Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks

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1 1 Localized Delanay Trianglation with Application in Ad Hoc Wireless Networks Xiang-Yang Li Gria Călinesc Peng-Jn Wan Y Wang Department of Compter Science, Illinois Institte of Technology, Chicago, IL s: {xli, calinesc, wan}@cs.iit.ed, wangy1@iit.ed. The research of the second athor was performed in part while isiting the Department of Combinatorics and Optimization of Uniersity of Waterloo, and partially spported by NSERC research grants. This paper is sbmitted as reglar paper for IEEE Transaction on Parallel and Distribted Systems. Xiang-Yang Li is the contact athor for this paper.

2 2 Abstract Seeral localized roting protocols garantee the deliery of the packets when the nderlying network topology is a planar graph. Typically, relatie neighborhood graph (RNG) or Gabriel graph (GG) is sed as sch planar strctre. Howeer, it is well-known that the spanning ratios of these two graphs are not bonded by any constant (een for niform randomly distribted points). Bose et al. recently deeloped a localized roting protocol, called FACE, that garantees that the distance traelled by the packets is within a constant factor of the minimm if Delanay trianglation of all wireless nodes are sed, in addition to garantee the deliery of the packets. Howeer, it is expensie to constrct the Delanay trianglation in a distribted manner. Gien a set of wireless nodes, we model the network as a nit-disk graph (UDG). in which a link exists only if the distance is at most the maximm transmission range. In this paper, we present a noel localized networking protocol that constrcts a planar 2.5-spanner of UDG, called the localized Delanay trianglation LDEL, as network topology. It contains all edges that are both in the nit-disk graph and the Delanay trianglation of all nodes. The total commnication cost of or networking protocol is O(n log n) bits, which is within a constant factor of the optimm to constrct any strctre in a distribted manner. Or experiments show that the deliery rates of some of the existing localized roting protocols are increased when localized Delanay trianglation is sed instead of seeral preiosly proposed topologies. Or simlations also show that the traelled distance of the packets is significantly less when the FACE roting algorithm is applied on LDEL than applied on GG. Keywords Delanay trianglation, localized method, planar, roting, spanner, topology control, wireless networks. I. INTRODUCTION We consider a wireless ad hoc network (or sensor network) with all nodes distribted in a two-dimensional plane. Assme that all wireless nodes hae distinctie identities and each static wireless node knows its position information 1 either throgh a low-power Global Position System (GPS) receier or throgh some other way. For simplicity, we also assme that all wireless nodes hae the same maximm transmission range and we normalize it to one nit. By one-hop broadcasting, each node can gather the location information of all nodes within the transmission range of. Conseqently, all wireless nodes V together define a nit-disk graph UDG(V ), which has an edge if and only if the Eclidean distance between and is less than one nit. Throghot this paper, a broadcast by a node means sends the message to all nodes within its transmission range. In wireless ad hoc networks the radio signal sent ot by a node 1 More specifically, it is enogh for or protocol when each node knows the relatie position of its one-hop neighbors. The relatie position of neighbors can be estimated by the direction of arrial and the strength of signal.

3 3 can be receied by all nodes within the transmission range of. The main commnication cost in wireless networks is to send ot the signal while the receiing cost of a message is neglected here. Conseqently, throghot this paper, we are interested in designing a protocol with small total nmber of messages. One of the key challenges in the design of ad hoc networks is the deelopment of dynamic roting protocols that can efficiently find rotes between two commnication nodes. In recent years, a ariety of roting protocols [1], [2], [3], [4], [5], [6] targeted specifically for ad hoc enironment hae been deeloped. See [7], [8] for a reiew of most roting protocols. Seeral researchers proposed another set of roting protocols, namely the localized roting, which select the next node to forward the packets based on the information in the packet header, and the position of its local neighbors. Bose and Morin [9] showed that seeral localized roting protocols garantee to delier the packets if the nderlying network topology is the Delanay trianglation of all wireless nodes. They also gae a localized roting protocol based on the Delanay trianglation sch that the total distance traelled by the packet is no more than a small constant factor of the distance between the sorce and the destination. Howeer, it is expensie to constrct the Delanay trianglation in a distribted manner, and roting based on it might not be possible since the Delanay trianglation can contain links longer than one nit. Seeral researchers also proposed to se some planar network topologies that can be constrcted efficiently in a distribted manner. Lin et al. [10] proposed the first localized algorithm that garantees deliery by memorizing past traffic at nodes. Bose at al. [11] proposed to se the Gabriel graph as nderlying strctre for the FACE roting method. Sbseqently, Karp et al. [12] discssed in detail of medim access layer and condcted experiments with moing nodes for Face roting method. Barriére et al. [13] extended the scheme on graphs which are fzzy nit graphs, that is, two nodes are connected if their distance is at most r, not connected if the distance is at least R, and may be connected otherwise. They showed that their algorithm works correctly if R 2r. Roting according to the right hand rle, which garantees deliery in planar graphs [9], is also sed when simple greedy-based roting heristics fail. Howeer, it is well-known [14], [15] that the spanning ratios of both RNG and GG are not bonded by any constant. Here, gien a graph H, a spanning sbgraph G of H is a t-spanner if the length of the shortest path connecting any two points in G is no more than t times the

