Landmark Selection and Greedy Landmark-Descent Routing for Sensor Networks

Transcription

1 1 Landmark Selection and Greedy Landmark-Decent Routing for Senor Network An Nguyen Nikola Miloavljević Qing Fang Jie Gao Leonida J. Guiba Department of Computer Science, Stanford Univerity. Department of Electrical Engineering, Stanford Univerity. Department of Computer Science, Stony Brook Univerity. Abtract We tudy the problem of landmark election for landmark-baed routing in a network of fixed wirele communication node. We preent a ditributed landmark election algorithm that doe not rely on global clock ynchronization, and a companion local greedy landmark-baed routing cheme. We aume no node location information, and that each node can communicate with ome of it geographic neighbor. Each node i named by it hop count ditance to a mall number of nearby landmark. Greedy routing at a node i performed to equalize it vector of landmark ditance to that of the detination. Thi i done by following the hortet path to the landmark that maximize the ratio of it ditance to the ource and the detination. In addition, we propoe a method to alleviate the difficulty in routing to detination near the boundarie by virtually expanding the network boundarie. The greedy routing, when combined with our landmark election cheme, ha a provable bounded path tretch relative to the bet path poible, and guarantee packet delivery in the continuou domain. In the dicrete domain, our imulation how that the landmark election cheme i effective, and the companion routing cheme perform well under realitic etting. Both the landmark election and greedy routing aume no pecific communication model and work with aymmetric link. Although ome of the analyi are non-trivial, the algorithm are imple, flexible and cot-effective enough to warrant a real-world deployment. I. INTRODUCTION The application emerging from the enornet community will depend, in part, on good deign and implementation of calable point-to-point routing. In thi pace, both geographic location-baed and location-free method have been propoed. In geographic routing, the node currently holding the packet forward the packet to it neighboring node that i cloet to the detination. Variou meaning of cloet are poible. The preence of hole in the enor field can caue thee greedy method to fail at node with no neighbor cloer to the detination. The failure of greedy forwarding i mainly due to the mimatch between naming/routing rule and the actual network connectivity when the node ditribution i pare, having hole or of complex hape. A variety of method have been propoed to overcome thi difficulty and to guarantee packet delivery, if at all poible. Probably the bet known approach among thee, the cheme of greedy routing combined with face routing [1], [2], [3] build a planar ubgraph of the connectivity graph and ue perimeter Acknowledgment: The author wih to acknowledge the upport of NSF grant CNS , CNS , ARO grant W911NF , and the upport of DoD Multidiciplinary Univerity Reearch Initiative (MURI) program adminitered by ONR under Grant N forwarding when greedy forwarding get tuck. While greedy forwarding i quite robut to enor location inaccuracie, perimeter routing fail in practice a delivery rate of only 68% i oberved [4], [5]. The problem come from the failure of the planarization of the communication graph in real world deployment. The relative neighborhood graph and the Gabriel graph are both connected planar ubgraph of unit dik graph (UDG) in theory. In reality they turn out to be diconnected due to directional or croing edge, a the conequence of location inaccuracy and the deviation of the real connectivity model from the UDG model. To olve thi problem depite the fact that accurate geographic location information i difficult and expenive to obtain, everal location-free routing cheme have been developed [6], [7], [8], [9]. Thee cheme build virtual coordinate ytem, repecting the true network connectivity, on which greedy meage delivery rule are developed. Several virtual coordinate routing ytem are landmarkbaed [7], [8], [9]. Generally peaking, a ubet of the node are elected a landmark. Every node record it hop count ditance to thee landmark. The landmark ditance are then ued to generate virtual coordinate for the node. Routing i typically guided in a greedy way by a potential function on the landmark-baed ditance. The major difference among thee cheme are motly in their potential function. Take two landmark-baed cheme a example. Gradient Landmark-baed Routing (GLIDER) [7] eparate global and local routing by partitioning the network into combinatorial Voronoi tile uch that the node inide the ame tile are cloet to the ame landmark. The tile adjacency graph i ued to guide global routing acro tile. Local routing within a tile i done by greedy decending on a potential function contructed uing ditance to a et of local landmark (the landmark in the adjacent tile). Such a potential function i local minimum free in the continuou domain. In another cheme, Beacon Vector Routing (BVR) [8] take a imilar but more heuritic approach and deign a potential function that depend on the ditance to the k landmark cloet to the detination, where k i a ytem parameter. Out of the k landmark, the one that are cloer to the detination than to the ource impoe a pulling force while the ret impoe a puhing force. The potential function i a combination of the two. Landmark-baed cheme are favored by their implicity and independence on the dimenion of the network. Unlike face routing in planar graph that require a planar deployment of enor node, landmark-baed routing cheme can be eaily extended to enor deployed in 3D.

