Distributed Planarization and Local Routing Strategies in Sensor Networks

Distributed Planarization and Lo cal Routing Strategies in Sensor Net w orks Florian Huc 1 , Aubin Jarry 2 , Pierre Leone 2 , and Jose Rolim 2 1 LPD, ´ Ecole P olytechnique F ´ ed ´ erale de Lausanne (EPFL) 2 Computer Science Departmen t, Universit y of Genev a Corresp onding author: pierre.leone@unige.ch Abstract. W e presen t an algorithm whic h computes a planar 2-spanner from an Unit Disk Graph when the no de density is sufficient. The comm unication complexity in terms of num b er of no de’s identifier sen t b y the algorithm is 6 n , while the computational complexity is O ( n∆ ), with ∆ the maximum degree of the communication graph. F urthermore, we present a simple and efficient routing algorithm dedicated to the computed graph. Last but not least, using traditional Euclidean co ordinates, our algorithm needs the broadcast of as few as 3 n no de’s identifiers. Under the hypothesis of sufficient no de density , no broadcast at all is needed, reducing the previous b est kno wn complexit y of an algorithm to compute a planar spanner of an Unit Disk Graph whic h w as of 5 n broadcasts. 1 In tro duction In many problems on net w orks, among which the problem of message routing [1–3], it is useful to know a planar subgraph of the comm unication graph. Although, all planar subgraphs are not equally in teresting: an usual requirement is that the length of a path b et w een t wo nodes is not to o muc h longer in the planar subgraph than in the original graph. In this paper, we prop ose a distribute d and simple way to c ompute such a planar sub gr aph of a unit disk gr aph when the no des of the communication graph are lo calized using the virtual r aw anchors c o or dinate system [4], instead of the stronger hypothesis of having the no des lo calized in a classical 2D co ordinate system. W e further prop ose a simple, efficient and light r outing algorithm that is dedicated to the constructed graph. This last contribution is related to the conjecture (partially solved in [5] and fully solved [6, 7]) that a 3-connected graph accepts an embedding suc h that the greedy routing algorithm 1 is guaran teed to deliver an y message. Indeed, w e prop ose a routing algorithm ( similar to the greedy routing), which ensures message deliv ery in the unit disk graph induced by an y set of no des, under some connectivit y assumptions. 1.1 Related work Planar graph and poset dimension [8] it is prov ed that a graph G = ( V , E ) is planar if and only if it has order-dimension at most three, where a graph has dimension d if and only if there exists a sequence < 1 , . . . , < d of total orders on V whose intersection is empty , and suc h that for each edge ( xy ) ∈ E and for eac h z ∈ V \ { x, y } , there is at least one order < j in the sequence suc h that x < j z and y < j z . It means that any three total orders whose intersection is empt y induce a planar graph, this graph being the subgraph of the complete graph obtained by keeping only the edges that satisfy the second condition. W e will refer to this graph as the Sc hnyder’s graph of the three total order, and w e note it G S chnyder < 1 ,< 2 ,< 3 or G S for short. 1 Giv en a distance among the no des, for instance the Euclidean distance, we call the gr e edy r outing algorithm , the algorithm whic h consists for a source x to forward a message to the no de that is the closest to the destination y in that distance Planar spanner m uch w ork ha v e b een dedicated to the construction of planar subgraphs. One of the first planarization technique is the use of Gabriel graphs, but, if connectivity is preserved, an edge may b e replaced by a path of unbounded length [9], whereas, as we mentioned previously , we wan t to a void this. The following well known definition catc hes the type of subgraphs w e are interested in: giv en a graph G and a subgraph G 0 , we say that G 0 is a k -sp anner if, for all pairs of no des x, y , a shortest path from a no de x to another no de y in G 0 , is at most k times longer than a shortest path b et ween these tw o no des in G ; the factor k is called the stretch. If G is the complete graph, we further sa y that G 0 is a geometric k -spanner. The Delaunay’s triangulation of a set of vertices V is a planar geometric spanner. Its stretc h factor is upp er b ounded by 1 . 