Visibility-Graph-based Shortest-Path Geographic Routing in Sensor Networks

Transcription

1 Athor manscript, pblished in "INFOCOM 2009 (2009)" Visibility-Graph-based Shortest-Path Geographic Roting in Sensor Networks Gang Tan Marin Bertier Anne-Marie Kermarrec INRIA/IRISA, Rennes, France. {gang.tan, marin.bertier, inria , ersion 1-5 No 2009 Abstract We stdy the problem of shortest-path geographic roting in a static sensor network. Existing algorithms often make roting decisions based on node information in local neighborhoods. Howeer, it is shown by Khn et al. that sch a design constraint reslts in a highly ndesirable lower bond for roting performance: if a best rote has length c, thenin the worst case a rote prodced by any localized algorithm has length Ω(c 2 ), which can be arbitrarily worse than the optimal. We present VIGOR, a VIsibility-Graph-based roting protocol that prodces rotes of length Θ(c). Or design is based on the constrction of a mch redced isibility graph, which gides nodes to find near-optimal paths. The per-node protocol oerheads in terms of state information and message transmission depend only on the complexity of the field s large topological featres, rather than on the network size. Simlation reslts show that or protocol dramatically otperforms localized protocols sch as GPSR and GOAFR+ in both aerage and worst cases, with reasonable extra oerheads. I. INTRODUCTION Geographic forwarding has been widely stdied as a roting strategy for large wireless networks, mainly de to its simplicity and scalability. Typically, the roting algorithm greedily adance to nodes that are progressiely closer to the destination. When reaching a local minimm, a face (or perimeter) roting method is employed to oercome the local minimm, and then the algorithm resmes greedy forwarding. For a network with niform and dense sensor deployment in a reglar region, sch a method prodces almost shortest paths with little oerhead [12]. Sch geographical algorithms, howeer, hae serios efficiency problems in sensor fields with complex topologies. In many of the real-world enironments the sensor field natrally has many obstacles/holes of arbitrary shape and size, casing the algorithm to rn into local minima freqently. Yet the algorithms are often satisfied with only being able to escape those local minima, withot regard to the efficiency of face roting. For instance, the GPSR protocol [12] always ses a right-hand rle to rote arond a hole, ignoring the possibility that there cold be a mch shorter face bondary path in the opposite direction. A series of sbseqent work [13] [15] has extended this strategy to offer shorter worst-case paths than GPSR. Unfortnately, as shown in [14], in the worst case a rote prodced by any localized algorithm has length Ω(c 2 ), c being the shortest possible path length. In a largescale network where c can be at the order of tens or hndreds, the mltiplicatie factor c clearly means an excessie waste of energy and, ery possibly, an nacceptable latency. s Physical network distance ector roting (e.g.,dsdv) Oerlay network (VON) VIG-FACE/VIG-GREEDY roting Fig. 1. Illstration for VIGOR. The oerlay network VON rns a distance ector algorithm to find shortest Eclidean distance paths, while in the physical network, a face algorithm VIG-FACE (or its greedy ariant VIG- GREEDY) realizes the roting between two oerlay nodes. The face algorithm is designed in sch a way that the hop cont between two (isible) nodes and is within a constant factor of their Eclidean distance on the plane. To improe this sitation, a ariety of optimization techniqes hae been proposed that se a limited amont of non-local information, for example by exploiting local face information [7], [16], or by sing some kind of abstract representation of the field topology [4], [8], [10]. Howeer, none of these heristics proides any constant bond on path stretch. In this context, a natral qestion arises: Is it possible for the protocol to achiee Θ(c) performance while remaining scalable? Being scalable reqires that the protocol s per-node oerhead mst be independent of, or grow ery slowly with, the increase in network size N. This paper presents the first protocol that achiees this goal. The protocol, called VIGOR, is based on isibility graphs (VGs), a strctre in comptational geometry that is often sed to find shortest paths in an enironment with obstacles. Using the information of detected hole bondaries, we deelop techniqes to constrct a mch redced VG, which captres the field s topology in a form of size proportional to the complexity of the field s large geometric featres (e.g., the nmber of holes aboe a certain size). The fact that sch featres are sally ery small compared with N allows s to bild a small oerlay network, called a Visibility-based Oerlay Network (VON), on top of the physical network. In the VON, nodes rn a distance-ector based protocol (e.g., DSDV [18]) to rote packets along paths of shortest Eclidean distance. Within this framework, one of or major contribtions is a face roting algorithm, called VIG-FACE, that rotes between two isible VON nodes. (See Figre 1 for an illstration.) This algorithm is proed to prodce paths with a constant hop t

