.-C. Yang, Hol rhaping rouing in largcal mobil a-hoc nwork, in Proc. of IEEE GLOBECOM, 9. [] S. P. Fk, A. Kröllr, D. Pfirr, S. Fichr, an C. Buchmann, Nighborhoo-ba opology rcogniion in nor nwork, Proc. of ALGOSENSORS, vol. 3, pp. 3 36, 4. [] Y. Wang an J. Gao, Bounary rcogniion in nor nwork by opological mho, in Proc. of ACM MOBICOM, 6. [3] H. Aiya an J. Wlch, Diribu Compuing: Funamnal, Simulaion an Avanc Topic. John Wily & Son, 4. [4] S. Rajbaum an J. Urruia, Som problm iribu compuaional gomry, in Inrnaional Colloquium on Srucural Informaion an Communicaion Complxiy (SIROCCO), 999. [] T. H. Cormn, C. E. Liron, R. L. Riv, an C. Sin, Inroucion o Algorihm. Th MIT Pr,. [6] Y. Wang an X.-Y. Li, Localiz conrucion of boun gr an planar pannr for wirl a hoc nwork, in Proc. of DIALM-POMC, 3. [7] K. Clarkon, Approximaion algorihm for hor pah moion planning, in Proc. of ACM STOC,