4 4 length of the shortest path connecting the two points in H. Moreoer, it was shown by Bose et al. [14] that the spanning ratio of Gabriel graph on a niformly random n points set in a sqare is almost srely at least Ω( log n/ log log n). Ths, no matter how good the roting method is, the spanning ratio achieed by applying the method on the Gabriel graph or on the relatie neighborhood graph is at least Ω( log n/ log log n) almost srely. Conseqently, to make a localized roting protocol efficient, we need constrct a planar spanner locally. Here, a strctre G can be constrcted locally if eery wireless node can decide the edges of G incident on by sing only the information of nodes within a constant hops. In this paper, we design a localized algorithm that constrcts a planar t-spanner for the nit-disk graph, sch that some of the localized roting protocols can be applied on it. We obtain a ale of approximately 2.5 for the constant t. Notice that the spanning ratio achieed by a specific roting method cold be mch larger than the spanning ratio of the nderlying strctre. Nonetheless, a strctre with a small spanning ratio is necessary for some roting method to possibly perform well. Or strctre is based on the Delanay trianglation. Gien a set of points V, the nit Delanay trianglation, denoted by UDel(V ), is the graph obtained by remoing all edges of the Delanay trianglation Del(V ) that are longer than one nit. It was proed in [16], [17] that UDel(V ) is a t-spanner of UDG(V ). We then gie a localized algorithm that constrcts a graph, called localized Delanay graph LDel (1 ) (V ). We proe that LDel (1 ) (V ) is a t-spanner by showing that it is a spergraph of UDel(V ). Additionally, we proe that LDel (1 ) (V ) has thickness two, i.e., it can be decomposed to two planar graphs. We then show how to make the graph LDel (1 ) (V ) planar efficiently withot losing the spanner property. The total commnication cost of or approach is O(n log n) bits, which is optimal within a constant factor. Notice that eery node has to send at least one message to its neighbors to notify its existence in any protocol, which implies that the commnication cost is at least n log n bits for any protocol. We assme a node ID can be represented by log n bits. The precise worst-case pper bond on the commnication costs to constrct or planar spanner is (37q +13p + 100)n bits, where q is the nmber of bits reqired to represent the niqe node IDs, and p is the nmber of bits reqired to represent the geometric position of a node. Or method operates in an asynchronos enironment, and we do not cont the time it takes to bild the strctre. We do not hae non-triial worst-case pper bonds on maintaining the strctre

5 5 de to changes in the network configration, bt beliee the aerage case (assming changes are random) is acceptable de to the fact that, becase the strctre is planar and planar graphs hae at most 3n 6 edges, the aerage degree of a node in the strctre is constant. Or belief is also based on the fact that inserting nodes in a Delanay trianglation in a random order reslts in a O(n log n) expected-time centralized algorithm [23]. When a node moes, the expected nmber of edges that are affected is a small constant. Ths, the aerage cost of pdating the strctre is small. The Gabriel graph can be constrcted with (q +p+1)n bits and is ery easy to maintain when the network changes. Ths, there is a trade-off between the better worst-case spanning ratio when applying the greedy roting schemes on or strctre and the higher cost of constrcting and maintaining it, compared to the Gabriel graph. Notice that the price of haing a strctre like local Delanay graph that has a constant bonded spanning ratio is the high cost of pdating the strctre when the nodes are moing. Althogh the aerage pdating cost is still a constant, it cold be ery large in the worst-case. We need a balance between worst-case spanning ratio and the worst-case pdating cost; also a balance between aerage-case spanning ratio and the aerage-case pdating cost. Crrently, we are inestigating the design of a strctre that achiees sch balance. Preiosly, there were some approaches proposed to approximate the Delanay trianglation locally. H [18] sed the Delanay trianglation to configre the wireless network topology sch that a planar graph with bonded node degree is compted. A major step in his method is that each node comptes all Delanay edges whose length is no more than the transmission range. It sed the Voronoi diagram of node to compte all sch Delanay edges. Howeer, this approach will not work always. A simple obseration is that in order to determine whether an edge belongs to the Delanay trianglation, we hae to check whether certain circles passing throgh and are empty (not containing wireless nodes in their interior). Obiosly, in the worst case, the circmradis of sch a circle cold be infinity een when the edge length is bonded, implying that we may hae to check all nodes. De to space limit, we omit the detail why the method in [18] will not work. Moreoer, it is nknown whether H s strctre has a constant spanning ratio. A related reslt was pblished by Gao et al. [17]. The conference ersion of this paper,

6 6 [16], was obtained independently and was sbmitted before [17] was pblished. Gao et al. [17] proposed another strctre, called restricted Delanay graph RDG and showed that it has good spanning ratio properties and described how to maintain it locally. They called any planar graph containing UDel(V ) as a restricted Delanay graph. They described a distribted algorithm to maintain a RDG sch that at the end of the algorithm, each node maintains a set of edges E() incident to. Those edges E() satisfy that (1) each edge in E() has length at most one nit; (2) the edges are consistent, i.e., an edge E() if and only if E(); (3) the graph obtained is planar; (4) UDel(V ) is in the nion of all edges E(). Their algorithm works as follows. First, each node acqires the position of its 1-hop neighbors N 1 () and comptes the Delanay trianglation Del(N 1 ()) on N 1 (), inclding itself. In the second step, each node sends Del(N 1 ()) to all of its neighbors. Let E() ={ Del(N 1 ())}. For each edge E(), and for each w N 1 (),ifand are in N 1 (w) and Del(N 1 (w)), then node deletes edge from E(). They proed that when the aboe steps are finished, the reslting edges E() satisfy the for properties listed aboe. The commnication cost cold be as large as Θ(n 2 ), and the comptation cost cold be as large as Θ(n 3 ), which are mch higher than ors. Recently, Li et al. [19] also proposed another strctre partial Delanay trianglation (PDT) which is a sbset of UDel(V ), for scatternet formation in Bletooth network. Unfortnately, PDT does not hae constant bonded spanning ratio. Bose et al. [11] and Karp et al. [12] proposed similar algorithms that rote the packets sing the Gabriel graph to garantee the deliery. Applying the roting methods proposed in [11], [12] on the planarized localized Delanay graph LDel (1 ) (V ), a better performance is expected becase the localized Delanay trianglation is denser compared to the Gabriel graph, bt still with O(n) edges. Or simlations show that the deliery rates of seeral localized roting protocols are increased when the localized Delanay trianglation is sed. In or experiments, seeral simple local roting heristics, applied on the localized Delanay trianglation, hae always sccessflly deliered the packets, while other heristics were sccessfl in oer 90% of the random instances. The greedy-based localized roting schemes may still fail to delier the packet on localized Delanay trianglation. Becase the constrcted topology is planar, we can se the right hand rle or Face roting to garantee the deliery of the packets from sorce node

