GOAL: A Parsimonious Geographic Routing Protocol for Large Scale Sensor Networks

Transcription

1 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 : A Parsimonious Geographic Rouing Proocol for Large Scale Sensor Neworks Myounggyu Won, Wei Zhang and Radu Soleru Absrac Geographic rouing is well suied for large scale sensor neworks, because is per node sae is independen of he nework size. However, due o he local minimum caused by holes/obsacles, he pah srech of geographic rouing may degrade up o O(c 2 ), where c is he pah lengh of he opimal roue. Recenly, a geographic rouing proocol based on he visibiliy graph (VIGOR) showed ha a consan pah srech can be achieved. The consan pah srech, however, is achieved a he cos of communicaion and sorage overhead, which makes he pracical deploymen of VIGOR in large scale sensor neworks challenging. To his end, we propose (Geomeric Rouing using Absraced Holes), a rouing proocol ha provably achieves a consan pah srech, wih lower communicaion and sorage overhead. To compacly describe holes, we develop a novel disribued convex hull algorihm, which improves he message complexiy O(n log 2 n) of sae of ar disribued convex hull algorihm o O(n log n). The concise represenaion of a hole is used by nodes o make locally opimal rouing decisions. Our heoreical analysis proves he correcness of he proposed algorihms and consan-srech propery of. Through exensive simulaions and experimens on a esbed wih 42 EPIC moes, we demonsrae he effeciveness of and is feasibiliy for resource consrained wireless sensor neworks; specifically, we show ha eliminaes par of communicaion overhead of VIGOR and reduces he memory overhead of VIGOR by up o 5%. Index Terms wireless sensor neworks, geographic rouing proocols, pah srech INTRODUCTION Geographic rouing proocols have araced significan aenion from he wireless sensor nework (WSN) research communiy, because hey are simple and scalable. In geographic rouing, a source node obains he locaion of a desinaion node from a locaion service [], or hrough a hash-funcion in a daa cenric sorage scheme [2]. A packe is hen forwarded o he neighbor ha is geographically closes o he desinaion. This greedy approach allows near-opimal pah lengh in uniform and dense neworks wihou obsacles (i.e., nework holes). However, Kuhn e. al. [3] proved ha, when holes are presen, he pah lengh may degrade up o O(c 2 ), where c is he opimal pah lengh, because of he local minimum phenomenon. In order o bypass he local minimum and ulimaely improve he pah srech, some geographic rouing proocols use he non-local informaion on a hole (i.e., he size, locaion, and shape of a hole) [4][5][6][7]. In hese proocols, holes are idenified firs. The size and shape of he idenified holes are hen propagaed o a subse of nodes, so ha hese nodes use he informaion when hey forward a packe o preven he packe from ending up in a local minimum. However, unless he informaion abou a hole is appropriaely absraced, such informaion M. Won, W. Zhang and R. Soleru are wih he Deparmen of Compuer Science and Engineering, Texas A&M Universiy, College Saion, TX {mgwon, soleru}@cse.amu.edu, charleyhuman@gmail.com can only be made known o a limied subse of nodes. For example, only he nodes locaed a he boundary of a hole called boundary nodes, or some of heir neighbors receive such informaion. Thus, a reacion o he local minimum for example, swiching o he face rouing mode is only acivaed when a packe reaches a node a a local minimum called he suck node, or some neighboring nodes of he suck node. This lae reacion problem resuls in a subopimal rouing pah. Recenly, Tan e. al. [8] proposed VIGOR, a geographic proocol ha achieves a consan pah srech. In VIGOR, a hole is represened as a polygon, which is used o build a visibiliy graph, a srucure ofen used in compuaional geomery o find he shores pah beween a source and a desinaion, given obsacles of polygonal shapes. However, VIGOR suffers from non-negligible proocol-relaed communicaion overhead. Firs, in order o build he visibiliy graph, he locaions of all Visibiliy-based Overlay Nework (VON) nodes (he verices of he polygons) mus be flooded o all nodes in he nework. Second, he VON nodes mus ieraively exchange messages among hemselves unil a complee rouing able is consruced. Third, for each source/desinaion pair, he source node mus send a conrol packe o he desinaion using a defaul rouing proocol (e.g., [9]) o find he enry poin among VON nodes, before i begins daa ransmission. Furhermore, each node in a nework mus sore he locaions of all VON nodes in a nework, incurring he sorage overhead. In his aricle, we propose a geographic rouing proocol ha achieves a consan pah srech, while signifi-

2 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 2 canly reducing he communicaion and sorage overhead of VIGOR. In our proocol, a hole is compacly described as a se of exreme poins of he convex hull covering he boundary nodes of he hole. A novel disribued convex hull algorihm is inroduced o build he convex hull of a hole. To he bes of knowledge, his disribued algorihm improves he message complexiy of he saeof-ar disribued convex hull algorihm [][], from O(n log 2 n) o O(n log n). A disribued exreme poins reducion algorihm furher reduces he size of he se of exreme poins. Consequenly, he locaions of only a few exreme poins of a convex hull are locally broadcas o nodes wihin h-hops from he hole. Based on his informaion on nearby convex hulls, a source node idenifies he inerfering holes wihin h-hops ha block he sraigh pah o he desinaion. A source node hen compues a se of inermediae desinaions ha guide a packe along he locally opimal pah. When a packe reaches an inermediae desinaion, he se of previous inermediae desinaions are updaed ieraively, improving he rouing pah. A roue o and from he nodes inside a convex hull is also handled efficienly, for guaraneed packe delivery. The conribuions of his aricle are as follows: We develop a geographic rouing proocol ha generaes a pah wih consan srech in neworks wih holes. The proocol has lower communicaion and sorage overhead compared wih he sae-of-he-ar proocol. We develop a disribued convex hull algorihm o efficienly reduce he size of daa ha describes a hole. To he bes of our knowledge, his disribued convex hull algorihm has smaller communicaion overhead, when compared wih he sae-of-ar algorihm. We presen a horough analysis o prove he correcness and consan pah srech propery of our rouing proocol. We perform exensive simulaions and experimens on real moes o confirm he effeciveness of our proocol, and is feasibiliy for pracical deploymen in resource consrained WSN. 2 RELATED WORK Rouing proocols for large scale WSNs can be largely caegorized ino geographic rouing proocols and hierarchical rouing proocols [2]. Recenly, Mao e. al. [2] proposed S4, a novel hierarchical rouing proocol, ha achieves he wors-case srech of 3 wih small per-node sae of O( N), where N is he oal number of nodes. Alhough S4 aains he desirable balance beween he pah srech and size of per-node sae, S4 fails o eliminae he dependence beween he per-node sae and nework size. In conras, geographic rouing suis paricularly well for resource consrained large-scale WSNs, because proocol s sae (i.e., he locaions of is immediae neighbors) is independen of he nework size. However, he pah srech of geographic rouing may degrade up o O(c 2 ), where c is he opimal