2 stretch between two nodes that are isible to each other. In conjnction with the distance ector algorithm at the oerlay leel, it enables VIGOR to achiee Θ(c) roting performance between general commnication pairs. The per-node oerhead depends only on the complexity of the large geometric featres, rather than on the network size. To improe the aerage-case efficiency of the protocol, we also deelop a greedy ersion of VIG-FACE, which leads to the protocol VIGOR-G. Simlation reslts show that VIGOR- G prodces near-optimal-length rotes and dramatically otperforms protocols sch as GPSR and GOAFR+ in both aerage and worst cases (by p to an order of magnitde), at low extra message and memory oerheads. A. Model II. MODEL AND PRELIMINARIES We assme that a total of N nodes, each with a distinct ID, are placed in the Eclidean plane R 2, forming a connected network. Eery node knows its own location, and a sorce knows the location of a destination throgh some location management system [6]. The network is planarized by some algorithm (e.g., CLDP [11]), so that no two edges cross each other in the graph. There is a maximm radio range, normalized to be 1, for all the nodes. The radio model does not need to be reglar. denotes the Eclidean length of the line segment ; D G (, ) denotes the shortest Eclidean path length between and on the graph G. B. Visibility Graphs and Shortest Paths It is shown [1] that the path of shortest Eclidean length between two points s and t in the plane aoiding polygonal holes/obstacles is a connected series of line segments, whose inner ertices are ertices of the holes. Conseqently, any edge of a shortest path is a straight line segment e between two ertices p and q of holes. Since e does not cross any hole, p and q are isible from each other. The graph formed by all the hole ertices and these isibility segments is called the Visibility Graph (VG) of the polygonal enironment; see Figre 2(a) for an example. In sch a graph, the set of ertices that can be seen by a certain ertex p is called p s isibility set, denoted Φ p. Likewise, a ertex p s edge isibility set is defined as the set of hole edges that can be seen by p, where an edge can be seen iff at least one point on that edge is isible. As sch, the problem of finding shortest obstacle-aoiding path between two points redces to how to constrct the VG and how to search for the shortest path in the VG. A Redced Visibility Graph (RVG) techniqe can be sed to redce the nmber of edges in a VG. The line segment xy is a tangent segment iff its prolongation is tangent to the holes at both x and y. A RVG is a sbset of VG that has all non-tangent segments remoed (see Figre 2(b)). It can be proed that the RVG contains the shortest path between a pair of sorce and destination points as well [1]. s Ob1 (a) Ob2 t s Ob1 (b) Ob2 Fig. 2. An example of (a) isibility graph and (b) redced isibility graph on a 2D plane. The gray regions are holes. The thick line segments (in red) represent the shortest path between s and t. C. Visibility-Graph-based Roting The basic idea of VG-based roting is to first identify the sensor nodes that represent the hole ertices, and then organize those nodes into a Visibility-based Oerlay Network (VON). If the VON is small enogh in size, we can afford sing a distance ector (e.g, the DSDV) protocol to maintain a shortest (Eclidean distance) oerlay roting table at each VON node. When a sorce nodes wants to rote to some destination node, it only needs to join the VON and rote on the VON. The VON path can sere as a reference for finding a shortest path in the nderlying network. Gien the framework, we need to address two technical challenges: how to constrct a small VON, and how to rote between two VON nodes efficiently. III. CONSTRUCTING A SMALL VON A. Detecting Holes In a planar graph, a face can be iewed as a closed (polygonal) region represented by a seqence of nodes and directed edges. In this seqence, a node can appear more than once bt edges cannot. The size of a face is defined as the nmber of edges in sch a seqence. A face is said to be a big face if its size is greater than η, where η is a system parameter, and a small face otherwise. A big face is also called a hole for its great impact on roting performance. An example of holes is gien in Figre 3(a), where η =15and the hole bondary nodes are shown in ble (darker color). Each node p has a set of adjacent faces. For each face F p in this set, p periodically rotes a probing packet along F p s bondary. This is done by roting to some fake point p within p s interior angle at F p and where no sensor node exists. Following the simple right-hand rle as widely sed by geographic protocols sch as GPSR and GOAFR+, sch a packet will finally create a loop [12], which can be detected by p when the packet first retrns. All the face probing packets are piggybacked with existing keep-alie beacons so no extra messages will be occrred. To redce message length, the face probing packets are dropped at a node that has a larger ID than the packet sorce node. As a reslt, only the node with the highest ID on a face F p can receie its own packet. Sch a node becomes F p s header. The face probing packet can easily carry the nmber of hops it has completed, so that the face header can learn t 2