7 7 to the destination when simple heristics fail. The experiments also show that seeral localized roting algorithms (notably, compass roting [20] and greedy roting) also reslt in a path whose length is within a small constant factor of the shortest path; we already know sch a path exists since the localized Delanay trianglation is a t-spanner. Notice that there is a difference between the spanning ratio of the nderlying strctre and the spanning ratio achieed by a specific roting algorithm. Obiosly, any roting method cannot achiee small spanning ratio when it is applied on a strctre with large spanning ratio. We also condct extensie simlations of the Face roting method [11] and the Greedy Face Roting method (applies the greedy roting wheneer possible and ses the Face roting if local minimm occrs) on the localized Delanay trianglation and the Gabriel graph. We fond that the worst-case spanning ratio when localized Delanay trianglation is sed is significantly less than the spanning ratio when Gabriel graph is sed althogh the aerage spanning ratio achieed by these two strctres are almost the same when the network has no more than 100 nodes. We expect or local Delanay trianglation to perform better when the nmber of nodes are significantly large since the spanning ratio of the Gabriel graph on a set of n random points is abot Ω( log n/ log log n). When n is small, Ω( log n/ log log n) is almost a small constant, bt when n is large, Ω( log n/ log log n) cannot be treated as a small constant anymore. The remaining of the paper is organized as follows. In Section II, we reiew some strctres that are often sed to constrct the topology for wireless networks. We define localized Delanay trianglations LDel (k) (V ) and stdy their properties in Section III. Section IV presents the first localized efficient algorithm that constrcts a planar graph, PLDel(V ), which contains UDel(V ) as a sbgraph. Ths, PLDel(V ) is a planar t-spanner. The correctness of or algorithm is jstified in the Appendix. We demonstrate the effectieness of the localized Delanay trianglation in Section V by stdying the performance of arios roting protocols on it. We conclde or paper and discss possible ftre research directions in Section VI. II. PRELIMINARIES A. Spanner Constrcting a spanner of a graph has been well stdied. Let Π G (, ) be the shortest path connecting and in a weighted graph G, and Π G (, ) be the length of Π G (, ). Then a

8 8 graph G is a t-spanner of a graph H if V (G) =V (H) and, for any two nodes and of V (H), Π H (, ) Π G (, ) t Π H (, ). With H nderstood, we also call t the length stretch factor of the spanner G. There are seeral geometrical strctres which are proed to be t-spanners for the Eclidean complete graph K(V ) of a point set V. For example, the Yao graph [21] and the θ-graph [22] hae been shown to be t-spanners. Howeer, both these two geometrical strctres are not garanteed to be planar in two dimensions. Let ϱ G (, ) be the path fond by a nicasting roting method ϱ from node to in a weighted graph G, and ϱ G (, ) be the length of the path. The spanning ratio achieed by a roting method ϱ is defined as max, ϱ G (, ) /. B. Delanay Trianglation We reiew the definition of Delanay trianglation [23]. We assme that all wireless nodes are gien as a set V of n nodes in a two dimensional space. Each node has some comptational power. We also assme that there are no for nodes of V that are co-circlar. A trianglation of V is a Delanay trianglation, denoted by Del(V ), if the circmcircle of each of its triangles does not contain any other nodes of V in its interior. A triangle is called the Delanay triangle if its circmcircle is empty of nodes of V. It is well-known that the Delanay trianglation Del(V ) is a planar t-spanner of the completed Eclidean graph K(V ). This is first proed by Dobkin, Friedman and Spowit [24] with pper bond π 5.08 on t. Then Kein and Gtwin [25], [22] improed the pper bond on t to be 2π 3cos π 6 = 4 3π The best known 9 lower bond on t is π/2, which is de to Chew [26], and it is widely belieed to be the actal pper bond also. C. Proximity Graphs Besides the Delanay trianglation, arios proximity sbgraphs of UDG can be defined [27], [28], [29], [30], [21] oer a set of n two-dimensional wireless nodes V. For conenience, let disk(, ) be the closed disk with diameter, let disk(,, w) be the circmcircle defined by the triangle w, and let B(, r) be the circle centered at with radis r. Let x() and y() be the ale of the x-coordinate and y-coordinate of a node respectiely. The constrained relatie neighborhood graph, denoted by RNG(V ), consists of all edges