pah lengh, due o he local minimum caused by opological complexiies like holes. Lile was known abou achieving a consan pah srech for geographic rouing proocols. Several geographic rouing proocols have been proposed o improve he pah srech, by idenifying he forwarding nodes ha may lead o a local minimum. In [5], a node uses he TENT rule o es wheher i is a suck node or no. If a node deermines ha i is a suck node, i iniiaes he BOUNDHOLE algorihm o build a rouing pah surrounding he hole by discovering a se of he boundary nodes o guide a packe ou of he local minimum. The boundary nodes are marked and used as landmarks for a fuure packe o bypass he hole, hereby eliminaing he need for implemening face rouing. However, he lae reacion problem (i.e., a deour is made a he boundary node) degrades he pah srech. Arad e. al. [7] use he angle beween wo adjacen neighbors o idenify he nodes ha may forward a packe o a local minimum. Nodes compare he angle wih a predefined hreshold; if he angle is greaer han he hreshold, he node is elevaed so ha his node is avoided by he greedy forwarding process. However, he decision on wheher a node is a a local minimum or no depends on he source and desinaion locaions. Thus, such heurisic approach resuls in he frequen failure of he algorihm. To remedy his problem, in [6], a nework is divided ino k regions. Each node mainains a vecor of size k, where each elemen indicaes wheher his node is a local minimum for he i-h region or no. To deermine he value of each elemen, he local minimum angle b is defined, and if he region is covered more han a cerain percenage by his angle, he region is considered o be he local minimum region. However, none of hese proocols provide he wors case pah srech guaranee. Several researchers proposed o propagae he informaion on a hole (i.e., he size, shape, and locaion of a hole) o a subse of nodes in a limied region. In [4], a suck node and is neighbors form an unsafe area of recangular shape. The esimaed shape of a hole is known o he nodes in his unsafe area. This disribued informaion model is used by forwarding nodes o avoid he local minimum. Alhough he propagaion of he parial informaion on a hole helps improve he pah srech, he lae reacion problem persiss, prevening he proocol from providing guaraneed srech. In [3], a hole is represened as an ellipse, and he absraced informaion on a hole is broadcas o nodes wihin h-hops from he boundary of he ellipse. However, he ellipse ofen fails o represen all hole shapes. Boh proocols [4][3] conribue o he reducion of pah srech, bu hey boh fail o provide a guaraneed consan pah srech.

3 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 3 Recenly, Tan e. al. [8] inroduced VIGOR, a geographic rouing proocol ha guaranees a consan pah srech. VIGOR finds a near opimal rouing pah by exploiing he visibiliy graph. A hole is represened as a se of VON nodes, and a virual overlay nework consising of hese VON nodes guides a packe along he close-o-shores pah ha bypasses holes. One noable aspec of VIGOR, when compared wih S4, is ha is per node sae is O(N von ), where N von is he oal number of VON nodes, hereby eliminaing he dependency of he per node sae on he nework size. However, VIGOR suffers from communicaion and sorage overhead, which makes he pracical deploymen of he proocol in large scale sensor neworks difficul. Firs, he locaions of all VON nodes mus be known o all nodes in a nework; Second, each VON node ieraively exchanges a message conaining is rouing able wih neighboring VON nodes unil he rouing able converges; Third, for each source/desinaion pair, he source node mus send a probing packe o he desinaion by using defaul rouing proocol (e.g., ), before acual daa ransmission begins. In [4], we proposed a geographic rouing proocol ha eliminaes he communicaion overhead for consrucing a rouing able and for seing up a rouing pah, while reaining he consan pah srech propery. A novel disribued convex hull consrucion algorihm absracs he informaion on a hole ino a se of exreme poins of he boundary nodes of he hole; he absraced informaion is used by forwarding nodes o make a rouing decision ha leads o a pah wih guaraneed srech. This aricle exends our previous work. Firs, in a compleely new se of simulaions we use a more realisic radio model [5][6], o more accuraely esimae pah sreches for differen rouing proocols. Second, simulaion scenarios are more sysemaically designed. We propose a new meric describing he complexiy of holes in he nework, and we design simulaion scenarios in an increasing order of his new meric. Third, we implemened our proposed rouing proocol on moe hardware and performed experimens on a esbed consising of 42 EPIC moes. In addiion, in Secion 7.3, we idenify a pracical issue for geographic rouing proocols ha achieve a consan srech. Specifically, he delivery raio for such proocols may significanly degrade when daa rae is relaively high, because such rouing proocols rely on a single bes pah. Thus, his aricle inroduces a new research problem called consan pah srech mulipah rouing, where, depending on he daa rae, he rouing proocol mus offer muliple pahs, each ensuring a guaraneed pah srech. 3 PRELIMINARIES Boundary node In his secion, we define he noaions and erms used hroughou his aricle. Figure illusraes hese noaions and erms. We consider a dense wireless sens p e 2 p e p ini p e 3 p e 4 Convex hull p e 5 Fig.. Illusraion of noaions and erms. Inerfering hole sor nework, consising of N nodes, denoed by a se V = {v, v 2,..., v N }, uniformly disribued in a wo dimensional space. We assume ha each node knows is locaion, which is given as wo-dimensional coordinaes, say (x, y). We furher assume ha he coordinaes (x, y) for each node are unique; ha is, we assume ha muliple nodes canno be deployed a he exacly same locaion. There are m holes in he nework, denoed by H, H 2,..., H m. Each hole H i is surrounded by a se of boundary nodes, denoed by P i = {p, p 2,..., p n }, p j V. A hole is eiher a closed cycle (p = p n ), or a chain (p p n ) ha forms a cycle wih he ouer boundary of a nework. We define one boundary node in each se P i, called he iniiaor, as follows: Definiion : An iniiaor, denoed by p ini, is a boundary node in P i wih he highes y coordinae. If here are several nodes wih he same highes y coordinae, he one wih he lowes x coordinae among hem is he iniiaor p ini. A hole H i is represened as a se of exreme poins of he convex hull covering he se of boundary nodes P i ; his se of exreme poins is denoed by Pe i = {p e, p 2 e,..., p n e } (i.e., Pe i P i ), where he exreme poin p j e is defined as follows (Noe ha we will use he erms node and poin inerchangeably): Definiion 2: An exreme poin is he corner poin of a convex hull. A pah beween source s and desinaion is denoed by s, and he lengh of he pah (i.e., as he number of hops) beween s and is denoed by s. A se of useful noaions are as follows: Definiion 3: Given any wo poins p and q, L( pq) is he se of poins on he lef hand side of a vecor pq, and R( pq) represens he se of poins on he righ hand side of pq. Given a pah s, he inerfering holes are he holes ha inersec wih line segmen s. The inerfering hole can be formally defined as follows: Definiion 4: Given a pah s, a hole represened by a se of exreme poins, P e, is called he inerfering hole iff here exiss some p k e P e, k P e such ha p k e L(s) and p k+ e R(s), or p k e R(s) and p k+ e L(s).