3 (a) holes and bonding bars (b) the final VON Fig. 3. The constrction of a VON for a network. With η =15, thevonhas 9 nodes, in contrast with the original network size of The darker circles and sqares represent hole bondary nodes and VON nodes, respectiely. For clarity the ot bondary nodes and their associated VON edges are not shown. the face s size. If F p s header finds that F p is a big face, then it sends a second packet arond F notifying all the nodes en rote of F p s size. After this, F p s header can start to generate a VON polygon for F p. B. Constrcting the VON Polygons A hole s header node initially takes itself as the first VON node on its crrent hole, and then sends ot a packet which will traerse the hole bondary in a clockwise direction. The packet has three fields: field 1 containing a seqence of locations of the VON nodes determined so far, field 2 containing a seqence of locations of all the nodes starting from last VON node p to crrent node, and field 3 containing a parameter ω>0. Eery time a node becomes a new VON node, it first appends its location to field 1 of the packet, and then replaces the content of field 2 with its location, and finally forwards the packet to the next node. Upon receiing the packet, a node first appends its location to field 2 of the packet, and then checks whether the following two conditions hold: (1) there exists a rectangle that coers all the locations contained in field 2. The rectangle, called a bonding bar, has a maximm width ω, and an nconstrained length (see Figre 3(a) for an illstration); (2) the seqence of locations in field 2 has a monotonic seqence of projection points on the length-nconstrained sides of the bonding bar. If these two conditions are met, then the packet proceeds to the next node on the bondary; otherwise the crrent node becomes a VON node. When the packet retrns to the hole header node, the header node can constrct a seqence of edges from the location seqence contained in field 1 of the packet, which forms a VON polygon. It is possible that the VON polygon generated has intersecting edges. In this case we need to make a finer approximation of the original hole by creating more VON nodes. Sppose that the header node has fond two intersecting edges 1 2 and 3 4. (The case of more intersecting edges can be handled similarly.) It performs the aboe hole approximation procedre a second time, with two changes: (1) the locations of 1, 2, 3 and 4 are recorded in the packet as a new field, and (2) ω ω/2 in field 3. The newly added field tells that new VON nodes need be created only between 1 and 2, and between 3 and 4, so nodes otside the two ranges only need to forward the packet. The decreased ω makes a bonding bar less able to accommodate many nodes, so that a new VON edge spans fewer nodes, and new VON nodes can be created. Note that as ω tends to zero, the created VON polygon tends to oerlap with the original hole bondary. The intersection aoiding procedre is repeated ntil there is no intersection between the VON edges in the ranges ( 1, 2 ) and ( 3, 4 ). Finally, an intersection-free polygon will be created at the hole header node. This intersection-free property can be garanteed since in the extreme case where the VON polygon redces to the original hole bondary, the VON polygon will be free of intersection, becase of the planar property of the nderlying network graph. Or experiments show that self-intersection happens rarely when the initial ω is appropriately chosen (e.g., ω =0.5). After determining a locally intersection-free VON polygon, a hole header node can broadcast the polygon (i.e., the seqence of its ertices) to the network. Note that the hole header node only needs to remember the VON nodes, rather than all the nodes, on its face. The broadcast at this stage, howeer, is not yet for constrcting the final VON; instead it is only for detecting possible intersections between VON polygons created from nearby holes. Eery hole header node periodically broadcasts its VON polygon information, and at the same time collects the broadcast messages from the network and checks if there are inter-hole intersections. Eery time an intersection has been fond, the hole header node performs the intersection aoiding procedre described earlier. The broadcast stops when a hole header node can find no intersection on its own VON polygon. After all the aboe procedres, the hole header nodes can start broadcasting the VON polygons to the network this time for completing the constrction of the VON. Eery node (not only the VON nodes) in the network maintains an edge set containing all the edges it can see, and pdates the set each time it receies a broadcast message. From this edge set, a node can determine the set of VON nodes it can see, that is, its isibility set Φ. The isibility relationship between VON nodes directly translates to VON edges, which, along with the VON polygons already identified, constitte the VON. An example of VON is gien in Figre 3(b). C. Redcing the Nmber of VON Edges In order to lower the message oerhead of the distance ector protocol, we se Yao graph [21] to redce the nmber of VON edges from O(Non) 2 to O(N on ), where N on denotes the nmber of nodes in the VON. The Yao graph with an integer parameter k>6is defined as follows. At each node, draw k eqally-separated rays from which define k cones. In each cone, remoe all edges, if there is any, except the shortest one incident on ; ties being broken arbitrarily. The reslting graph is known to be a spanner sbgraph of the original graph [21]. Using the idea of Yao graph, we propose an algorithm called VON-SPARSIFY to redce the nmber of VON edges. In this algorithm, when a node picks a neighbor in a certain cone 3