9 9 sch that 1 and there is no point w S sch that w < and w <. The constrained Gabriel graph, denoted by GG(V ), consists of all edges sch that 1 and the interior of disk(, ) does not contain any node from V. The constrained Yao graph with an integer parameter k 6, denoted by YG k (V ), is defined as follows. At each node, anyk eqal-separated rays originated at define k cones. In each cone, choose the closest node to with distance at most one, if there is any, and add a directed link. Ties are broken arbitrarily. Let YG k (V ) be the ndirected graph obtained by ignoring the direction of each link in YG k (V ). Bose et al. [14] showed that the length stretch factor of RNG(V ) is at most n 1 and the length stretch factor of GG(V ) is at most 4π 2n 4. Seeral papers [31], [32], [27] showed that 3 1 the Yao graph YG k (V ) has length stretch factor at most. Howeer, the Yao graph is 1 2sin π k not garanteed to be planar. The relatie neighborhood graph and the Gabriel graph are planar graphs, bt they are not a spanner for the nit-disk graph. In this paper, we are interested in locally constrcting a planar graph that is a spanner of the nit-disk graph. D. Localized Roting Algorithms Let N k () be the set of nodes of V that are within k hops distance of in the nit-disk graph UDG(V ). A node N k () is called the k-neighbor of the node. Usally, here the constant k is 1 or 2, which will be omitted if it is clear from the context. In this paper, we always assme that each node of V knows its location and identity. Then, after one broadcast by eery node, each node of V knows the location and identity information of all nodes in N 1 (). The total commnication cost of all nodes to do so is O(n log n) bits. A distribted algorithm is a localized algorithm if it ses only the information of all k-local nodes of each node pls the information of a constant nmber of additional nodes. In this paper, we concentrate on the case k = 1. That is, a node ses only the information of the 1-hop neighbors. A graph G can be constrcted locally in the ad hoc wireless enironment if each wireless node can compte the edges of G incident on by sing only the location information of all its k-local nodes. In this paper, we design a localized algorithm that constrcts a planar t-spanner for the nit-disk graph UDG(V ) sch that some localized roting protocols can be applied on it. The localized constrction of the strctre is attractie for wireless ad hoc networks de to efficient pdating of the strctre in mobile enironment. Assme a packet is crrently at node, and the destination node is t. Seeral localized roting

10 10 algorithms that se jst the local information of to rote packets (i.e., find the next node of ) were deeloped. Kranakis et al. [20] proposed to se the compass roting, which basically finds the next relay node sch that the angle t is the smallest among all neighbors of in a gien topology. Lin et al. [10], Bose et al. [11], and Karp et al. [12] proposed similar greedy roting methods, in which node forwards the packet to its neighbor in a gien topology which is closest to t. Recently, Bose at al.[33], [9], [11] proposed seeral localized roting algorithm that rote a packet from a sorce node s to a destination node t. Specifically, Bose and Morin [9] proposed a localized roting method based on the Delanay trianglation. They showed that the distance traelled by the packet is within a small constant factor of the distance between s and t. They also proed that the compass roting and the greedy roting method garantee to delier the packet if the Delanay trianglation is sed. III. LOCAL DELAUNAY TRIANGULATION In this section, we define a new topology, called local Delanay trianglation, which can be constrcted in a localized manner. We first introdce some geometric strctres and notations to be sed in this section. All angles are measred in radians and take ales in the range [0,π]. For any three points p 1, p 2, and p 3, the angle between the two rays p 1 p 2 and p 1 p 3 is denoted by p 3 p 1 p 2 or p 2 p 1 p 3. The closed infinite area inside the angle p 3 p 1 p 2, also referred to as a sector, is denoted by p 3 p 1 p 2. The triangle determined by p 1, p 2, and p 3 is denoted by p 1 p 2 p 3. An edge is called constrained Gabriel edge (or simply Gabriel edge here) if 1 and the open disk sing as diameter does not contain any node from V. It is well known [23] that the constrained Gabriel graph is a sbgraph of the Delanay trianglation, more precisely, GG(V ) UDel(V ). Recall that a triangle w belongs to the Delanay trianglation Del(V ) if its circmcircle disk(,, w) does not contain any other node of V in its interior. To simplify the proofs, from now on we assme that there are no for nodes of V co-circmcircle. If for nodes are on the same circle, a ery small random pertrbation to their coordinates allows the assmption aboe withot casing any problems in the actal network. It is easy to show that nodes, and w together can not decide if they can form a triangle w in Del(V ) by sing only their local information. We say a node x can see another node y if xy 1. The following definition is one of the key ingredients of or localized algorithm. Definition 1: A triangle w satisfies k-localized Delanay property if the interior of the

11 11 circmcircle disk(,, w) does not contain any node of V that is a k-neighbor of,,orw; and all edges of the triangle w hae length no more than one nit. Triangle w is called a k-localized Delanay triangle. Triangle w is called localized Delanay if it is a k-localized Delanay triangle for some constant integer k 1. Definition 2: The k-localized Delanay graph oer a node set V, denoted by LDel (k) (V ), has exactly all Gabriel edges and the edges of all k-localized Delanay triangles. When it is clear from the context, we will omit the integer k in or notation of LDel (k) (V ). Or original conjectre was that LDel (1 ) (V ) is a planar graph and ths we can easily constrct a planar t-spanner of UDG(V ) by sing a localized approach. Unfortnately, as we will show later, the edges of the graph LDel (1 ) (V ) may intersect. While LDel (1 ) (V ) is a t-spanner, its constrction is a little bit more complicated than some other non-planar t-spanners, sch as the Yao strctre [21] and the θ-graph [22]. Bt we can make LDel (1 ) (V ) planar efficiently, a reslt we describe later in this paper. The k-localized Delanay graph LDel (k) (V ) oer a node set V satisfies a monotone property: LDel (k+1 ) (V ) is always a sbgraph of LDel (k) (V ) for any positie integer k. A. LDel (k) (V ) is a t-spanner Theorem 1: Graph UDel(V ) is a sbgraph of the k-localized Delanay graph LDel (k) (V ). PROOF. We proe the theorem by showing that each edge of the nit Delanay trianglation graph UDel(V ) appears in the localized Delanay graph LDel (k) (V ). For each edge of UDel(V ), the following fie cases are possible (see Figre 1 for illstrations). 0 1 w w w w w w w z z z z Case 1 Case 2.1 Case 2.2 Case 3 Case 4 Case 5.1 Case 5.2 Fig. 1. The neighborhood configration of edge. Dashed lines (solid lines) denote edges with length > 1( 1). Case 1: there is a triangle w incident on sch that all edges of w hae length at most one nit. Becase the circmcircle disk(,, w) is empty of nodes of V, triangle w satisfies the k-localized Delanay property and ths edge belongs to LDel (k) (V ). Case 2: each of the two triangles incident on has only one edge longer than one nit.