4 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 4 In paricular, we define ha wo nodes are visible o each ohers iff here are no inerfering holes beween hem. 4 : GEOMETRIC ROUTING USING ABSTRACTED HOLES This secion provides he deails of he proposed geographic rouing proocol. Saring from he overview of he proocol, he following subsecions describe he deails of each componen of he proocol. 4. Proocol Overview consiss of wo main componens: hole absracion and packe forwarding. The objecive of he hole absracion process is o concisely represen holes in a nework. In his process, a hole is represened as he convex hull of he boundary nodes surrounding he hole. The hole absracion process is comprised of hree phases. In he firs phase, boundary nodes surrounding a hole are idenified. The second phase consrucs he convex hull of he boundary nodes for each hole; specifically, his phase finds a se of exreme poins from he boundary nodes by using a probing packe ha makes a single raversal along he boundary of a hole. In he las phase, he locaions of exreme poins are broadcas o nodes wihin h-hops from each hole. When he hole absracion process finishes, each node knows he locaions of exreme poins for holes wihin h- hops from i. Nodes use hese locaions o make a rouing decision. To be more specific, a source node idenifies inerfering holes and runs our forwarding algorihm o find an inermediae desinaion among all he exreme poins ha belong o he inerfering holes. The source node hen sends a packe o he inermediae desinaion; upon receiving he packe, he node a he inermediae desinaion runs he same forwarding algorihm o deermine he nex inermediae desinaion. This process is repeaed unil he packe reaches a desinaion. The deails of he wo componens are presened in he following subsecions. 4.2 DCC: Disribued Convex Hull Consrucion This secion describes he deails of he hole absracion process. We adop he boundary node deecion scheme used in VIGOR [8] for implemening he firs phase of his process. This boundary node deecion scheme, when i is done, allows each boundary node o have he noion of a lef and righ neighboring boundary node. Once boundary nodes are found, we selec he iniiaor among he boundary nodes for each hole. If boundary nodes form a cycle, p ini is eleced using an exising leader elecion algorihm on a ring opology, which has message complexiy O(n log n), where n is he number of boundary nodes [7]. This leader elecion algorihm is based on wo assumpions: () each node has unique ID; and (2) each Algorihm DCC (code for p ini ) : if hop coun = hen 2: P e P e {p ini } 3: send a probing packe in couner-clockwise. 4: else 5: if N pi = or p ini R( 6: erminae. 7: end if 8: end if Algorihm 2 DCC (code for p i ) p Pe e p ) hen : for each p l e P e, l P e do 2: if m, l + m P e, p m e L( p l ep i ) hen 3: P e P e \ {p l+ e, p l+2 e,..., p Pe e } 4: end if 5: end for p Pe e 6: if p i R( p i+ ) hen 7: P e P e {p i } 8: // EPRA 9: if P e > Threshold hen : find p i crs wih minimum d i crs : p i e p i crs 2: P e P e \ p i+ e 3: end if 4: // End of EPRA 5: forward a probing packe o p i+. 6: else 7: forward a probing packe o p i+. 8: end if node has he noion of a lef and righ neighbor. The firs assumpion corresponds o he unique coordinaes of each node. The second assumpion is saisfied by he boundary node deecion scheme. If boundary nodes form a chain, one of he wo boundary nodes a each end of he chain is eleced as p ini. Specifically, if node p finds ha i is he boundary node a he end of he chain by checking he number of neighboring boundary nodes, node p sends a message conaining is coordinaes o boundary node q a he oher end of he chain. If node q s y coordinae is greaer han node p s y coordinae, node q becomes p ini. If node q s y coordinae is he same as node p s y coordinae, x coordinaes are compared, and if node q s x coordinae is smaller, hen node q elecs iself as p ini. Once p ini is eleced, p ini sars he DCC algorihm. Algorihm describes he pseudo code for p ini. p ini adds is locaion o he se P e as he firs exreme poin p e, and piggybacks he se P e on a probing packe. This packe is sen o p ini s lef neighboring boundary node, saring o raverse he boundary nodes of he hole in a couner-clockwise direcion (Line 2-3). Upon receiving

5 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 5 p i+3 p i+2 p i+ p i p e l P crs d crs P e 2 P e P e 6 S Inerfering Holes m2 P m The larger convex hull P 2 m 3 S P P 2 m ' Inerfering Hole Hole Fig. 2. Illusraion of DCC. Y (meers) Boundary poin Exreme poin X (meers) Y (meers) P e 3 2 d crs 2 P crs P e 4 P e 5 Fig. 3. Illusraion of EPRA Boundary poin Exreme poin X (meers) Fig. 4. The igh se of exreme poins; reduced exreme poins. he packe, a boundary node p i examines wheher i is an exreme poin, by execuing he code for p i depiced in Algorihm 2. Figure 2 illusraes an example describing how exreme poins are idenified. For each boundary node p i, if p i R( p l ep i+ ), where p l e is he mos recenly seleced exreme poin, hen p i is he farhes (w.r.. he disance he probing packe raveled) visible boundary node from p l e so far, because nex boundary node p i+ is no visible from p l e; hus, p i is seleced as he nex exreme poin and added o he se P e ; p i hen forwards he probing packe conaining he se P e o he nex boundary node p i+ (Line 6-8). Noe ha, a his poin, P e is {p l e, p i }, and he mos recen exreme poin p Pe e is p i. Similarly, p i+ is added o he se P e, because p i+ R( p i p i+2 ) However, p i+2 is no an exreme poin, because p i+2 / R( p i+ p i+3 ). Thus, so far, P e = {p l e, p i, p i+ }. Each ime he DCC algorihm checks wheher a given boundary node is an exreme poin (i.e., he farhes visible node from he mos recenly seleced exreme poin), he DCC algorihm also ensures ha he se of exreme poins P e is updaed when he curren boundary node is farhes from any exising exreme poin in he se P e. For example, if a boundary node p i+3 ha is visible from p l e is found, all previous