998 [10], and low er b ounded strictly by π / 2 [11], the exact stretc h b eing unkno wn. In [12], the authors efficiently construct a planar 2.5-spanner of UDG that contains the edges of length 1 of the Delauna y’s triangulation of a set of no des V . The complexity of this construction is improv ed in [13], in whic h an algorithm needing 5 broadcast is prop osed. An other construction of spanner of Unit Disk Graph is prop osed in [14] with stretc h > 2. Other spanners exist, in particular a w ay to construct a 2-spanner from a complete graph is prop osed in [15]. In terestingly , it is shown in [16] that three different constructions lead to the same planar geometric 2-spanner. These three constructions are the half- θ 6 -graphs, the triangular- distance Delaunay triangulation (TD Delaunay graphs) and the geo desic embeddings. In [17], the authors further prop ose a planar spanner with bounded degree. W e refer the interested reader to the recent survey of Bose and Smid [18]. Greedy embedding When one consider building a spanner, one usually does not fo cus on preserving easy routing prop erties suc h as greedy routing. It means that even if the greedy routing algorithm delivers a message to the destination in the original comm unication graph, it has no guarantee to succeed in an usual spanner. Preserving suc h a prop ert y would be of great interest. This problem is related to the following conjecture [19]: given a 3-connected graph, do es an embedding exist suc h that the greedy routing algorithm is guaranteed to deliv er any message ? In [5], it is prov ed that the conjecture is true if the graph is a plane triangulation by using Schn yder’s c haracterization of planar graphs [8]. Later, the conjecture was pro ved for every 3-connected graphs in [6, 7]. 1.2 Summary of results VRA C F or all the results presented in this pap er, w e assume that the nodes are lo calized using the virtual ra w anchors co ordinate system (VRAC, [4]), or a simple v ariant of it. Supp osing that the no des are lo calized in these co ordinate systems, is a strictly weak er h yp othesis than the hypothesis that the nodes are lo calized in a traditional 2D co ordinate system. Indeed, if the no des are lo calized in a traditional 2D co ordinate system, it is p ossible to compute their co ordinates in the virtual ra w anc hors coordinate systems, while the conv erse is imp ossible. F urthermore, this co ordinate system is exp ected to be easier to implement in practice. Planar subgraph and spanner As mentioned previously , a graph of order-dimension 3 is planar [8]. Hence, to planarize a graph, it is sufficient to select edges that corresp ond to three total orders. This tec hnique has already b een used, and, for instance, the half- θ 6 graph mentioned in the previous section can b e constructed along this line. How ev er, sev eral issues app ears. First, this technique may need imp ortant computations. Second, the three orders b eing total, the computation ma y not b e feasible lo cally . Third, the computed planar graph is a subgraph of the complete graph, and may not b e a subgraph of the communication graph. In this pap er, w e address these issues when the communication graph is a Unit Disk Graph. T o do so, we prop ose three total orders based on the VRAC co ordinates using which we can construct a planar gr aph suc h that if the no de densit y is high enough, 1. it needs only comparison (no other op erations of an y type), 2. it is a 2-spanner 3. constructing it requires to broadcast at most 6 n no des iden tifiers. 2 In particular, our result improv es the result of [12] by construction a spanner with stretch factor 2 v ersus 2.5. Plus, using the VRAC co ordinates, our algorithm induces the broadcasts of at most 6 n no de’s identifiers (excluding the one needed for the neighborho o d discov ery), and has computational complexity O ( n∆ ), for ∆ the maximum degree of G . F urthermore, using traditional Euclidean co ordinates, it needs the broadcast of as few as 3 n no de’s iden tifiers that can b e done in a single communication round. As when the densit y is high enough, the constructed graph is a planar 2-spanner, our work answers the op en pr oblem numb er 22 of [18] . If this work is inspired b y the pap er of Schn yder [8], we stress out that the constructed graphs are not necessarily subgraphs of G S , the planar graph induced b y the three total orders as when follo wing Sc hnyder’s theory . In more details, in Section 3 we construct a first subgraph e G from G b y using only each no de’s neigh b ors. If we only keep the edges of length at most 2 r / √ 5 ≈ 0 . 