12 12 Case 3: one triangle w incident on has only one edge with length larger than one nit and the other triangle z has two edges with length larger than one nit. Case 4: each of the two triangles incident on has two edges longer than one nit. We proe the cases 2, 3, and 4 together. Assme the two triangles are w and z. Let H,w be the half-plane that is diided by and contains node w. Then edge is not the longest edge in triangle w and ths the angle w < π. This implies that the circmcircle 2 disk(,, w) contains disk(, ) H,w. Similarly, the other half of disk(, ) is contained inside the circmcircle disk(,, z). Both disk(,, w) and disk(,, z) do not contain any node of V inside. It implies that disk(, ) is empty, i.e., edge is a Gabriel edge. Conseqently, edge will be inserted to LDel (k) (V ). Case 5: there is only one triangle incident on and it has at least one edge with length larger than one nit. Similar to cases 2-4, we can show that disk(, ) is empty and therefore edge will be inserted to LDel (k) (V ) as a Gabriel edge. B. LDel (1 ) (V ) may be non-planar The definition of the 1-localized Delanay triangle does not preent two triangles from intersecting or preent a Gabriel edge from intersecting a triangle. Figre 2 gies sch an example with 6 nodes {,, w, x, y, z} that LDel (1 ) (V ) is not a planar graph. Here = xy 1, and y = y > 1. Node x is ot of circmcircle of disk(,, w). Triangle w is a 1-localized Delanay triangle. If the node z does not exist, edge xy is an Gabriel edge. The triangle w intersects the Gabriel edge xy if z does not exist, otherwise it intersects the 1-localized Delanay triangle xyz. The example illstrated by Figre 2 also implies that a triangle in LDel (1 ) (V ) can intersect many other edges (by creating eqal-length Gabriel edges x 1 y 1, x 2 y 2,, which are parallel to Gabriel edge xy). C. LDel (1 ) (V ) has thickness 2 In this sbsection, we claim that LDel (1 ) (V ) has thickness two, or in other words, its edges can be partitioned in two planar graphs. From Eler s formla, it follows that a simple planar graph with n nodes has at most 3n 6 edges, and therefore LDel (1 ) (V ) has at most 6n edges. The proof of the following theorem is in Appendix.

13 13 w x z y Fig. 2. LDel (1 ) (V ) is not planar. Theorem 2: Graph LDel (1 ) (V ) has thickness 2. Or constrction algorithm of LDel (1 ) (V ) below also implies that LDel (1 ) (V ) has a linear nmber of edges. Theorem 2 gies a better constant in the analysis of the planarization algorithm. D. LDel (k) (V ), k 2, is planar Althogh we gae an example showing that LDel (1 ) (V ) is not always a planar graph, we will show that LDel (k) (V ), k 2, is always planar. Theorem 1 implies that each edge of UDel(V ) is either a Gabriel edge or forms a 1-localized Delanay triangle with some edges from UDel(V ). Obiosly, any two edges in UDel(V ) do not intersect. Ths, each possible intersection in LDel (k) (V ) is cased by at least one edge of some localized Delanay triangle. We begin the proof that LDel (k) (V ), k 2, is planar by giing some simple facts and lemmas. REMARK: If a Gabriel edge intersects an edge xy, then xy does not belong to UDel(V ). Lemma 3: If a Gabriel edge w intersects a localized Delanay triangle xyz, then and w can not be both otside the circmcircle disk(x, y, z). PROOF. Let c be the circmcenter of the triangle xyz. Then at least one of the x, y, and z mst be on the different side of line w erss the center c; let s say x. If both and w are otside the circmcircle disk(x, y, z), then wx > π. Ths, x is inside disk(, w), which 2 contradicts that w is a Gabriel edge. Lemma 4: Let w and xyz be two triangles of LDel (k) (V ), k 1, and assme the edge w intersects the triangle xyz and that disk(,, w) does not contain any of the nodes of {x,

14 14 y, z}. Then disk(x, y, z) contains either or w. See the appendix for the proof. The aboe lemma garantees that if two k-localized Delanay triangles w and xyz intersect, then either disk(,, w) or disk(x, y, z) iolates the Delanay property by jst considering the nodes {,, w, x, y, z}. The following theorem gies more insight regarding localized Delanay graphs and it is neer sed later in the paper. Its proof is immediate from Theorem 7. Theorem 5: LDel (2 ) (V ) is a planar graph. In conclsion, we defined a seqence of localized Delanay graphs LDel (k) (V ), where 1 k n. All graphs are t-spanner of the nit-disk graph with the following properties: UDel(V ) LDel (k) (V ), for all 1 k n; LDel (k+1 ) (V ) LDel (k) (V ), for all 1 k n; LDel (k) (V ) are planar graphs for all 2 k n; LDel (1 ) (V ) is not always planar. IV. LOCALIZED ALGORITHM In this section, we stdy how to locally constrct a planar t-spanner of UDG(V ). We se q to denote the nmber of bits reqired to represent the niqe node IDs, and p to denote the nmber of bits reqired to represent the geometric position of a node. When sing big-oh notation, we make the reasonable assmption that both q and p are O(log n). Althogh the graph UDel(V ) is a t-spanner for UDG(V ), it cannot be constrcted locally. We can constrct LDel (2 ) (V ), which is garanteed to be a planar spanner of UDel(V ), bt the total commnication cost of this approach cold be O(m log n) bits, where m is the nmber of edges in UDG(V ) and cold be as large as O(n 2 ). In order to redce the total commnication cost to O(n log n) bits, we do not constrct LDel (2 ) (V ), and instead we extract a planar graph PLDel(V ) ot of LDel (1 ) (V ). A. Algorithm Recall that LDel (1 ) (V ) is not garanteed to be a planar graph. Or algorithm first constrcts LDel (1 ) (V ) and then remoes edges from it to make it planar. The reslt of the algorithm is called PLDel(V ) and we show later that it is planar and that it contains UDel(V ) as a sbgraph (and therefore it is a t-spanner of UDG(V )).