exreme poins idenified afer p l e (i.e., p i and p i+ ) are deleed from he se P e (Line -5). The above process is repeaed unil he probing packe eiher reurns o he iniiaor, or reaches he end of he chain (code for p ini : Line 5-7). The DCC algorihm requires a single raversal of he probing packe; hus, Fig. 5. Illusraion of : Seps -4; Sep 5. ogeher wih he p ini elecion procedure, he message complexiy is O(n log n), a beer bound han he message complexiy O(n log 2 n) of he currenly known disribued convex hull algorihm on a ring opology [][]. The DCC algorihm generaes a igh convex hull. This means ha for a smooh hole, he DCC algorihm migh generae many exreme poins (e.g., if he hole is a perfec circle, all he boundary nodes will be seleced as exreme poins). A large number of exreme poins will degrade he sysem performance; hus, in some cases, he number of exreme poins mus be conrolled. To his end, we develop he Exreme Poins Reducion Algorihm (EPRA). EPRA limis he oal number of exreme poins by a user defined hreshold. I does no incur addiional communicaion overhead, because i operaes as par of he DCC algorihm. EPRA is embedded in he DCC algorihm, as shown in Algorihm 2 (Line 8-4). Figure 3 illusraes an example ha shows how EPRA works. We define p i crs as he inersecion of wo lines p i ep i+ e and p i+2 e p i+3 e, where i P e, and p Pe + e = p e, p Pe +2 e = p 2 e,... The Euclidean disance (p i crs, p i+ e p i+2 e ) denoed by d i crs, where (p, uv) is a line segmen connecing p and p s projecion on line uv, is compued for each p i crs. When he probing packe reaches poin p j R( p Pe e p j+ ), a candidae for an exreme poin, and he number of exreme poins found so far is greaer han he predefined hreshold (Line 9), he p i crs values for previously found exreme poins are compued; and p i crs wih he smalles corresponding d i crs value is assigned as a new exreme poin; and exising exreme poins p i+ e, and p i+2 e are removed from P e (Line -2). This process is repeaed as he probing packe raverses he boundary nodes. Figure 4 shows an example of he exreme poins generaed by he DCC algorihm, and Figure 4 depics he resuls when DCC algorihm is inegraed wih EPRA. When he DCC algorihm complees, he iniiaor for a hole has he locaions of all exreme poins for he hole. The iniiaor hen broadcass hese locaions o nodes wihin h-hops from he hole. 4.3 Forwarding Algorihm When he hole absracion process finishes, he locaions of exreme poins are made known o nodes wihin h-

6 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 6 v v a s m H 3 m H 2 m H s α δ d m δ H A s α B δ d m δ H d R c u (c) w v b Fig. 6. Illusraion of recursive runs of our rouing algorihm. Fig. 7. Symbols for correcness proof; bounding region represening possible locaions for inermediae desinaions; (c) final bounding region R. hops from a hole. Using his informaion, source node s makes a rouing decision by following he seps described below: Sep : Source s idenifies inerfering holes based on he locaions of received exreme poins, source s, and desinaion. If here is no inerfering hole, source s forwards a packe using a geographic forwarding algorihm. Sep 2: If here are inerfering holes, source s compues a larger convex hull of he se of poins P consising of he exreme poins of he inerfering holes, source s, and desinaion. Figure 5 shows an example of such larger convex hull. This new convex hull can be easily consruced by applying an exising cenralized convex hull algorihm o he se P [8]. Source s hen considers wo possible pahs: one along he upper par of he hull, denoed by P in Figure 5, and he oher pah along he lower par of he hull, denoed by P 2. Sep 3: Source s selecs he shorer pah beween P and P 2. Source s hen uses he exreme poins along he seleced pah as he inermediae desinaions. This sep is depiced in Figure 5 wih he inermediae desinaions, denoed by m, m 2, and m 3. Sep 4: In his sep, source s checks wheher here exis inerfering holes for pah sm. If here is no inerfering hole, he packe is forwarded o m using simple geographic forwarding. Oherwise, Sep 5 is execued, where new inermediae desinaions are found for pah sm. Upon receiving he packe, m becomes a new source s; and he forwarding algorihm reruns from Sep, o reflec he new vision of m. Sep 5: Source s firs uses Seps and 2 o find he wo possible pahs connecing s and m. Figure 5 shows such pahs. If he lengh of one pah is longer han s + m, source s selecs he oher pah. If boh pahs are shorer han s + m, he pah ha is closer o line sm is chosen. Source s hen sends a packe o m using Sep 4. Now we show ha has low compuaional overhead o jusify he feasibiliy for pracical deploymen of. For Sep, each se of exreme poins Pe i is scanned o check wheher any hole H i wihin h- hops inersecs wih line segmen s. The compuaional complexiy of Sep is hus O(N ex ), where N ex is he oal number of exreme poins in he nework. Sep 2 can be easily implemened using an exising cenralized convex hull algorihm. We noe ha he bes performance of currenly known cenralized convex hull algorihms is O(N ex log N ex ). The wors case happens when all holes inerfere wih pah s. Such an exreme case rarely happens, making he average complexiy of lower han O(N ex log N ex ). However, as shown in Figure 6, Seps 4 and 5 of he forwarding algorihm migh be recursively run if here is an inerfering hole, H 2, for pah sm, hen anoher inerfering hole, H 3, for pah sm, and so on. In Secion 5.2, we will show ha he number of such ieraions is bounded by a consan C. Consequenly, he compuaional complexiy of for each node is O(N ex log N ex ). 5 PROTOCOL ANALYSIS 5. Correcness of Convex Hull Consrucion This secion proves he correcness of he DCC algorihm; ha is, we shall show ha given a hole (i.e., a se of boundary nodes), our algorihm finds all exreme poins ha belong o he convex hull of he hole s boundary nodes. We firs show ha he iniiaor node is an exreme poin. Lemma : p ini is an exreme poin. Proof: If boundary nodes form a chain, he claim rivially holds. So, we consider only he case where boundary nodes form a cycle. Assume, by conradicion, ha p ini is no an exreme poin. By definiion, he y-coordinae of p ini is larger han any oher exreme poins. Thus, p ini is no covered by he convex hull, which is a conradicion. Noe ha if here is an exreme poin wih he same y- coordinae as p ini, he x coordinaes of all exreme poins are greaer han p ini. Thus, p ini is no covered by he convex hull, a conradicion. As described in Secion 4.2, he DCC algorihm searches for he farhes visible node from he las discovered exreme poin. The following lemma shows ha such farhes visible node is he nex exreme poin. Lemma 2: Given an exreme poin p i e, he farhes visible boundary node from p i e, say p i+ e poin. Proof: Assume by conradicion ha p i+ e, is he nex exreme is no he nex exreme poin; ha is, here exiss an exreme poin (p i+ e )