4 C, it first checks whether C contains a isible node sch that is a tangent edge. Only when this condition holds does establish a link to the nearest isible node in C (here need not be tangent). Since non-tangent edges neer appear in a shortest path in the VG, their remoal will not affect the spanner property of the generated sbgraph. The Yao graph techniqe is also applied to non-von nodes. For each non-von node s, this will select at most k nodes ot of s s isibility set Φ s for establishing (irtal) links. These nodes define s s selectie isibility set, denoted Φ s. D. Redcing the Nmber of VON Polygons Increasing the parameter η can redce the nmber of VON polygons generated, yielding a smaller VON. For example, in Figre 3, increasing η =15to 40 will reslt in a smaller VON withot the smaller trianglar hole. No dobt there is a tradeoff between the size of VON and roting performance, which will be analyzed in the next section. IV. THE VIGOR ROUTING PROTOCOL The VIGOR protocol integrates two components: the distance ector algorithm which is responsible of shortest path (in terms of Eclidean distance) roting on the VON, and a face algorithm that rotes between two VON nodes in the physical network. Gien a sorce node s and a destination node t, VIGOR first needs a procedre to determine whether the rote needs to go throgh the VON, and if it does, at which node s the rote enters the VON and at which node t it exits the VON. A. Establishment of VON Rotes The node s first checks whether t is isible. It is easy to erify that t is isible to s iff st does not intersect any edge from s s edge isibility set. If t is fond to be isible, then s = s and t = t, which means that the rote does not need to go throgh the VON; otherwise s needs some extra commnication to determine s and t. First, s sends its isibility set Φ s (piggyback with its first packet) to t. Since a shortest path between s and t (or stpath for short) has not yet been established, the packet has to follow a sb-optimal path. This path can be generated sing a simple existing algorithm sch as GPSR or GOAFR+. We adopt the latter in the paper. When t receies the packet containing Φ s, it forwards the set to each member of t s selectie isibility set Φ t.(note that Φ t k.) Next, each Φ t ses its VON roting table to select a VON node Φ s sch that D = s + D on (, )+ t is the smallest. Here D on (, ) means the shortest distance between and in the VON. Each then sends a packet containing the chosen and calclated D back to t. All these commnications se the GOAFR+ algorithm. Now, t picks the sch that D is the smallest, lets t = and s be the node chosen by, records t and s to a packet, and sends it to s this time along the VON path. After receiing this packet, s knows s and t. In ftre transmissions, eery packet from s will contain the nodes s ;; q F1 F2 p' p Fig. 4. The H-distance set that coers the nodes isited by VIG-FACE s roting between two isible VON nodes and. F 1 and F 2 are small faces and F 3 is a big face, B is a bonding bar of F 3,andB is a rectanglar sb-area of B that contains a path between p and q. The small oal-shaped area is Ψ(p,q,ω). and t. This information will be sed by the distance ector algorithm for roting on the VON. B. Roting Between Two Visible Nodes VIGOR ses a pre face roting algorithm to rote from two isible nodes and. Here and can be the commnication sorce/destination nodes or VON nodes. The algorithm ses a concept called the H-Distance Set. AnH-Distance Set of a line segment on the plane, denoted Ψ(,, H), isthesetof points whose distance from is no greater than H. Here the distance of a point from is the smallest distance between that point and all points on. Geometrically, Ψ(,, H) corresponds to an oal-shaped area as illstrated in Figre 4. The face roting algorithm of VIGOR, referred to as VIG-FACE, consists of the following steps. Algorithm VIG-FACE (, ) Let p be the crrently isited node; p nearest be the closest point to the rote has met so far on ; F p be p s crrent face (i.e., p s adjacent face intersected by p nearest); p next be the next node to isit on F p.while is not directly reachable, repeat the following face roting process (in the clockwise direction by defalt): 1) If F p is a big face, then start exploring the bondary of F p. At each node, check whether the rote will a) enconter a VON node other than and p, or b) go beyond the bondary of Ψ(,, ω), or c) cross at a point that is farther from than p nearest. If so, trn back and explore the bondary of F p in the opposite direction. Contine ntil reaching a node q sch that qq next intersects at a point p closer to than p nearest. Letp q if q next does not lie on, orp q next otherwise; let p nearest p. 2) If F p is not a big face, then explore the whole bondary of F p, at the same time pdating p nearest; after retrning to p, adance to the node q sch that qq next intersects line at p nearest. Letp q if q next does not lie on, orp q next otherwise. Figre 4 gies an example of the algorithm roting throgh three consectie faces. C. VIGOR is Correct We examine VIG-FACE(,) s behaior on the two types of faces. We first assme that the VON is correctly constrcted; the handling of exceptions will be discssed later. When the crrent face F p is not a big face, the algorithm ensres that a F3 B' B H w 4

5 positie progress will be made toward after roting on F p. When F p is a big face, the following lemma shows that the algorithm will also make a progress toward. (The proofs of all lemmas and theorems can be fond in [19]). Lemma 1: Sppose p and q are two neighboring intersection points between the VON edge and a big face F b p. Then there exists at least one p q-path on F b p s bondary that does not contain an intermediate VON node. Sppose VIG-FACE(,) starts traersing the big face F b p from a node adjacent to p, that is, from a node p sch that pp next crosses at p (or from p itself if p is a real node). Lemma 1 sggests that there mst be a pq-path that does not contain an intermediate VON node, ths jstifying the trning-back operation of VIG-FACE when it enconters a VON node other than and p. Lemma 1 also implies that both p and q are coered by the same bonding bar B. Becase the bondary nodes in B hae