15 15 We assme that when a node sends ot a message, all neighboring nodes will receie this message immediately. Algorithm 1: Localized Unit Delanay Trianglation 1. Each wireless node broadcasts an annonce message with its identity and location to its neighbors N 1 () and listens to the messages from other nodes in N 1 (). 2. Assme that node gathered the location information of N 1 (). It comptes the Delanay trianglation Del(N 1 ()) of its 1-neighbors N 1 (), inclding itself. 3. For each edge of Del(N 1 ()), let w and z be two triangles incident on. Edge is a Gabriel edge if both angles w and z are less than π/2. Node marks all Gabriel edges, which will neer be deleted. 4. Each node finds all triangles w from Del(N 1 ()) sch that all three edges of w hae length at most one nit and w π. The node sorts the edges of sch triangles in 3 clockwise order and broadcast a proposal message. The proposal message contains s ID, followed by the IDs of the nodes with w as aboe. A single bit sent in between two IDs of and w indicates if w is as aboe, and we say that is proposing that w be added to LDel (1 ) (V ). Then node listens to the messages from other nodes in N 1 (). 5. Each node after receiing a proposal message from node, comptes if is proposing adding triangles w and z to LDel (1 ) (V ). Then comptes if any of the triangles proposed by other nodes belongs to Del(N 1 ()). After compting all the triangles, broadcast an accept message which contains the IDs of, the clockwise list of ertices sch that compted that for some w the triangle w belongs to Del(N 1 ()), and a bit indicating if two sch consectie, form a triangle of Del(N 1 ()). 6. A node adds the edges and w to its set of incident edges if the triangle w is in Del(N 1 ()) and both and w hae sent either a propose message or an accept message for triangle w. It trns ot (see below) that the edges added by all the ertices in this step form LDel (1 ) (V ). At this moment, the planarization phase starts. 7. Each node broadcast a check message with its ID and the position of all ertices sch that LDel (1 ) (V ). 8. Based on the check messages sent by the nodes in N 1 (), comptes for eery 1-local

16 16 Delanay triangle w if any node is inside the circmcircle of w. If sch a node is fond, discards the triangle w. Then the node sorts clockwise the edges which are either Gabriel or belong to a triangle of LDel (1 ) (V ) which was not discarded. 9. Each node broadcast a alie message, consisting of its ID, followed by the IDs of nodes incident to and a single bit in between two sch nodes and w telling if w LDel (1 ) (V ) and w was not discarded. 10. Node keeps the edge in its set of incident edges if it is a Gabriel edge, or if there is a triangle w LDel (1 ) (V ) which appears in the alie messages of,, and w. This implicitly creates the graph PLDel(V ) and finishes the algorithm. We first claim that the graph constrcted at the end of Step 6 of the aboe algorithm is LDel (1 ) (V ). Indeed, for each triangle w of LDel (1 ) (V ), one of its interior angle is at least π/3 and w is in Del(N 1 ()), Del(N 1 ()) and Del(N 1 (w)). So one of the nodes amongst {,, w} will broadcast the message proposal(,, w) to form a 1-localized Delanay triangle w and the other two nodes will accept the proposal. Ths, LDel (1 ) (V ) is a sbgraph of the constrcted graph. Obiosly, the constrcted graph is also a sbgraph of LDel (1 ) (V ) by definition, which in trn implies that they are the same. Second, we show that PLDel(V ) is indeed a planar graph. Theorem 6: PLDel(V ) is a planar graph. PROOF. Two Gabriel edges do not intersect. Then eery intersection mst inole a localized Delanay triangle xyz which was broadcast alie by all three x, y, and z. Assme that an edge w intersects a 1-localized Delanay triangle xyz on an edge xy. Edge w is either a Gabriel edge or an edge of another 1-localized Delanay triangle, say w. In either case, either Lemma 3 or Lemma 4 implies that either or w is inside the disk(x, y, z). By symmetry we assme w is inside the disk(x, y, z). The triangle ineqality implies that x + wy < w + xy 2. The fact that triangle xyz is in PLDel(V ) implies that w / N 1 (x) N 1 (y) N 1 (z). Ths, wy > 1, which implies that x < 1. In other words, N 1 (x). When broadcasts its check message, x does find ot the existence of a node inside disk(x, y, z). Then x does not broadcast xyz in its alie message. We obtained a contradiction, ths completing the proof.