7 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 7 ha is closer o p i e han p i+ e. If (p i+ e ) L( p i ep i+ e ), a hole is reshaped as a concave hull. If (p i+ e ) R( p i ep i+ e ), or if (p i+ e ) is on he line p i ep i+ e, hen p i+ e is no visible from p i e. The following lemma shows ha he farhes visible node for he las exreme poin is he iniiaor, creaing a cycle of exreme poins. Lemma 3: p ini is he farhes visible boundary node of p n e when P e = {p e, p 2 e,..., p n e }. Proof: Since he y-coordinae of p ini is he highes among all boundary nodes (or he x-coordinae of p ini is he lowes among all boundary nodes), all boundary nodes on p ini s lef hand side are no visible from p n e. Thus, p ini is he farhes visible node from p n e. Now we are ready for he correcness proof. Theorem : Given a hole H i, DCC finds all exreme poins, say Pe i = {p e, p 2 e,..., p n e }, of he convex hull covering he boundary nodes of H i. Proof: By Lemma, p e = p ini, and subsequen exreme poins are deermined by Lemma 2. Lasly, by Lemma 3, p e is he farhes visible node from p n e. Thus, connecing all p i e, i n (i.e., p e p n e - p e), we ge a convex hull. Now we prove ha here are no more exreme poins. Assume by conradicion ha here is one more exreme poin in P e. Wihou loss of generaliy, assume ha a poin p e is beween wo exreme poins, p i e and p i+ e for some i, i P e. By Lemma 2, p i+ e is he farhes visible node of p i. Consider he case where p e L( p i ep i+ e ). In his case, he resuling polygon becomes concave. If p e R( p i ep i+ e ) or p e is on he line p i ep i+ e, hen, p i+ e is no longer visible from p i e. 5.2 Correcness and Consan Pah Srech of In his secion, we prove he correcness of, and show ha a pah generaed by has a consan pah srech. Consider pah sm in Figure 7, where m is he firs inermediae desinaion for pah P on he upper hull of he larger convex hull (recall Figure 5), and H is an inerfering hole for pah s. Before we presen our main proof, we firs define some symbols and heir geomeric properies. Le α be he angle beween wo line segmens sm and s. The range of α is α < π, because if α > π, hen m would have been in R( s), being a poin for pah P 2, a pah on he lower hull of he larger convex hull. The erm d refers o he Euclidean disance from source s o he firs inerfering hole (i.e., he lengh of line segmen s ). d is smaller han r h, where r is he communicaion radius of a node, and h is he number of hops wihin which he absraced informaion abou hole H is broadcas. δ represens he maximum heigh of an inerfering hole for sm. Noe ha he heigh of an inerfering hole for sm, denoed by δ, is smaller han δ, because if no, s will choose a pah s m. δ can be expressed as d sin α, where α π. Theorem 2: is correc. Proof: In order o prove guaraneed packe delivery, i suffices o show ha a packe is successfully roued from source s o is firs inermediae desinaion m, because when a packe arrives a he firs inermediae desinaion m, m becomes source s; and s applies he same forwarding algorihm from Sep o forward a packe o is nex inermediae desinaion. Therefore, if we prove a successful delivery from s o m, wihou loops or arbirarily long pahs, a guaraneed delivery can be proved by inducion. The heigh of an inerfering hole δ for pah sm mus be smaller han δ, because oherwise such a hole would have been deeced as an inerfering hole for pah s (i.e., δ δ). Therefore, we obain he upper bound for he possible posiions of a new inermediae desinaion, which is depiced as a doed line A in Figure 7. Nex le poin v be he new inermediae desinaion ha belongs o he inerfering hole for sm. One observaion is ha sv + vm mus be smaller han d+ m, because if sv + vm > d + m, hen s would have seleced a pah s m. Thus, he possible locaions of a new inermediae desinaion (i.e., he locaion of poin v) mus be bounded by an ellipse denoed by B having s and m as foci and passing hrough poin. Considering he wo boundaries we compued and he range of he angle beween sm and sv (i.e., o π), he possible locaions of a new inermediae desinaion are bounded by region R as shown in Figure 7(c). This region canno be arbirarily large, since δ is a mos d which depends on he consan parameer h. Theorem 3: has consan srech. Proof: Wihou loss of generaliy, we represen our nework as a Uni Disk Graph (UDG). More precisely, we adop he k bounded degree uni disk graph where he degree of each node is bounded by k [9]. However, k bounded degree uni disk graph can be consruced from a general uni disk graph [3]. As shown in Theorem 2, for any pair of inermediae poins u and v, including source s and desinaion, possible locaions for an inermediae desinaion for pah uv are bounded by some region R. According o Kuhn e. al. [3], he oal number of nodes N R in region R is bounded by (k + ) 8 π (A(R) + p(r) + π), where A(R) is he area of region R, and p(r) is he perimeer of region R. Thus, he oal number of nodes in R is bounded as follows: N R (k+) 8 π {(3 d sin α+2) uv +2d sin α (d+)+6d+π} 2 (k + ) 8 π {(3 2 d + 2) uv + 2d2 + 8d + π} ( < α < π) (k + ) 8 π {(3 2 r h + 2) uv + 2d2 + 8d + π} (d < r h).