2 performance of BVR conitently and ignificantly compared to that uing a random landmark election. Fig. 1. A typical enor field and routing path between 2 node in it. An important iue in all landmark-baed cheme that ha not been addreed i landmark election. Obviouly, poorly elected landmark can reult in poor routing performance. Simulation often aume that landmark are randomly elected or manually placed along network boundarie [7], [8]. It remain a challenging open problem, both in theory and in practice, to undertand the criteria for landmark election and their influence on routing performance. In thi paper we propoe a ditributed landmark election protocol and an accompanying imple routing cheme, Greedy Landmark Decent Routing (GLDR). In the continuou domain, GLDR guarantee packet delivery with provably bounded path tretch relative to the bet path poible. It exhibit good routing performance in network, a verified by our imulation. In our cheme, landmark are elected by a randomized and ditributed proce that enure approximately uniform pacing and good eparation of landmark, for a given eparation parameter. We how that only a contant number of nearet landmark of a node, denoted by the addreing landmark, i needed to addre the node for routing. In practice, a few a 6 nearet landmark are enough to obtain a high routing ucce rate. The routing i imple: intuitively, when the ource i far away, the addreing landmark that i cloet to the detination applie a pulling force until the packet get cloe to the detination. At thi point, the algorithm route the packet toward a ingle landmark elected by a local rule until the ditance to the landmark equal the ditance from the detination to that landmark. Then a new landmark i elected and the above proce repeat. We evaluated the performance on low-degree graph on randomly placed node and oberved favorable performance in both routing tretch and delivery rate compared to BVR. The propoed landmark election protocol can be combined with other landmark-baed routing method. We how by imulation that our landmark election protocol improve the II. LANDMARK SELECTION The ucce of GLDR depend on making good choice of the et of landmark. We build a hortet path tree rooted at each landmark via flooding in the initialization phae. Intuitively, GLDR allow the landmark near the detination to impoe a pulling force on the packet following the hortet path. A packet en route i therefore under the force from a et of nearby landmark. By chooing only one pulling force at a time according to a local rule, and with a et of well elected landmark, we can navigate the packet in the network. Therefore, a deirable landmark election algorithm hould give a et of landmark that are evenly pread out, and each node in the network, if not near a field boundary, i well urrounded by a mall et of nearby landmark. We how that we can ue an r-ampling to elect a good et of landmark. We how, in the next ection, how the r-ampling propertie are eential to guarantee the greedy routing ucce, and to yield a theoretically bounded per-hop tretch factor (of 5 for realitic etting). In fact, by incurring higher cot in node addre compoition, even lower tretch could be achieved. For detail, ee ection III-B. We propoe a ditributed protocol to implement an approximate r-ampling in a practical etting. A. r-ampling in the Continuou Domain Let B(p, r) denote the ball (dik) of radiu r centered at p. Definition 1: Given a continuou domain S, an r-ampling of S i a et of point R S uch that each point in S i within ditance r of ome point in R, i.e. S p R B(p, r). Definition 2: We call a point p an interior point if a ball centered at p with radiu 4r i completely inide S, i.e. B(p, 4r) S. We are intereted in landmark that form an r-ampling and would like to reduce the total number of landmark needed for a given enor field. For a 2-D infinite continuou domain, the bet reult poible i to place landmark at point uch that they form a hexagonal ampling pattern [10]. Thi could be achieved by a global equential algorithm. Another trategy i to ue random ampling. How many independent ample (in fact landmark), elected uniformly at random from S, are ufficient to yield an r-ampling with a confidence level? We divide the domain into quare grid cell of area Θ(r 2 ) and olve for the number of landmark neceary to guarantee at leat one landmark in each grid cell. Thi reduce our quetion to the coupon collector problem. By a well known reult [11], for m equally ized grid cell, the expected number of landmark needed i m(ln m+o(1)). The logarithmic factor characterize the expected non-uniformity of a uniform random ditribution. In other word, the overampling rate of the randomized landmark election algorithm with repect to the determinitic optimal algorithm i about logarithmic in the number of grid cell.