monotonic projections on B s length-nconstrained edges, this frther means that there exists a p q-path whose nodes are all coered by a shorter bar B, which is intersected by at p and q, shown as a shaded area in Figre 4. It is possible to erify that B Ψ(p,q,ω) Ψ(,, ω). Hence, there exists a p q-path on F b p s bondary whose nodes are all coered by Ψ(,, ω). This explains why VIG-FACE trns back when it hits the bondary of Ψ(,, ω). Now that a path leading to q is garanteed to be fond in the region scoped by Ψ(,, ω), VIG-FACE can finally make its way to q, making a positie progress toward. In smmary, VIG-FACE makes a positie progress toward eery time it completes execting on a face. This ensres that will be finally reached becase there are a finite nmber of faces in the network. Together with the correctness of the distance ector algorithm (e.g., [18]) this establishes the correctness of VIGOR. VIGOR s correctness is robst to network dynamics. In normal cases, for VIG-FACE(, ) the two nodes and are isible. When nodes fail or new nodes are added to the network, may be no longer isible to, ormaylose its role as a VON node. This will by no means affect the correctness of VIG-FACE, since the algorithm will still follow the principle of face roting traerse a face, make a progress, traerse the next face, and so on. The only thing affected is the direction chosen for face roting, which may mislead the packet into longer (bt iable) face bondary paths and reslt in sboptimal performance. In the case that the node fails or becomes nreachable, the algorithm can detect this by checking whether it has completed traersing a face withot getting closer to. The roting error will be reported to, which can find an alternatie VON rote in its roting table and try roting again. D. VIGOR Has Constant Stretch In this section we analyze the performance of VIGOR. We adopt the widely sed nit disk graph (UDG) model [12] to represent the network. (This assmption is needed only in this section.) To aoid pathologic network topologies (e.g., one in which a node has O(N) degree), we employ an alternatie planarization algorithm deeloped by Wang and Li [20] that garantees a bonded degree χ for eery node. Sch a localized algorithm also generates a spanner, meaning that for any two nodes, their shortest path length in the sbgraph is within a constant factor δ, called the stretch factor, of their shortest path length in the original graph. The reslting graph is referred to as a WL graph. The main reslt is the following theorem (see a complete proof in [19]). Theorem 1: Let L VIG (s, t) be the hop cont needed by VIGOR to rote from node s to node t, and L OPT (s, t) be the hop cont of a shortest possible path between s and t in the network, then L VIG 64δ(χ + 1)(H + 1)(H +1+2/π) (s, t) L OPT (s, t), 1 2sin(π/k) where H = max{ω, η/2}. Proof: (Sketch) We first consider the case where both s and t are not within any VON polygons, which sffices to show the main idea of the proof. Consider the non-triial case in which st > 1. First, we look at the algorithm s behaior between two adjacent VON nodes (or between an endpoint and a VON node). Again consider the non-triial case > 1. Forabig face F b, we already know from the correctness analysis that the algorithm can rote arond it withot going beyond the region scoped by Ψ(,, ω). For a small face F s, which has at most η bondary edges (and hence a perimeter at most η), the algorithm can go at most η/2 far away from, becase otherwise F s wold hae a perimeter greater than η. Ths, Ψ(,, η/2) sffices to bond the region isited by the algorithm when exploring a small face. Combining these two cases gies that the nodes isited by the algorithm while it rotes between and are bonded by the H- distance set of, where H = max{ω, η/2}. Now we se a techniqe de to Khn et al. [13] (Lemma 4.1) to bond the nmber of nodes within that region. Applying this reslt to Ψ(,, max{ω, η/2}) yields an pper bond of the nmber of nodes isited by VIG-FACE: ( N VIG (, ) 16(χ + 1)(H +1) H ), π where H = max{ω, η/2}. From the description of VIG-FACE, we can see that the algorithm isits any node of a small face at most twice. Also, the p nearest has the effect of preenting the algorithm from traersing any edge of a big face more than twice. Ths for any particlar face, the algorithm isits any node at most two times. Considering the fact that a node can belong to at most two adjacent faces that are intersected by the line, we can conclde that a node in the region Ψ(,, max{ω, η/2}) will be isited at most for times by the algorithm. Therefore, the hop cont needed by VIG-FACE(, ) is: L VIG (, ) 4N VIG (, ) where H = max{ω, η/2}. Next, consider an st-path that aoids the VON polygons; that is, does not contain a point lying inside any VON 5