8 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 8 d/2 C s Opimal pah d - h d >> Pah of B h A hole d/2 s d - h d >> Fig. 8. he single-hole case; he muli-hole case. By he assumpion of dense and uniform disribuion of nodes and he propery of greedy forwarding, a packe is forwarded ouward from a poin s a each sep of he algorihm. This implies ha each node in region R is visied a mos once. Thus, he oal number of hops H R in R is bounded by N R (i.e., H R N R ). Now consider all (u, v) pairs beween s, and assume ha he sysem parameer h is chosen as he maximum hop coun of he nework so ha all nodes in a nework know he locaions of exreme poins. The oal number of hops from s o, H s is hen given as follows: H s (u,v) s R δ B δ h A hole [(k + ) 8 π {(3 2 r h + 2) uv + 2d2 + 8d + π}]. ((k + ) 8 π {(3 2 r h + 2) (u,v) s uv + 2d 2 + 8d + π}. where (u,v) uv is he shores pah in he Visibiliy s Graph [2]. By [8], he shores pah beween s and in he Visibiliy Graph is bounded by some consan facor of Euclidean disance beween s and as he following: (u,v) s uv sin( π ϵ ) s. Therefore, we ge: H s C s + C 2. C = (k + ) 8 π (3 2 r h + 2) sin( π ϵ ). C 2 = 2(r h) 2 + 8(r h) + π. 5.3 Average Pah Srech of We showed ha he wors-case pah srech of is consan. Now we heoreically analyze he average pah srech of when he sysem parameer h can vary. In his analysis, we consider a d d square region in which nodes are uniformly and densely deployed (i.e., a pah beween wo nodes can be hough of as a line segmen connecing he wo nodes). This analysis for a nework region of square shape can be easily exended o a recangular-shaped nework; and arbirarily shaped nework region can be approximaed by a recangular shape. We assume ha each node has circular communicaion range wih radius, and holes are absraced as convex hulls. We firs consider he case where here is a single hole in a nework. As shown in Figure 8, he area of riangle ABC, which represens he degree of deviaion from he opimal pah, are maximized when: i) source s is locaed in he middle of one side of he square; ii) desinaion is locaed in he middle of oher side of he square ha faces he side ha has s; and iii) he hole wih widh is locaed along he side ha has. The following lemma proves he average pah srech of for he singlehole case. Lemma 4: The average srech λ of for he single hole case is d Proof: The pah lengh of opimal pah is 2 and he pah lengh of is (d h) + Thus, he pah srech is f(h) = (d h)+ 4 + d2 + d 2, d h2 + d 2. d 2 4 +h2 + d 2 d 2 4 +d2 + d 2, and he average pah srech λ for inpu h is given as follows: d h= λ = f(h) + d + 5 because ( 5+)d << and d h= d 2 4 +h2 + d 2 d 2 4 +d2 + d 2 d. Now we invesigae he average srech for he mulihole case. A key observaion is ha he muli-hole case can be considered as a series of single-hole cases for each inerfering hole for s for he following reasons: () each inermediae desinaion makes a new rouing decision by rerunning he forwarding algorihm, and (2) if here are more han wo inerfering holes wihin h-hops, hey are considered as a single convex hull covering all he holes. Consider Figure 8. When a packe reaches a node a B, he packe is deoured o he inermediae desinaion a A. One difference from he single-hole case is ha here migh be oher holes ha inerfere wih he pah from B o A. However, as proven in Theorem 2, he deviaion of he pah BA is bounded by he region R. Thus, he average srech of pah BA, λ becomes: λ = d h= (d h)+ d 2 4 +h2 + d 2 4 +d2 d 2 4 +h2 d Using his propery, we obain he following resul. Theorem 4: The average pah srech of is Proof: Given source s and desinaion, assume ha s visis n inermediae desinaions, denoed by i,..., i n. The average pah srech from s o he firs inermediae desinaion i is a mos 2.5. Thus, he pah lengh of si is

9 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 9 (c) (d) Fig. 9. Differen hole deploymen schemes: hole Scenario ; hole Scenario 2; (c) hole Scenario 3; and (d) hole Scenario 4; These hole schemes are arranged in an increasing order of he concaviy of deployed holes. 2.5d, assuming ha opimal pah lengh is d. Now we consider he pah i and he possible inerfering holes for he pah as a single-hole case. We similarly find ha he pah lengh of i i 2 is a mos 2.5d 2, where d 2 is he opimal pah lengh of i i 2. If we repea his process for all he remaining inermediae desinaions, he oal average pah srech for s is 2.5(d+ +dn )+dn d + +d n Discussion I is imporan o remark ha he consan and average pah srech resuls we have proved for are when source and desinaion nodes are ouside convex hulls, which is precisely wha our moivaing applicaion requires. More precisely, we are developing a disaser managemen applicaion [2] consising of WSN, adhoc and delay oleran neworks. In our applicaion, and many ohers, holes ypically have regular shapes, hus having igh convex hulls. This resuls in very few nodes falling inside convex hulls. In order o handle he source/desinaion node inside a convex hull, our proocol uses he cycle of boundary nodes o guide a packe o eiher leave a convex hull, or reach he desinaion inside a convex hull (i is imporan o remark ha he consan and average pah srech bounds do no apply o his special siuaion). There are hree cases considers: Case : Source s is inside a convex hull. Source s compues he shores pah based on he rouing proocol, and i sends he packe o he firs inermediae desinaion. If source s has a clear pah o he firs inermediae node, he packe is roued o he firs inermediae desinaion by using greedy forwarding. However, if he packe is blocked by a hole, hen he packe would reach one of he boundary nodes. The packe hen sars a couner clockwise raversal along he boundary nodes unil greedy rouing o he firs inermediae node can be resumed. Upon receiving he packe, he firs inermediae desinaion resumes he rouing. Case 2: Desinaion is inside a convex hull. This case is similarly handled as Case. A packe is forwarded along a se of inermediae desinaions previously deermined by our rouing proocol. Upon reaching he las inermediae desinaion, he packe is greedily forwarded o desinaion. If he las inermediae desinaion has a clear pah o desinaion, geographic forwarding is sufficien for he packe o reach he desinaion. Oherwise, he packe would reach one of he boundary nodes. Then, he packe sars raversing he se of boundary nodes in a couner clockwise direcion unil greedy forwarding o desinaion can be resumed. Case 3: Boh source s and desinaion are inside convex hulls. This case can be simply handled as a combinaion of Case and Case 2. An idea for handling all possible source/desinaion pairs in a cohesive