3 B. r-ampling in the Dicrete Cae Definition 1 and 2 have their dicrete-cae analog, with the ditance between a pair of node defined a the length of the hortet path between them. Definition 3: The r-neighborhood of node c, denoted by B(c, r), i the et of node at mot r hop away from c. We aume in thi ubection that the enor field i nice, in a ene that any neighborhood of radiu r in the enor field contain at leat αr 2 node for ome contant α. To obtain an r-ampling, we need to have at leat one landmark in every r-neighborhood of any interior node. One way to carry out the ampling i to ue the following equential greedy procedure. Algorithm 1: All node are initially unmarked. While there are unmarked node, repeatedly pick an unmarked node, make it a landmark, and mark all node in it r-neighborhood. Thi algorithm produce a et of landmark with a minimum ditance of r-hop apart, and each non-ampled node being in the r-neighborhood of at leat one landmark. Since the landmark r/2-neighborhood are dijoint, there are at mot 4n/(αr 2 ) landmark. On the other hand, in an ad hoc network, any equential algorithm i prohibitively expenive. A parallel algorithm i thu much more preferable. Thi motivate the following randomized algorithm that could be executed in parallel. Algorithm 2: Each node make itelf a landmark with a probability p where p i ome contant. We have the following reult: Theorem 1: The et of landmark produced by Algorithm 2 i an r-ampling with probability if p 1 (1 αr2 16n ) 4 αr 2. Proof: Firt elect a ubet of node which we call center by following the procedure decribed in Algorithm 1, ubtituting the r-neighborhood with r 2-neighborhood. By the previou analyi, there are at mot m = 16n/(αr 2 ) center. If we make ure that there i a landmark in each center r 2-neighborhood, the et of landmark would be an r- ampling. Let E i be the event that no node i ampled in center i r 2 -neighborhood, then Pr[E i] (1 p) αr2 /4. The probability that Algorithm 2 produce an r-ampling i given by Pr [ m i=1 ( E i)] (1 (1 p) αr2 /4 ) n/(αr2 /16). Setting thi probability to be at leat and olve for p we obtain the deired reult. Corollary 1: The expected over-ampling rate of the parallel random ampling algorithm to that of the equential determinitic algorithm i therefore at leat: n [1 (1 αr2 16n ) 4 αr 2 ] 4n αr2 4 αr 2 [ = k 1 [ ( 1 16n 1 αr 2 ) 4 αr 2 ] ( ) ] 1/k f 4m Where k = αr 2 /4 i the lower bound on the number of node in an r 2-neighborhood, f = 1. For a fixed confidence level and a large network, a the r- neighborhood ize grow, the over-ampling rate of the parallel algorithm to that of the equential algorithm grow linearly with the local neighborhood ize (which i quadratic in r). The performance gap between the equential algorithm and the randomized parallel algorithm i rather ignificant. Thi reult motivated u to deign a protocol that find the middle ground between the two algorithm. C. The Protocol During the network initialization phae, we elect a et of landmark by computing an approximate r-ampling of the network node, and the hortet path tree rooted at thee landmark. We propoe to ue a randomized, ditributed cheme that combine both procee. Firt, each node in the network generate a number v uniformly at random in [0, 1). Each node then wait for Kv time, where K i a network parameter, and the unit of time i the time it take to end a packet from a node to it neighbor. If during thi waiting period, a node i not uppreed by any other node, it declare itelf a landmark. The new landmark broadcat it tatu to it r-neighborhood, uppreing all non-landmark node in it. When K i large, the above cheme effectively give u a equential algorithm for computing an r-ampling. When K i mall, ome upreing meage arrive too late due to the delay incurred in communication. Conequently, additional landmark are elected. When the node are ditributed approximately regularly (a defined below), we can upper bound the expected number of landmark. Definition 4: We ay that the node ditribution i (α, β)- regular (β α 1) if the r -neighborhood contain between αr 2 and βr 2 node, for any r R. Theorem 2: If the node ditribution i (α, β)-regular, then any r-neighborhood contain, in expectation, at mot (9β/Kr β 1) landmark. Proof: See Appendix VI-A. Theorem 2 how that by adjuting K, we can tradeoff between the over-ampling rate and the time require to complete the ampling proce. For K = Θ(r), the overampling rate grow linearly with r. To have a contant overampling rate, we need K = Θ(r 2 ). III. LANDMARK-DESCENT THE GREEDY RULE To route a packet to a detination, the ource node elect, out of the landmark that addre the detination, the one that maximize the ratio of ditance to ource and detination. It then move toward thi landmark along the hortet path, until the landmark become equally ditant from the ource and the detination, at which point, the proce repeat. Effectively, the ource i trying to equalize it vector of landmark ditance to that of the detination, by only reducing the ditance, i.e. going toward landmark, not away from them. If the ource and the detination coincide, thee vector are indeed equal. However, the convere i not true becaue multiple node may have the ame hop count ditance to the ame et of landmark. Neverthele, in the continuou domain we will how that not only doe the convere hold, but the ditance vector can be made equal uing a imple greedy trategy, i.e. decreaing the coordinate that maximize the dicrepancy (the ratio to the correponding detination coordinate).