6 polygons (bt can toch the bondary of the VON polygons). We denote its minimm Eclidean length by D R2 (s, t). Let G on be the graph corresponding to the VON as defined in Section III, G on+ be G on agmented by adding s and t, as well as the edges determined by the isibility relationship between s, t and all the VON nodes, to G on. Applying the VON-SPARSIFY algorithm to G on and G on+ generates two sbgraphs, YG on and YG on+, respectiely. By combining Clarkson s indction techniqe [5] with Lkoszki s reslt [17] (Lemma 2.2), we can proe that D YGon+ (s, t) 1 1 2sin(π/k) DR2 (s, t). When VIGOR determines s s VON entry and exit nodes, it actally finds a shortest st-path ia the edges in the nion of YG on+ s edge set and the set of isibility edges between s and all nodes in Φ s, which is a sperset of YG on+ s edge set. Since the addition of edges to a graph does not increase the shortest path length between any two nodes in that graph, the shortest st-path length fond by VIGOR, denoted D VIG (s, t), is no greater than the shortest st-path in YG on+. Hence, D VIG (s, t) D YGon+ 1 (s, t) 1 2sin(π/k) (s, t). DR2 Let P(s, t) be the edge set of the VON path from s to t, then we hae L VIG (s, t) = (,) P(s,t) L VIG (, ) 64(χ + 1)(H + 1)(H +1+2/π) (,) P(s,t) = 64(χ + 1)(H + 1)(H +1+2/π)D VIG (s, t) 64(χ + 1)(H + 1)(H +1+2/π) D R2 (s, t), 1 2sin(π/k) where H = max{ω, η/2}. Last, consider a shortest st-path on the WL-graph, whose Eclidean length is denoted by D WL (s, t). We can proe that D R2 (s, t) D WL (s, t). Denote the minimm length of an st-path on the UDG by D UDG (s, t). Then we hae D R2 (s, t) D WL (s, t) δd UDG (s, t) δl OPT (s, t) where δ is the stretch factor of the WL-graph. Combined with (1), this proes the theorem. If either s or t is within some VON polygon, the VON roting needs to be slightly modified in order to ensre optimality. The details and proof are omitted here. The stretch reslt presents a salient featre of or protocol. To the best of or knowledge, no prior geographic roting algorithm has achieed a constant bond (een in a UDG graph). Note that the correctness of or protocol does not need this UDG assmption. V. VIGOR-G: A GREEDY+FACE PROTOCOL In this section, we present a protocol, named VIGOR-G, that combines greedy roting and face roting. This protocol also rns on a planar (not necessarily a UDG) network. (1) VIGOR-G works in the same way as VIGOR except that VIGOR-G incorporates a greedy algorithm, referred to as VIG- GREEDY, to rote between two isible VON nodes. This algorithm tries to rote greedily toward the destination wheneer possible, that is, by forwarding the message to the neighbor located closest to. When reaching a local minimm with respect to, the algorithm employs a face roting to oercome the local minimm. When VIG-GREEDY reaches a local minimm at a node p, it ses the following two rles to determine the right face roting direction based on the information of p s associated VON edge P ccw P cw, where P ccw and P cw represent p s VON node neighbors conter-clockwise and clockwise on the face, respectiely. (If more than one sch edges exist in the case of a complex VON polygon, then one is picked randomly.) Note that P ccw and P cw both belong to p s isibility set. 1) Direction Rle 1: Ifp is not a VON node, then calclate its projection point p on the line P ccw P cw.if p is anglarly closer to p P ccw than to p P cw, then take the conter-clockwise direction, otherwise take the clockwise direction. In Figre 5(a), the grey half disk represents the anglar range for p to choose a conterclockwise direction. 2) Direction Rle 2: Ifp is a VON node, then if p is anglarly closer to pp ccw than to pp cw (Figre 5(b)), take the conter-clockwise direction; otherwise take the clockwise direction. It is possible that the line p crosses a VON polygon (Figre 5(c)), which may lead the packet into the wrong direction and create a big detor. VIG-GREEDY determines whether this is happening by checking whether p intersects any of p s associated VON edges. For a particlar VON edge P ccw P cw,ifp is on the right of this edge, it means that p is inside the VON polygon that contains P ccw P cw ; in this case the algorithm first calclates p s projection p on P ccw P cw and then checks whether p intersects P ccw P cw. The aboe procedre is performed eery time the algorithm hits a big face bondary node except dring the face roting. The rle for choosing a direction in this case is as follows: Direction Rle 3:Ifp (or p in the case that p is within the VON polygon that contains P ccw P cw ) intersects P ccw P cw, then take a conter-clockwise direction if p is on the left of the line, otherwise take a clockwise direction. After determining the face roting direction, the algorithm rotes along the face bondary ntil a node q sch that qq next intersect p at some point closer to than p. When VIG-GREEDY starts face roting at node p, the next face to traerse is the face intersected by p, where is next VON node on the path. This garantees that a positie progress toward is made after eery face roting procedre. According to Frey and Stojmenoic s reslt (Theorem 4) [9], the face roting procedre along with the greedy steps progressie forwarding can garantee deliery in a planar graph. Together with the correctness of the distance ector algorithm, this means that VIGOR-G is correct in general planar graphs. VIGOR-G handles network dynamics or roting exceptions in 6

7 P cw p' P ccw p P cw p P ccw (a) (b) (c) P cw p P ccw Fig. 5. Heristics for determining face roting direction in VIG-GREEDY. The sqares represent VON nodes. and are the sorce and destination nodes, p is the crrent node, p is the projection node of p on the line P ccw P cw. the same way as VIGOR. Exected with an incorrect VON, it behaes like a traditional greedy+face protocol with face roting directions chosen in a sb-optimal way. VI. SIMULATIONS In this section we ealate the performance of VIGOR-G throgh simlations. We focs on the algorithm s behaior on the topological leel. Comparisons are made with GPSR [12], GOAFR+ [13], and a centralized shortest-path algorithm, which seres as a baseline. The sensor network is deployed in a 1000m 1000m sqare field with irreglar holes. Figre 6 shows for example field layots. The nmber of sensors ranges from Sensor nodes are placed on a grid and then pertrbed with a random shift following a normal distribtion [4]. The generated networks remain connected. For the