manner (i.e., including source/desinaion inside convex hulls), which we are currenly exploring, is o use visibiliy-graph-based rouing echnique, i.e., VIGOR, only for rouing a packe inside a convex hull; concepually, he idea is o decompose he global visibiliy graph ino muliple subgraphs, each subgraph represening he inner srucure of each convex hull; he smaller size of he visibiliy graph reduces he communicaion overhead for building rouing ables, and for seing up a rouing pah. However, his idea faces some challenges: firs, i requires higher implemenaion overhead, because we need o combine wo independen rouing proocols ino a resource consrained node; hus, exracing common feaures of he wo rouing proocols, removing conflicing or unnecessary par of he proocol, and efficienly merging hem are imporan o design an efficien rouing proocol; second, his idea fails o compleely remove he communicaion overhead of VIGOR. Thus, he developmen of his idea remains as our fuure work. 6 SIMULATION RESULTS In his secion we presen performance evaluaion resuls of our proocol execuing in large scale sensor neworks. These large scale sensor nework resuls are obained hrough simulaions. In Secion 7 we will provide experimenal resuls for our proocol, obained hrough a real hardware implemenaion and evaluaion a real esbed consising of 42 EPIC moes. We implemened VIGOR [8],, [9], GOAFR + [22], and he cenralized shores pah rouing

10 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 Toal number of x on perimeer rouing mode (K) Scenario Scenario2 Scenario3 Scenario4 Fig.. Toal number of packe ransmissions in perimeer-rouing mode. This measure represens he degree of concaviy of a hole. Y (meers) DoI=. DoI= X (meers) Y (meers) DoI=. DoI= X (meers) Fig.. Illusraions of radio ranges for differen DOI values: Radio range wih DOI=.2; and Radio range wih DOI=.4. We adop a radio model based on he degree of irregulariy (DOI). proocol) in C++, since we are focusing on he opological behavior of rouing proocols. For his se of simulaions, we randomly deployed 3, nodes in a wo dimensional nework of,,m 2 region. Holes wih varying sizes and shapes are sraegically designed; specifically, we considered four nework configuraions, each having differen degrees of concaviy of deployed holes, as shown in Figures 9, 9, 9(c), and 9(d). The degree of concaviy is quanified by measuring he oal number of packe ransmissions in perimeer-rouing mode, given he daa ransmissions beween randomly seleced, source/desinaion pairs. Figure shows he degree of concaviy for each nework configuraion. As shown, he four nework scenarios are arranged in he increasing order of he concaviy of he deployed holes. In modeling he physical layer of each node, we adoped he radio model from [5][6]. This model is useful o accoun for he realisic communicaion channels. He e. al. [5][6] defines he degree of irregulariy (DOI) as he maximum radio range variaion in he direcion of radio propagaion. Figure and Figure show he radio range for DOI=.2 and DOI=.4, respecively. The defaul communicaion radius was se o 3m wih DOI=.4; he corresponding average node densiy was approximaely 9. We compared our proocol,, wih VIGOR,, and GOAFR + ; specifically, we measured and compared average pah srech, maximum pah srech, communicaion overhead, and sorage overhead among he four proocols. The focus of hese comparisons is o show: () he low pah srech of, (2) he low communicaion and sorage overhead of ; and (3) he impac of imporan parameers. In paricular, he comparisons wih and GOAFR + serve as a base line; ha is, he resuls of such comparisons are used for represening how much he rouing proocols wih consan pah srech (i.e., and VIGOR) can improve he performance compared wih widely used geographic rouing proocols. For his se of experimens, we varied he following parameers: h, δ, communicaion radius r, and he percenage of locaion errors p. The erm h is he number of hops from he boundary of a hole, wihin which nodes receive he informaion on he hole; δ refers o he widh of a bounding box used o find VON nodes [8]; he percenage of locaion errors p is used o simulae he error in he locaion of a node; ha is, he locaion of a node, represened as wo-dimensional coordinaes (x, y), may change o (x ± p, y ± p ) wih random probabiliy, where p = {z R : z > and z < r p}. 6. Hop Srech In his se of experimens, we measured he pah sreches for differen rouing proocols. Given a source and a desinaion, we define he pah srech as he following: pah srech = measured hop coun minimum hop coun, where he measured hop coun is he number of hops along he rouing pah connecing he source and desinaion; he minimum hop coun means he hop coun for he shores pah connecing he source and desinaion; he minimum hop coun is measured by using a cenralized shores pah rouing proocol. We se δ o 3m, and h o a sufficienly large number (e.g., h = for all four scenarios) o allow he absraced informaion on a hole o reach all nodes in he nework. We randomly seleced, source/desinaion pairs, and he pah srech for each pair was calculaed. Figures 2, 2, 2(c), and 2(d) depic he CDF of calculaed pah sreches for Scenario, Scenario 2, Scenario 3, and Scenario 4, respecively. We observe ha, regardless of he nework scenarios wih he differen level of concaviy, he pah sreches of boh VIGOR and are close o. The difference beween he pah srech of VIGOR and ha of is negligible. The main reason for such low pah sreches of he consansrech rouing proocols is ha hose proocols, by using non-local informaion (e.g., absraced informaion on exising holes), allow forwarding nodes o choose a neighbor such ha he packe does no end up in a