4 In the dicrete domain, our experimental reult how that uch a greedy trategy ucceed with high probability. However, it i poible that the packet get tuck into a loop before reaching it detination. Baed on our imulation, by remembering the ID of the lat 8 extreme node (node at which the packet topped proceeding to it current landmark), we can detect uch a ituation and declare the packet get tuck. Once a packet get tuck, it i greedily forwarded to the detination uing L 1 and L norm, and if the detination i till not reached, a coped flooding i ued to deliver the packet. In the following ection, we firt decribe the continuou verion of the landmark decent routing and provide theoretical reult. We then decribe the routing algorithm in a dicrete network. We aume continuou, imply connected (without hole ) enor domain with an outer boundary in the form of a cloed curve. d 4r Fig. 2. Landmark decent can alway proceed if d i an interior point. l d l Fig. 3. Landmark decent alway make progre if l > dl. A. Greedy Succe in the Continuou Domain Suppoe there i a et of landmark node in the domain. The algorithm will ue only ditance to landmark node for routing to non-landmark point. Each detination i addreed by the vector of ditance from a ubet of landmark. We will ee later, by theory and imulation, that thi number can be made mall (independent of the network ize). In landmark-decent, the ource i trying to equalize it vector of landmark ditance to that of the detination. If the ource and the detination coincide, thee vector are indeed equal. In thi ection we how that under certain condition on the landmark placement the convere i alo true, i.e. the detination i the only point with thi property. Furthermore, we how that the vector can be made equal uing a imple greedy trategy. For implicity of notation, throughout we ue, d and l to denote the ource, detination and landmark point that are ued in one routing tep. Furthermore, we denote by the point to which the ource move after the tep ha been taken. The following lemma ay that unle it ha reached the detination, the ource can alway find a dicrepancy in the coordinate, i.e. the proce can never terminate prematurely. Thi i under the condition that the detination be an interior point. Lemma 1: If d i an interior point, then a long a d, there i a landmark l that atifie l > dl. Proof: Aume d, and conider the interection of B(d, 4r) and the open halfplane defined by the line through d perpendicular to d, and not containing (Figure 2). Thi area i contained in the domain, ince d i an interior point, and it contain a ball of radiu r. Therefore it contain a landmark l, by the r-ampling condition. Clearly dl > 90 > dl, o l > dl. The next reult ay that by equalizing the coordinate, the ource indeed get cloer to the detination in term of the traight-line ditance. Lemma 2: If l > dl, then d < d. Fig. 5. Upper bound on R/r for variou κ. Proof: Refer to Figure 3. Since d l i iocele, d l < 90, o d > 90. Thi implie d < 90, hence d < d. From the previou two reult it i obviou that the routing proce eventually terminate by reaching the detination, if the latter i an interior point. We ummarize thi part in the following theorem. Theorem 3: If d i an interior point, the landmark decent routing to d i alway ucceful. B. Bounded Stretch in the Continuou Domain In thi ection, we analyze the theoretical performance of our routing approach. Let be the current point, d the detination, l a landmark, ρ = l / dl, and φ = π dl (ee Figure 4). Aume that ρ > 1 and let be the point on the line l uch that l = dl. The ratio κ = /( d d ) we call the one-tep tretch factor of routing from to d uing l. ρx φ d Fig. 4. Bounding the one-tep tretch κ by a function of ρ or φ. We firt relate κ to the minimum ditance between d and it addreing landmark (including l). Lemma 3: If the landmark form an r-ampling and the ditance between d and it addreing landmark i at leat R, then f(κ) R/r, where f = u w, u = 1 1 κ 2, v = 1 1 κ, w = (1 u 2 ) + 2u 2 (1 v 2 ) + 2u (1 v 2 )(1 u 2 v 2 ). Proof: See Appendix VI-B α x l

5 It i traightforward to how that for κ > 1, f(κ) i monotonically decreaing with range (2, ). It follow that it invere f 1 exit, and κ f 1 (R/r) for R > 2r. The plot of f i hown in Figure 5. Apparently, the lower the tretch upper-bound, the farther away the addreing landmark need to be. To have an upper bound of 5 on the per-hop tretch, we need each node to be addreed by it ditance to landmark that are at leat 5r away. We will how in the next corollary that the number of addreing landmark for each node i an cale-free contant. We have thu proved: Theorem 4: If R > 2r, then κ i bounded. In the ret of the paper we aume R = 2r for implicity. Now we can bound the number of addreing landmark, a quantity eential for calability of the ytem, under the regularity aumption of node ditribution in the previou ection. Corollary 2: If the node ditribution i (α, β)-regular, then the expected number of addreing landmark for any node i at mot (144β/Kr β 1)p. Proof: Not difficult, but omitted for brevity. C. Greedy Landmark Decend Routing (GLDR) in Network Each GLDR packet carrie in it header, (i) the addre of the detination, (ii) the ID of the landmark toward which the packet i forwarded, (iii) a lit of recently encountered extreme node. The addre of a detination i a vector of ditance to it addreing landmark, together with it ID (multiple node may have the ame et of nearet landmark). A packet header i updated only at extreme node, node with the ame ditance to the current landmark a the detination. At an extreme node, if there i no landmark that i farther from the node than from the detination or if the extreme node i found in the packet header, the packet i tuck. If not, the current extreme node i added to the header. The packet i then forwarded toward the landmark that maximize the ratio of ditance to the current node and to the detination. If a packet get tuck, we reort to flooding. A thi i expenive, we actually forward the packet greedily uing L 1 norm, then uing L norm. If the detination i till not reached, we perform coped flooding. The peudo code to implement the algorithm i hown in III-C. To keep the packet header mall, we do not tore all extreme node that we encountered and only