constrction of VONs, the Yao-graph parameter k is set to 7. The maximm width of a bonding bar ω is 30 meters. The threshold size for a face to be a hole, η, issetto 15 by defalt. Each sensor has a commnication range of 60 meters. We hae considered both UDG and qasi-udg radio models. Since the adantage of VIGOR/VIGOR-G comes from its captring large topological featres, local ariation in connectiity has little effect on its global behaior, and ths does not fndamentally change the comparatie reslts. We therefore only report on the case of UDG radio model. Hop Stretch We randomly pick 3000 pairs of sorce and destination nodes in the network and rn the for roting algorithms for those pairs. For each pair, the path length in terms of hop cont prodced by the centralized algorithm is taken as the baseline, and the other three algorithms are compared in terms of path stretch, that is, the ratio of a path s length to the baseline ale. Figre 7 plots the stretch ale distribtions of the algorithms nder the for field layots in Figre 6. All the for plots show that VIGOR-G has a significant smaller stretch than GPSR and GOAFR+. For instance, in Figre 7(b), VIGOR-G has a mean stretch of 1.05, in contrast with GPSR s 4.10 and GOAFR+ s This indicates that VIGOR-G achiees near-optimal performance in aerage case, and considerably otperforms the other two representatie algorithms. The worse-case behaior of the algorithms can be obsered from the maximm stretch ales. With this metric, an een bigger difference can be fond between these algorithms. Throghot the experiments, the stretch of VIGOR-G remains below 2.0, while the other two hae a stretch arying widely. For instance, in Figre 7(b), GPSR prodces a stretch as high as 56.17, nearly 30 times that of VIGOR-G s peak ale of In other cases, GPSR and GOAFR+ consistently prodce a maximm stretch nearly an order of magnitde higher than that of VIGOR-G. The great adantage of VIGOR-G in the worst case also sggests that or heristics are sccessfl. Protocol Message Oerhead In addition to the keep-alie beacons between neighbors, the VIGOR-G protocol reqires extra messages to (re)constrct the VON, broadcast the VON polygons, and to rn the distance ector algorithm on the VON. Assming an inter-node beacon interal of b = 1 second [3], [12] and taking the beacon message oerhead as a baseline, we simlate the maintenance of VON and compare the protocol s oerhead with the baseline. The maintenance of VON inoles all the bondary nodes of holes. Since face probing packets are all piggybacked with the keep-alie beacons, we only need to consider the packet issed by hole header nodes. We assme that a hole header reconstrcts its VON polygon and broadcast the polygon eery b 1 time, where b 1 is adjstable. For the message oerhead of the distance ector algorithm rnning on the VON, we take the DSDV protocol [18] as a rnning example and borrow the experimental setting from [3]. In [3], the periodic rote pdate interal for DSDV is set to 15 seconds. Taking into accont the triggered pdates, the effectie rate of protocol message transmission is one pdate per node per second. We let the pdate messages be transmitted oer the VON edges, which correspond to the paths generated by VIGOR-G in the nderlying network. Table I shows the protocol oerhead measred in nmber of messages per second for arying b 1 ; the baseline oerhead is gien in the parenthesis. It can be seen that for all the settings, VIGOR-G has an oerhead within a moderate factor of the baseline. For instance, for the field topology Figre 6(a) with b 1 =15(the same as the VON rote pdate interal), the protocol oerhead is 4849 messages per second, which translates to 1.4 messages per node per second. We beliee this is a reasonable oerhead for a single node. It shold also be noted that the settings in [3] are for a dynamic network with high node mobility. In a static setting, the pdate of the VON strctre and the VON roting tables can be performed at a mch lower freqency, which leads to an een smaller protocol oerhead. Memory Reqirement VIGOR-G reqires eery VON node to maintain a roting table, which is of size O(N on ). In or experiments, the sizes of VONs are at the order of tens. Eery non-von node in the network is also reqired to remember an edge isibility set (which also contains its node isibility information). Theoretically, this set is of size O(N on ); in practice, it is sally mch smaller than N on. 7

8 8 (a) N = 3356, ag degree = 10.1 (b) N = 3166, ag degree = 9.6 (c) N = 3732, ag degree = 9.32 (d) N = 2958, ag degree = 10.0 Fig. 6. For 1000m 1000m sensor fields with holes. The red and ble lines show seeral examples of paths generated by GPSR and VIGOR-G, respectiely. inria , ersion 1-5 No 2009 (a) Stretch distribtion: Fig.6(a) (b) Stretch distribtion: Fig.6(b) (c) Stretch distribtion: Fig.6(c) (d) Stretch distribtion: Fig.6(d) Fig. 7. Path stretch distribtions of VIGOR-G, GPSR, and GOAFR+ nder the field layots in Figre 6. b 1 =10 b 1 =15 b 1 =20 b 1 =30 Fig. 6(a) 5084(3356) 4849(3356) 4731(3356) 4613(3356) Fig. 6(b) 5797(3166) 5561(3166) 5443(3166) 5325(3166) Fig. 6(c) 8553(3732) 8264(3732) 8120(3732) 7976(3732) Fig. 6(d) 4780(2958) 4569(2958) 4464(2958) 4358(2958) TABLE I NUMBER OF PROTOCOL MESSAGES PER SECOND AS COMPARED WITH THE BASELINE BEACON MESSAGE OVERHEAD (IN PARENTHESIS). Intitiely, the smaller η is, the more VON polygons there will be in the VON, and the closer the generated path length will be to the optimal. On the other hand, the larger VON reqires more messages transmission and memory to maintain. Figre 8 shows the VON constrcted for the field layot