17 17 Note that any triangle of LDel (1 ) (V ) not kept by the algorithm is not a triangle of LDel (2 ) (V ). Therefore we hae: Theorem 7: PLDel(V ) is a spergraph of LDel (2 ) (V ). Next we careflly analyze the commnication cost of the algorithm. There are fie types of messages and each incldes a few bits to describe which type. Disregarding the few other bits sed by eery message, we hae: 1. annonce messages contain in total n(p + q) bits 2. proposal messages contain in total 11nq bits. Indeed, if the proposal of a node incldes the nodes 1, 2,..., j, then for eery 1 i j, either i i+1 π 3 or i 1 i π 3 (where for conenience we assme 0 = j and 1 = j+1 ). Immediate geometric argments imply that j Each accept message contains the triangle incident with some ertex. Those triangle are either big or small as defined in the proof of Theorem 2. The graph gien by big triangles and the graph gien by small triangles are planar. Using Eler s formla we obtain that there are at most 4n triangles accepted by two nodes. In total, at most 6n triangles are implicitly proposed. Conting mltiplicity (some proposed triangles are accepted once and some are accepted twice), a total of 10n triangles are accepted - each contribting one ID to the commnication cost. As each accept message also contains the ID of the sender, a total of at most 11n IDs are broadcast by accept messages. 4. Each check message contains the ID of a node and a nmber of positions corresponding to s neighbors in LDel (1 ) (V ). AsLDel (1 ) (V ) has thickness two, it has at most 6n edges and therefore a total of at most 12n positions are broadcast in check messages. 5. Each alie message contains the ID of a node and a nmber IDs corresponding to s neighbors in LDel (1 ) (V ), and bits indicating whether consectie neighbors make a triangle not yet discarded. As aboe, there are at most 12n IDs broadcast. From the preios discssion, Theorem 7, Theorem 1, and the reslts of [16], [17] regarding the spanning ratio of UDel(V ), we obtain: Theorem 8: PLDel(V ) is planar 4 3π-spanner of UDG(V ), and can be constrcted with 9 total commnication cost of n(37q +13p + 100) bits, where q is the nmber of bits reqired to represent a node ID and p is the nmber of bits reqired to represent the position of a node.

18 18 Recently, Călinesc [34] presented a localized method sch that all wireless nodes collectiely find the 2-hop neighbors N 2 () for eery node with O(n log n) commnication complexity with the assmption that the geometric location of eery node is known. The knowledge of the 2-hop neighbors information allows the direct constrction of LDel 2 (V ). Howeer, the hidden constant of [34] is mch larger than the constant presented here. V. ROUTING In this section, we discss how to rote packets on the constrcted graph. Recently, Bose and Morin [9] first proposed a localized roting algorithm that rotes a packet sing the Delanay trianglation and garantees the distance traelled by the packet is no more than a small constant factor of the distance between the sorce and the destination nodes. Howeer this algorithm has a major deficiency by reqiring the constrction of the Delanay trianglation and the Voronoi diagram of all wireless nodes, which cold be ery expensie in distribted compting. Bose et al. [11] also proposed another method, called Face roting, that rotes the packets sing the Gabriel graph to garantee the deliery. The Gabriel graph is a sbgraph of PLDel(V ). Ths, if we apply the roting method proposed in [11] on the newly proposed planar graph PLDel(V ), we expect to achiee better performance becase PLDel(V ) is denser than the Gabriel graph (bt still with O(n) edges). The constrcted local Delanay trianglation not only garantees that the length of the shortest path connecting any two wireless nodes is at most a constant factor of the minimm in the nit-disk graph, it also garantees that the energy consmed by the path is also minimm, as it incldes the Gabriel graph (see [35], [27]). Moreoer, becase the constrcted topology is planar, a localized roting algorithm sing the right hand rle garantees the deliery of the packets from sorce node to the destination node. We stdy the following roting algorithms on the graphs proposed in this paper. Compass Roting(Cmp) Let t be the destination node. Crrent node finds the next relay node sch that the angle t is the smallest among all neighbors of in a gien topology. See [20]. Random Compass Roting(RndCmp) Let be the crrent node and t be the destination node. Let 1 be the node on the aboe of line t sch that 1 t is the smallest among all sch neighbors of. Similarly, we define 2 to be nodes below line t that minimizes the angle 2 t. Then node randomly choose 1 or 2 to forward the packet. See [20].

19 19 t 1 t t 2 Compass Random Compass Greedy t α t α t Most Forwarding Nearest Neighbor Farthest Neighbor Fig. 3. Shaded area is empty and is next node. Greedy Roting(Grdy) Let t be the destination node. Crrent node finds the next relay node sch that the distance t is the smallest among all neighbors of in a gien topology. See [11]. Most Forwarding Roting (MFR) Crrent node finds the next relay node sch that t is the smallest among all neighbors of in a gien topology, where is the projection of on segment t. See [10]. Nearest Neighbor Roting (NN) Gien a parameter angle α, node finds the nearest node as forwarding node among all neighbors of in a gien topology sch that t α. Farthest Neighbor Roting (FN) Gien a parameter angle α, node finds the farthest node as forwarding node among all neighbors of in a gien topology sch that t α. It is shown in [11], [20] that the compass roting, random compass roting and the greedy roting garantee to delier the packets from the sorce to the destination if Delanay trianglation is sed as network topology. They proed this by showing that the distance from the selected forwarding node to the destination node t is less than the distance from crrent node to t. Howeer, the same proof cannot be carried oer when the network topology is Yao graph, Gabriel graph, relatie neighborhood graph, and the localized Delanay trianglation. We present or experimental reslts of arios roting methods on different network topologies. Figre 4 illstrates some network topologies discssed in this paper. Recall that Gabriel graph, relatie neighborhood graph, Delanay trianglation, LDel (2 ) (V ), and PLDel(V ) are always planar graphs. The Yao strctre, Delanay trianglation, LDel (2 ) (V ), and PLDel(V ) are always a t-spanner of the nit-disk graph. We se integer parameter k =8in constrcting the