keep the lat 8 recently encountered extreme node in the header. While there i no theoretical proof that uch a hort lit uffice to detect looping, in our extenive imulation we never encountered a loop. Thi i becaue looping i uually caued by too mall a number of landmark being available for routing. We would like to emphaize that GLDR i imple and eay to implement. A local greedy rule guide the greedy routing at each tep. The protocol i tatele unle loop occur in the route, which rarely happen. D. Handling the Boundary Effect Recall that the reult we dicued up to now apply to interior node only. For the non-interior node, the routing 1: if p.mode = GLDR then 2: if dit(p.betbae) > dit(p.det) then 3: FORWARD p toward p.betbae 4: ele 5: if current node i in p.extremenode then 6: SET p.mode = L 1 7: ele 8: RECORD current node into p.extremenode 9: COMPUTE new bae p.betbae 10: if p.betbae = null then 11: SET p.mode = L 1 12: ele 13: FORWARD p toward p.betbae 14: end if 15: end if 16: end if 17: end if 18: if p.mode = L 1 then 19: if ome neighbor nb i cloer to p.det uing L 1 then 20: FORWARD p to nb 21: end if 22: SET p.mode = L 23: end if 24: if p.mode = L then 25: if ome neighbor nb i cloer to p.det uing L then 26: FORWARD p to nb 27: end if 28: FLOOD to find p.det 29: end if Algorithm 1. Peudocode for routing a packet p. i more challenging. Intuitively peaking, for virtual coordinate baed method, routing to detination near the network boundarie i more difficult becaue there are le and/or only one-ided reference point to navigate the packet. Specifically in our cae, the aumption of our continuou domain reult (Lemma 2, Theorem 3, and 4) are not even approximately atified near the boundary. Thi etback i common in all virtual coordinate baed method. To alleviate thi boundary effect, we propoe to virtually expand the enor field. Suppoe that we duplicate the noninterior part of the network, and place it immediately below the original o that they meet along the boundary. The expanded network i obtained by unfolding the replica. Thi effectively provide pulling force on packet toward the boundary. We firt extend the definition of hortet path and ditance. Path between the original (real) and replicated (virtual) node are required to viit a boundary node, and are thu called reflected path. The ditance between two uch node (reflected ditance) i the length of hortet reflected path connecting them. The reflected and the tandard traightline ditance between x and y are denoted by xy and xy, repectively. Now we can define an augmented landmark election procedure and the matching routing algorithm to be eentially the ame a in the bae cae. Thi i poible becaue the algorithm operate only on hortet path. If the boundary i not very curved compared to r, it locally look like a traight line (Figure 6 left). Thu uing a virtual landmark i almot the ame a uing it mirror image with repect to the boundary, viewed a a real landmark.

6 x l l Fig. 6. Left: When the boundary i locally traight, replacing a virtual landmark l by a reflected real landmark l preerve the ditance involved (e.g. l = l ), and doe not change the routing tep. For a general low-curvature boundary, the change will be mall. Right: Aumption on the curvature of the boundary. From the above dicuion it hould be clear that the reult will depend on ome meaure of curvature of the boundary. A the boundary i not alway known in a realitic etting, and even more complex iue arie when there are large hole [12], [13] in the network, we at thi tage tudy thi method more or le a a theoretic intereting poibility rather than a practical olution to ad-hoc network. In particular, we aume that the boundary i a mooth curve uch that the two tangential ball of radiu R at any point x on on the curve, do not interect the curve anywhere except in x (Figure 6 right). In practice the boundary can often be approximated by uch a curve to within an accuracy comparable to dicretization error. Theorem 5: If R i large enough compared to r, the augmented-gldr guarantee packet delivery along a path of bounded tretch factor. In particular, if R 320r, and the tretch factor of the bae cae routing i κ < 2.5 (cf. Theorem 4), then the tretch factor of the augmented-gldr i at mot 7κ/(5 2κ). Proof: The key idea i to replace a virtual landmark ued in a given routing tep by a carefully deigned real landmark which reult in a imilar routing tep. The bae cae reult can then be applied to the latter. For technical detail, ee Appendix VI-C. The reaon for uing virtual landmark i to compenate for the lack of addreing landmark for non-interior node along the field boundary. They are ued, and only incur extra torage cot, in a local neighborhood along the boundary. Specifically, virtual landmark need only to flood 12r hop inward from the boundary, to let node in thi region learn their ditance to virtual landmark, and non-interior node learn their local addree. It i eay to ee that only node within 12r hop away from the boundary need to learn their ditance to virtual landmark: all the other node are interior, o they mut have at leat one real addreing landmark; after only one tep of landmark decent (alway to a real landmark), the new ource poition will be within 8r hop from d (for it uffice to pick any of d real addreing landmark, which are all at mot 4r hop away from d), i.e. at mot 12r hop away from the boundary. IV. SIMULATIONS We implemented our landmark election and routing algorithm in java, for heuritic tudy. We imulate the network at the network level, not taking into account networking iue uch a packet lo or delay. R x R A. Default imulation etup In our imulation, we generated 3200 node randomly in a unit quare, ee Figure 1. The communication radiu wa et to make the average node degree roughly 10. The field wa fairly dene, but till had many mall hole, which made greedy routing challenging. We generated the landmark to form an 10-ampling. The default K wa For a given enor field, we ran our landmark election cheme 100 time, and for each et of landmark obtained, we generated 320 pair of random ource and detination point and computed the route between them uing that et of landmark. When routing, we ued the ditance to the nearet 10 landmark of a node a it addre. In each imulation, we