Figre 6(c) nder different η. When η =15, the VON has a size of 92; increasing η to 55 yields a mch sparser VON of only 49 nodes; see Figre 8(b). We ary η to examine the tradeoff between path qality and protocol oerhead in terms of N on and the total nmber of VON maintenance messages. The trend agrees well with the intition discssed aboe. For example, when η = 25, the aerage stretch is 1.083; for η = 100, the aerage stretch increases slightly to For these two cases, the message oerhead is 8162 and 7136 messages per second, respectiely. The reslts sggest that allowing only a small extra oerhead can significantly improe roting performance as compared with a localized algorithm. (a) VON with η =15 (b) VON with η =55 Fig. 8. Effect of η on the VON. For instance, in or experiments, a non-von node needs to remember at most 21 isible edges, each represented by two points, in the for field layots in Figre 6. These reqirements are nlikely to be a concern for crrent sensor hardware. Effect of the Parameter η The parameter η determines whether a face is a hole, ths needs to participate in the VON. VII. DISCUSSION Scaling Isses The per-node protocol oerhead of VIGOR(-G) is O(N on ), where N on is the nmber of VON nodes. N on indeed reflects the field s geometric complexity, embodied by the nmber, size, and shape of holes aboe a certain size. What jstifies the VIGOR protocol is the fact that in real-world enironments, the geometric complexity is often low enogh compared with the network size, allowing the protocol to rn withot oerloading sensor nodes. For example, the nmber of bildings, or the nmber of bondary edges of bildings, in a factory or a camps, is sally ery small compared with the nmber of sensors that cold be deployed in sch enironments. For een more geometrically complex

9 fields and larger systems, choosing sitable parameters η and ω proides a conenient way to tradeoff between protocol oerheads and path qality. Network Dynamics We expect that in a static sensor network, topology changes are sally local and happen infreqently (de to, e.g., depletion of energy or external damages), so salient, global featres like large holes in the commnication graph are rather stable nder those local changes. This jstifies a low-freqency pdate of the isibility graph which leads to low oerheads. Een if the VON becomes ery inaccrate de to massie network errors, only the performance will be affected; the protocol will remain correct. VIII. RELATED WORK Many efforts hae been made to improe the performance of localized face roting algorithms or their greedy ariants. Khn et al. [14] propose a localized algorithm that can achiee asymptotically optimal performance in a UDG network. In [13], they frther propose GOAFR+, a protocol that achiees good aerage-case performance while presering worst-case optimality. In [14], Khn et al. proe that any localized algorithm can prodce path length qadratic of the optimal, which is ndesirable in a large-scale network. Ths, in order to garantee finding a reasonably short path, sing a certain amont of non-local information is ineitable. In this direction, Leong et al. [16] hae proposed Path Vector Face Roting, which achiees sperior performance to GPSR by haing nodes remember some information abot their local faces. A similar approach has also been sggested in [7] to spport path migration and improe path qality. Both sets of research, howeer, only consider optimization on local faces. When it comes to a network with faces of arbitrarily complex shape and large size, the limited horizon makes them nable to find a globally short path. In [2], Arad et al. present a Node Eleation Ad-hoc Roting (NEAR) scheme that assigns irtal coordinates to nodes so that local minima can be aoided more efficiently. Very recently, Jiang et al. [10] propose a noel information model in which a hole and its affected nodes are identified as an nsafe area. In so doing the algorithm can sae many steps by aoiding entering an area that does not contain a path to the destination. While these heristics are shown to be ery beneficial for roting performance, neither is proed to proide a bond on path stretch. Brck et al. [4] propose MAP, a naming and geographic roting protocol that prodces globally short paths sing medial axis graphs (MAGs). After a preprocessing stage, a MAG is stored at each node in the network. With MAG, roting is first planned on the abstract medial axis graph, and then realized in each canonical region. In [8], a protocol called GLIDER with a similar principle to that of MAP is proposed. MAP and GLIDER share seeral featres with VIGOR: they all try to captre the field s large geometric and topological featres in a sccinct form, and the roting is performed at two leels: the abstraction leel, and the leel of nderlying networks. One major difference is that VIGOR focses on shortest path roting and has proable performance, whereas MAP and GLIDER hae other focses and proide no garantee in this regard. IX. CONCLUSIONS We hae presented isibility-graph-based geographic protocols for shortest path roting in sensor networks. VGs facilitate roting throgh topologically complex enironments along paths with shortest Eclidean lengths. The proposed protocols show excellent performance with low extra oerheads in addition to the reqirement by basic geographic roting protocols sch as GPSR. ACKNOWLEDGMENT We wold like to thank Daide Frey and Cigdem Sengl for comments on early drafts of the paper. We also thank the anonymos reiewers for their helpfl feedback. REFERENCES [1] H. Alt and E. 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