20 20 GG RNG Yao Del LDel (2) PLDel Fig. 4. Varios planar network topologies (except Yao). Yao graph. In the experimental reslts presented here, we choose total n =50wireless nodes which are distribted randomly in a sqare area with side length 100 meters. Each node are specified by a random x-coordinate ale and a random y-coordinate ale. The transmission radis of each wireless node is set as 30 meters. We randomly select 10% of nodes as sorce nodes; and for eery sorce node, we randomly choose 10% of nodes as destination nodes. The statistics are compted oer 10 different node configrations. Interestingly, we fond that when the nderlying network topology is Yao graph, LDel (2 ) (V ), orpldel(v ), the compass roting, random compass roting and the greedy roting deliered the packets in all or experiments. Table I illstrates the deliery rates of different localized roting protocols on arios network topologies. For nearest neighbor roting and farthest neighbor roting, we choose the angle α = π/3. The LDel (2 ) (V ) and PLDel(V ) graphs are preferred oer the Yao graph becase we can apply the right hand rle when preios simple heristic localized roting fails. The reason that the compass, random compass, and greedy methods are able to garantee the deliery of the packets in or simlations may be follows: when the transmission range of a node is large enogh, the localized Delanay trianglation of a randomly niformly distribted point set is almost the same as the Delanay trianglation (this can be proed [36]). We already know that these three methods garantee deliery if Delanay trianglation is sed. RNG and GG hae small deliery rate when these simple localized heristics are sed since these two strctres hae less edges than other strctres and therefore each node often has less choices of nodes to relay messages. Table II illstrates the maximm ratios of Π(s, t) / st, where Π(s, t) is the path traersed by the packet sing different localized roting protocols on arios network topologies from sorce s to destination t. We only consider the pairs when a path is fond by the roting method. In or experiment, we fond that the ratios Π(s, t) / st are small. We also condcted extensie simlations of the Face roting method [11] and the Greedy- Face-Greedy (GFG) roting method [37] on Gabriel graph and the local Delanay trianglation

21 21 TABLE I The deliery rate of different localized roting methods on arios network topologies. Yao RNG GG Del LDel (2) PLDel NN FN MFR Cmp RndCmp Grdy TABLE II The maximm spanning ratio of different localized roting methods on arios network topologies. Yao RNG GG Del LDel (2) PLDel NN FN MFR Cmp RndCmp Grdy LDel 1 (V ). We choose n =20, 30,, 90, 100 nodes randomly and niformly distribted in a sqare of length 100 meters. The niform transmission range of nodes are set as r, where r aries from 30, 40, 50, 60, 70 meters. Since it will be hard to control the network density directly, we change the network density by indirectly change the nmber of nodes in the network or the transmission range of the nodes. Large nmber of nodes or larger transmission range will reslt in a denser network. See Tables III and IV. For a gien point set, we randomly select 10% nodes as sorces and 10% nodes as targets. The maximm and aerage spanning ratio is compted for all chosen pair of nodes. Gien n and r, we generate 10 sets of random n points. We fond that the spanning ratio of the Face roting method is significantly smaller when the local Delanay trianglation is sed instead of the Gabriel graph. This may be de to the fact that local Delanay

22 22 trianglation has more edges, ths the nmber of faces traersed by the Face roting algorithm is often smaller when LDel is sed than when GG is sed. The aerage spanning ratios of the GFG method on Gabriel graph and the local Delanay trianglation are similar when the nmber of nodes in the network is no more than 100. We expect or local Delanay trianglation to perform better when the nmber of nodes are significantly large since the spanning ratio of the Gabriel graph on a set of n random points is abot Ω( log n/ log log n). When n is small, Ω( log n/ log log n) is almost a small constant, bt when n is large, Ω( log n/ log log n) cannot be treated as a small constant anymore. Notice that, or local Delanay trianglation garantees that there is always a path to connect any two nodes with length no more than 2.5 times the length of the shortest path connecting them in the original UDG. TABLE III The maximm spanning ratio of Face (oer the fraction line) and GFG (nder the fraction line) roting methods on Gabriel graph and local Delanay trianglation. n n / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /2.7 On Gabriel graph / / / / /2.9 On local Delanay trianglation VI. CONCLUSION It is well-known that Delanay trianglation Del(V ) is a t-spanner of the completed graph K(V ). In this paper, we define seeral new strctres and then gae a localized algorithm that

23 23 TABLE IV The aerage spanning ratio of Face (oer the fraction line) and GFG (nder the fraction line) roting methods on Gabriel graph and local Delanay trianglation. n / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /1.2 On Gabriel graph n / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /1.2 On local Delanay trianglation constrcts a graph, namely PLDel(V ). We proed that PLDel(V ) is a planar graph and it is a t-spanner by showing that UDel(V ) is a sbgraph of PLDel(V ). The total nmber of messages sent by all nodes in or algorithm is O(n log n) bits. Or experiments showed that the deliery rates of existing localized roting protocols are increased when localized Delanay trianglation is sed instead of seeral preiosly proposed planar topologies. Or simlations also shows that the traelled distance of the packets is significantly less when the FACE roting algorithm is applied on LDEL than applied on GG. We proed that the shortest path in PLDel(V ) connecting any two nodes and is at most a constant factor of the shortest path connecting and in UDG. It remain open designing a localized algorithm sch that the path traersed by a packet from to has length within a constant of the shortest path connecting and in UDG. REFERENCES [1] Daid B Johnson and Daid A Maltz, Dynamic sorce roting in ad hoc wireless networks, in Mobile Compting, Imielinski and Korth, Eds., ol Klwer Academic Pblishers, [2] S. Mrthy and J. Garcia-Lna-Acees, An efficient roting protocol for wireless networks, ACM Mobile Networks and Applications Jornal, Special isse on Roting in Mobile Commnication Networks, ol. 1, no. 2, 1996.