varied a ingle parameter and kept the other parameter a in the above default etting. We collected tatitic on all the path computed for different parameter we were intereted in. For comparion, we ran BVR and computed tatitic on the ame path on the ame enor field. In thi default etup, there were on average 28.2 landmark. Our routing algorithm achieved a ucce rate of 90.9% without flooding. When the routing failed, the hop ditance from the node where the algorithm terminated to the detination wa 4.47 on average, with a tandard deviation of B. Succeful delivery rate We conidered the performance of the routing algorithm by varying the number of routing landmark, the r-ampling parameter (landmark eparation ditance, roughly), and the average network degree. The ucce rate i hown in Figure 7. Firt we gradually increaed the number of routing landmark and found that, a expected, the ucce rate alo increaed, from 85% with only 6 landmark, up to 92% with 18 landmark. We oberved that the node where GLDR failed to delivered were often near the boundary o that they were not well urrounded by their routing landmark. GLDR mut often rely on greedy routing uing L 1 norm and L norm to reach boundary node, and increaing the number of routing landmark doe little to help for thi phae. When the number of routing landmark wa mall, GLDR wa more ucceful than BVR; a the number increaed, the performance gap narrowed. BVR random landmark election often produce highly irregular landmark and may not work well when only a few routing landmark are ued. We alo examined the influence of the total number of landmark. We varied the eparation ditance r in the landmark election algorithm between 6 and 18, and the number of landmark from around 60 to around 20. The performance improved with more landmark, motly becaue more node near the boundary were well covered by the extra landmark. Finally, we varied the communication radiu o that the number of communication neighbor, averaged over all node, wa between 6 and 18. A the average degree increaed, fewer landmark were ued, o that the landmark denity remained relatively fixed. GLDR performance improved when network degree wa high, a the network became more regular.

7 Fig. 7. The ucce rate of GLDR i high for variou etting. GLDR compare favorably to BVR. In all cae, the ucce rate of GLDR wa conitently over 80%, and frequently over 90%, which i good, conidering the relatively low degree of the network. C. Remaining path and tretch factor We varied the denity of the landmark by changing the landmark eparation ditance parameter between 6 and 18 in the landmark election algorithm. For pair of node on which the routing got tuck, we computed the hop count from the terminating node to the detination. Thi meaure gave u ome idea on how much we need to flood in order to reach the detination. When we could route uccefully, we alo computed the hortet path between the ource and the detination, and computed the tretch factor. The average remaining path (Figure 8) wa comparable with the eparation ditance between landmark. The tretch factor of the path (Figure 9) wa quite mall, between 1.04 and A typical route wa about 30 hop, which mean that our route were only 1 3 hop longer than the hortet path. GLDR compared favorably to BVR in thee meaure. D. Landmark election We meaured the effectivene of the landmark election algorithm in GLDR. For thi purpoe, we ran the election algorithm for variou value of K, ee Table I. A K increaed, the number of landmark decreaed, a landmark had more time to uppre nearby node, preventing them from becoming landmark. From Table I, we noted that when K increaed from 600 to 3, 000, the number of landmark increaed, but the routing performance tayed relatively contant. Thi ugget that tightly clutered landmark might not contribute much to the routing quality, and that the K value hould not be too mall. On the other hand, when K increaed above 1, 200, the number of landmark tarted to tabilize. A the network initialization time wa dominated by K, K hould be mall, at the expene of having more landmark in the network. We compared the performance of GLDR with landmark choen uniformly at random, and according to II-C. The ucce rate degraded ignificantly with random landmark, a there were more node that are not urrounded well by landmark. Thu, our landmark election i crucial for effectivene of GLDR. E. Improving BVR with r-ampling landmark election Our landmark election cheme can be ued to improve other landmark-baed algorithm. For example, it helped improve the ucce rate of BVR (Figure 10). The performance Fig. 11. Virtual landmark improve routing performance. of BVR with our landmark election cheme i very imilar to GLDR. F. Improving routing performance uing virtual landmark We teted the idea of virtual node and landmark for routing to node near the enor field boundary (Section III-D). To implify the implementation, we elected virtual landmark to coincide with real landmark which are near the boundary (intead of chooing them in the ame way a in the bae cae, only uing a different notion of ditance for ome landmark pair). We found that the virtual landmark improve the routing quality of GLDR, ee Figure 11. V. CONCLUSION We propoed a practical landmark election protocol that uniformly ample a network of enor. Combined with a imple greedy routing baed on ditance to landmark, we can achieve good routing performance which i provable in the continuou domain and manifeted in our extenive network imulation. GLDR, while comparable to BVR in performance categorie uch a packet delivery rate and routing path length, ha lower routing overhead. Further, the landmark election algorithm i alo beneficial to ome other landmark-baed routing cheme. Our experiment howed that BVR performance improve ignificantly with a et of landmark/beacon elected uing thi algorithm. Landmark election i eldom a tand-alone problem, but rather trongly tied to the overlying application, be it point-topoint network routing or robot navigation. Thi paper provide a practical olution in the context of routing in wirele ad-hoc network. The general problem remain important and challenging. We hope thi paper could encourage more reearch interet in thi direction in the networking community.