11 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 CDF GOAFER+.3 VIGOR Pah srech CDF GOAFER+.2 VIGOR Pah srech CDF GOAFER+.4 VIGOR Pah srech (c) CDF GOAFER+.2 VIGOR Pah srech (d) Fig. 2. The CDF graphs of pah sreches for each deploymen scenarios: Scenario ; Scenario 2; (c) Scenario 3; and (d) Scenario 4. local minimum. In conras o he small pah sreches of consan-srech rouing proocols, he classic geographic rouing proocols, and GOAFR +, exhibi large pah sreches. We also observe ha, as he degree of hole-concaviy increases, he pah sreches of and GOAFR + deeriorae. In paricular, a nework scenario wih small holes wih low concaviy (i.e., Scenario ) shows ha he pah sreches of and GOAFR + do no degrade much, because he chances for a packe being suck in a local minimum is relaively smaller compared wih oher scenarios having holes wih high concaviy. We summarize he saisical daa of pah sreches by means of he average and maximum pah srech. The average and maximum srech are ofen used o measure he average and wors-case performance of a rouing proocol, respecively. Figure 3 depics he average sreches for differen rouing proocols in he four scenarios. As shown, he average pah sreches of and VIGOR are small, regardless of he hole scheme, as boh proocols preven a packe from falling ino a local minimum. Compared wih he classic geographic rouing proocols (e.g., and GOAFR + ), he average pah srech of is up o 5% smaller in our nework configuraions. We also observe ha he concaviy of deployed holes influences he performance of a rouing proocol. As shown in Figure 3, he average pah srech increases when here are holes wih more complex shapes in a nework. The difference in he wors-case performance beween radiional geographic rouing proocols and is much larger as shown in Figure 3; he maximum srech of is smaller han and GOAFR + by up o 3,8%. Addiionally, we observe ha, similar wih he resuls for he average pah srech, he maximum pah srech increases, as here are holes wih more complex shapes in he nework. 6.2 Communicaion Overhead Secion 6. shows ha he pah srech of is as small as ha of VIGOR. Besides he consan pah srech propery, he srengh of lies in he reduced communicaion and sorage overhead compared wih VIGOR. In his secion, we shall analyze he communicaion overhead of VIGOR o verify he energy efficiency of. Average pah srech GOAFER+ VIGOR Scenario Scenario2 Scenario3 Scenario4 Max pah srech GOAFER+ VIGOR Scenario Scenario2 Scenario3 Scenario4 Fig. 3. Saisical summary of pah sreches: Average hop srech; and Maximum hop srech. achieves significan improvemens in boh average and maximum pah srech. The main source of VIGOR s communicaion overhead comes from wo aspecs of he proocol: firs, each VON node mus ieraively exchange rouing ables wih oher VON nodes unil is rouing able converges; second, for each source/desinaion pair, he source node mus send a conrol packe o he desinaion using he defaul rouing proocol (e.g., ) o find he enry poin among he VON nodes; ha is, he source node mus join he overlay nework of he VON nodes, by sending a conrol packe o he desinaion, before he packe ransmission begins. We firs measured he oal number of packe ransmissions used for building rouing ables by varying δ. The parameer δ defines he widh of a bounding box ha is used o consruc VON polygons. If his value is large, we can reduce he number of oal VON nodes, hereby reducing he communicaion overhead for consrucing he rouing ables. However, if δ is large, he edges of VON polygons may inersec; furhermore, he pah srech may deeriorae, because he boundary of a hole is no precisely represened. Figure 4 depics he resuls. As shown, each nework scenario suffers from a large number of daa packe ransmissions o build rouing ables. We also find ha his communicaion overhead decreases when δ increases, a he cos of imprecise represenaion of he boundary of a hole ha leads o increased pah srech. We hen measured he oal number of conrol packe ransmissions used for seing up a rouing pah. Similar wih he experimenal configuraions in Secion 6.,

12 TRANSACTIONS ON PARALLEL AND DISTRIBUTED COMPUTING, VOL. X, NO. X, SEPTEMBER 2 2 Number of ransmissions for rouing able build-up Scenario Scenario 2 Scenario 3 Scenario δ (m) Fig. 4. Communicaion overhead due o rouing able consrucion. VON nodes ieraively exchange messages unil rouing ables are buil. TABLE Sorage overhead of VIGOR and Average pah srech δ VIGOR VIGOR Num of TX for pah-se up (k) r=3 r=5 r=7 r=9 Scenario Scenario2 Scenario3 Scenario4 Fig. 5. Communicaion overhead due o pah se-up. VIGOR requires each source/desinaion pair o exchange a conrol packe, using he underlying defaul geographic rouing proocol, before acual daa ransmission occurs. we randomly chose, source/desinaion pairs. Figure 5 shows he resuls. One observaion is ha his communicaion overhead is influenced by he concaviy of deployed holes. The main reason is ha VIGOR uses he underlying defaul geographic rouing proocol for ransmiing his conrol packe o a desinaion, and he pah srech of he geographic rouing proocol degrades when here are holes of complex shapes. The communicaion radius r of a node is anoher facor ha affecs his ype of communicaion overhead. As shown in Figure 5, larger communicaion radius allows nodes o use fewer packe ransmissions. Finally, i is imporan o menion ha eliminaes hese wo ypes of communicaion overhead of VIGOR, hereby achieving is energy efficiency, and is feasibiliy for pracical deploymen. 6.3 Sorage Overhead In he previous secion, we analyzed he communicaion overhead of VIGOR. In his secion, we sudy he sorage overhead of VIGOR. In VIGOR, each VON node mainains a rouing able; each enry of he rouing able specifies he bes neighboring VON node for a given desinaion VON node. In addiion, each non-von node has o mainain he visibiliy se, a se of visible VON nodes h Fig. 6. The impac of sysem parameer h. The parameer h defines he number of hops from a hole wihin which he informaion abou he hole is disribued. In order o compue his visibiliy se, each non-von node has o know all he locaions of VON nodes. Thus, addiional memory requiremen for nodes in VIGOR is heoreically O(V von ), where V von is he number of VON nodes. Compared wih VIGOR, he memory overhead of is heoreically O(V ex ), where V ex refers o he number of oal exreme poins in a nework, because each node has o mainain he locaions of all he exreme poins when he h value is sufficienly large (noe ha he memory overhead of may be reduced by adjusing he h value a he cos of increased pah srech). We noe ha V ex is smaller han V von, because he exreme poins are he subse of he VON nodes. Therefore, reduces he sorage overhead. Table I compares he memory overhead of wih ha of VIGOR in he nework scenario shown in Figure 9(d). Specifically, Table I compares he oal number of VON nodes of VIGOR, and he oal number of exreme poins of on each row of he able wih varying δ values. As expeced, he number of VON nodes is larger han he number of exreme poins. In addiion, he number of VON nodes decreases as we increase he δ value, and obviously, is no influenced by he δ value. If he δ value is sufficienly large, he number of VON nodes may be reasonably small; however, his is possible a he cos of worse pah srech, possibly wih crossing edges, because imprecise represenaion of he boundary of a hole increases he chances of using he face rouing when a packe is roued beween wo neighboring VON nodes. 6.4 Impac of h The sysem parameer h is he number of hops from he boundary of a hole, wihin which nodes receive hole