8 Fig. 8. Unucceful route ended up near the detination. Fig. 9. Succeful route had low tretch factor. Fig. 10. Good landmark election cheme can be ued to improve the performance of BVR. TABLE I THERE ARE FEWER LANDMARKS WHEN K INCREASES. K #LM GLDR No LMS REFERENCES [1] P. Boe, P. Morin, I. Stojmenovic, and J. Urrutia, Routing with guaranteed delivery in ad hoc wirele network, Wirele Network, vol. 7, no. 6, pp , [Online]. Available: citeeer.it.pu.edu/boe01routing.html [2] B. Karp and H. Kung, GPSR: Greedy perimeter tatele routing for wirele network, in Proc. of the ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom), 2000, pp [3] F. Kuhn, R. Wattenhofer, Y. Zhang, and A. Zollinger, Geometric ad-hoc routing: Of theory and practice, in Proc. 22 nd ACM Int. Sympoium on the Principle of Ditributed Computing (PODC), 2003, pp [4] Y.-J. Kim, R. Govindan, B. Karp, and S. Shenker, Geographic routing made practical, in Proceeding of the 2nd USENIX Sympoium on Networked Sytem Deign and Implementation (NSDI 05), Boton, MA, USA, May [5] B. K. S. S. Young-Jin Kim, Rameh Govindan, On the pitfall of geographic face routing, in Proceeding of the Third ACM/SIGMOBILE International Workhop on Foundation of Mobile Computing, September [6] A. Rao, S. Ratnaamy, C. Papadimitriou, S. Shenker, and I. Stoica, Geographic routing without location information, in Proc. 9th Annual International Conference on Mobile Computing and Networking (Mobi- Com 2003). San Diego: ACM Pre, September 2003, pp [7] Q. Fang, J. Gao, L. Guiba, V. de Silva, and L. Zhang, GLIDER: Gradient landmark-baed ditributed routing for enor network, in Proc. of the 24th Conference of the IEEE Communication Society (INFOCOM), March [8] R. Foneca, S. Ratnaamy, J. Zhao, C. T. Ee, D. Culler, S. Shenker, and I. Stoica, Beacon vector routing: Scalable point-to-point routing in wirele enornet, [9] A. Caruo, A. Urpi, S. Chea, and S. De, GPS free coordinate aignment and routing in wirele enor network, in Proc. of the 24th Conference of the IEEE Communication Society (INFOCOM), March [10] R. William, The Geometrical Foundation of Natural Structure: A Source Book of Deign. New York: Dover Publication, [11] R. Motwani and P. Raghavan, Randomized Algorithm. Cambridge Univerity Pre, [12] A. Kroller, S. Fekete, D. Pfiterer, and S. Ficher, Determinitic boundary recognition and topology extraction for large enor network, in SODA 06: Proceeding of the eventeenth annual ACM-SIAM ympoium on Dicrete algorithm. New York, NY, USA: ACM Pre, 2006, pp [13] Q. Fang, J. Gao, and L. J. Guiba, Locating and bypaing hole in enor network, Mobile Network and Application, vol. 11, no. 2, pp , [14] N. Amenta and M. W. Bern, Surface recontruction by Voronoi filtering, in Sympoium on Computational Geometry, 1998, pp [Online]. Available: citeeer.it.pu.edu/article/amenta98urface.html A. Proof of Lemma 2 VI. APPENDIX Conider any node c. Let T be the number of landmark in B(r, c). Suppoe that T t, where t t := β (3+ α) 2. Let 2ρ+1 be the α mallet odd upper bound on the minimum inter-landmark ditance. ρ-neighborhood of the landmark are dijoint, and contained in B(c, r + ρ). It follow ρ argument. Uing t βr 2 we find 1 αr/ r, by a tandard packing αt/β 1 pαt/β 1, hence 2ρ + 1 (2 + α)r p αt/β 1. (1) Since t t, the right hand ide of (1) i at mot r, o the two landmark p and q which realize the minimum inter-landmark ditance are at mot r hop from each other. Hence v p v q (2 + α)r K( p αt/β 1). (2) The probability of thi event i at mot the value of the right hand ide (which i at mot 1, auming K r). By definition, E[T ] (t 1) + P βr 2 t=t Pr[T t]. We bound the um uing (2). βr X 2 (2 + α)r t=t K( p αt/β 1) (2 + α)r K 2β( 2 α + 1)r ln(r α) K r α Zβr 2 t=t «. dt p αt/β 1 We can implify the expreion uing r 1, α 1, and ln x x 1 2 (for x > 0), to obtain the deired reult. B. Proof of Lemma 3 Lemma 4: κ p ρ/(ρ 1). Proof: Without lo of generality aume d = 1. Let dl = x and ld = α. It follow l = ρx and d = 2x in(α/2) (ee Figure 4). We have (ρ 1)x κ = 1 2x in α h 2 1 = ρ 2 x 2 + x 2 2ρx in 2 α i 2 (definition of κ), (coine law dl).

9 l τ φ g φ a d o b d x l R θ l θ l Fig. 12. Bounding the one-tep tretch factor. d y z Eliminating in(α/2), κ = (ρ 1)x 1 p (1 (ρ 1) 2 x 2 )/ρ. (3) 1, 1 ρ+1 Triangle inequality ρx 1 x implie x [ ]. Tediou ρ 1 but traightforward calculation how that the right hand ide of (3) i minimized over thi interval for x = 1/ p ρ(ρ 1), at which point κ = p ρ/(ρ 1). Lemma 5: If φ < π/2, then κ 1/(1 in φ). Proof: Let be uch that i parallel to dl and d i perpendicular to both and dl. Let be the interection of d and, ee Figure 4. Note that d L = dl < π/2, and when when φ < π/2, it i eay to ee that i between and d, and i alo between and. From triangle inequality, = L dl d. We alo have that d d d = d in φ. Thu κ = /( d d ) 1/(1 in φ). Note that a approache d, ρ approache 1 and φ may or may not be acute. Thu both Lemma 4 and 5 do not yield any bound on κ. We now retrict the routing landmark of d to only landmark that are further than ome threhold ditance R from d. We alo aume that we elect the landmark L uch that the ratio ρ for L i the larget among thoe. Let o be the center and τ be the radiu of the Apoloniu circle of d with ratio ρ, the circle C of all point p atifying p / pd = l / ld = ρ. It i well-known that o i on the line d. Let a and b be the interection of thi circle with d, with a i between d and b i outide (Figure 12). By the choice of l, there i no routing landmark of d inide C. From ρ = / d = b / bd, it i eay to ee that da / db = (ρ 1)(ρ + 1) < 1, and thu d i between a and o. By Lemma 4, da / db 1/(2κ 2 1) = (1 u)/(1+u), and thu da τ(1 u), do τu, db τ(1 + u). Let d be between d and o uch that d b = τ(1 + u). Let g = ( db dl )/2. Since there i a dik of radiu g empty of landmark (Figure 12), we have r g ( d b d l )/2 = τ(1 + u x)/2, where x = d l /τ can be bounded a follow. Lemma 6: For w a in the tatement of Lemma 3, x w. Proof: Suppoe x > w. Then co φ co φ = x2 (1 u 2 ) > 0, 2x o φ < π/2. By Lemma 5, in φ 1 1/κ = v, and thu in 2 φ + co 2 φ (x2 (1 u 2 )) 2 + v 2 > 1, 4x 2 a contradiction. Since R dl 2τ, r R(1 + u x)/4. From Lemma 6, R/r 4/(1 + u w) =: f(κ). C. Proof of Theorem 5 We give only the proof for the cae when, d, are real and l i virtual (Figure 13), the other cae are imilar. Let x and y be the boundary point where the hortet path l and dl cro the boundary. Let l be uch that l = l and dl = dl Fig. 13. virtual. Analyi of the augmented-gldr when, d, are real, and l i (in general, there are two uch point ymmetric with repect to the line d, but we define l the be the one on the ame ide of the line a the local boundary). Notice that l wa picked a a routing landmark becaue the ditance (poibly reflected) among, d and l atify certain relation. By contruction,, d and l preerve thi et of ditance. It follow that if the bae cae protocol i applied to, d and l, then Theorem 4 provide a tretch guarantee, but for a new ource poition, which i different than in general. The true one-tep tretch i d d + d d + κ, (4) which tend to κ a / become maller. Thu if we prove that / 1, we are done. We bound the ratio a follow. Let θ be the (mall) angle between the boundary normal at x and y. Let l and z be the mirror image of l about the tangent to the boundary at x and y, repectively. We find that dl dl, pecifically dl dz zl (triangle inequality) dl dl max{ ll, lz }θ 2 l θ. Now, dl and dl have two pair of ide of the ame length ( d and l ) and one which differ only lightly (at mot 2 l θ). The mall change in the length implie a mall change in the internal angle, more preciely l l 2 dl dl l 2 2 l θ l 4θ. (multiplying the fraction by 2 i far too loppy, but illutrate the point). Hence 4θ. So we have bounded the ratio by a multiple of the angle θ between the normal, which we expect to be mall if the curvature i low. Formally, we know that x and y are cloe at mot 8r apart (both are within 4r from l) and we invoke the following well known geometric lemma. Lemma 7 (Amenta and Bern [14]): The angle between the normal whoe footpoint are at mot ρ R/4 apart i at mot 4ρ/R. It follow that if R 320r, then θ 4 8r/(320r), hence / 0.4. Subtituting into (4) we find that the one-tep tretch i bounded. Notice that the proof o far did not aume (or require) guaranteed delivery, a it wa concerned only with a ingle routing tep. However, we can ue the ingle-tep analyi to prove that even in the augmented cae the detination i alway reached. A in the oneided cae, it uffice to prove, in the ame etting a above, that d > d. But thi imply follow by the following chain of inequalitie d d d d (κ 4θ), which i poitive, under our low curvature requirement. Once we have the delivery guarantee, we can extend the one-tep tretch bound to the entire path. Thi complete the proof.