8944 r , the subgraph e G is planar. When the no de densit y is to o small, the obtained graph may not b e connected, and, to av oid this, we in tro duce virtual edges (edges that are not edges of the connexion graph). In Section 4, changing slightly the VRA C co ordinates system, w e pro v e that e G 0 is a subgraph of the half- θ 6 graph whic h is equals to G S . It implies that e G 0 is planar, but it gives no result on its stretch. Nonetheless, we pro ve that it v erifies 1) the length of a shortest path in e G 0 is at most t wice the length of a shortest path in G , and 2) a virtual edge corresponds to a path of tw o edges in G . When the node densit y increases, the virtual edges disapp ear, hence, e G 0 is a planar 2-spanner of G when the no de density is high enough. All these results hold even when the constructed graph is not equal to the half- θ 6 graph. Finally , using the VRAC co ordinates, our algorithm needs tw o round of comm unications and induces the broadcasts of 6 n no de’s iden tifiers on top of the one needed for the neigh b orhoo d disco v ery . Using Euclidean co ordinates, w e can reduce this to the broadcasts 3 n no de’s identifiers that can b e p erformed in a single round of comm unication, and no messages are exchanged at all when the densit y is high enough. Routing In [5], it is prov ed that a plane triangulation has an em bedding in whic h the greedy routing algorithm is guaran teed to deliver any message. In our work, w e assume given the embedding in the plane, so, it means that, instead of choosing an embedding for the nodes, w e look at the dual problem, that is designing a routing algorithm (as close as p ossible to the greedy routing algorithm) whic h guarantees deliv ery . 2 The mo del 2.1 Comm unication mo del W e consider a wireless net work in which tw o no des can communicate if they are at distance at most r , the comm unication radius. W e can normalize the distances so that r = 1, in which case w e hav e Unit Disk Graphs (UDG). How ever, w e will keep men tioning r , as we believe it carries useful information. The use of the UDG mo del for the communication links is sub ject to caution from a practical p oint of view. W e quickly men tion the recen t pap er [20] that discusses how protocols that are prov ed v alid under the UDG mo del can b e turned to v alid proto cols in the more realistic SINR model. Another wa y of extending the results of this pap er to more general communication mo dels is to use basic prop erties of such mo dels like the conv exity of the region where the comm unication can happ en [21]. Indeed, it seems to us that most of the argumen ts that w e use are related to this prop erty . The comm unication graph is giv en by the structure ( V , E ) where V is the set of no des and E , the set of edges, i.e. the set of couples of no des that can comm unicate together directly . W e will use virtual edges. A virtual edge is an edge b et ween tw o nodes x and y such that ( xy ) 6∈ E , but with a path from x to y of edges of E . Finally , w e do not consider the impact of interferences or collisions during wireless comm unication. 3 2.2 Co ordinate system W e use the virtual ra w anc hor co ordinate system [4] with three anchors A 1 , A 2 , A 3 . It means that eac h no de kno ws its distances to the three anchors, distances which form the no de coordinates. I.e. the co ordinates of no de x is the v ector ( d ( x, A 1 ) , d ( x, A 2 ) , d ( x, A 3 )). Definition 1. The c o or dinates of a no de x is a ve ctor ( x 1 , x 2 , x 3 ) = ( d ( x, A 1 ) , d ( x, A 2 ) , d ( x, A 3 )) Throughout the paper, w e supp ose that all no des la y inside the triangle defined by the three anchors on a 2D-plane, this area is denoted A . W e use t wo differen t distances to define the co ordinate system. In Section 3, we use the Euclidean distance for the distance function d . Given tw o p oin ts x and y , we note | xy | the Euclidean distance from x to y , throughout the paper. In Section 4, we extend the results using for the distance d ( x, A 1 ) , d ( x, A 2 ) , d ( x, A 3 ), the heights of the triangles \ A 2 xA 3 , \ A 1 xA 3 and, \ A 1 xA 2 resp ectiv ely . W e will note this distance d h ( x, A i ) or d h A i ( x ) for 1 ≤ i ≤ 3. W e further suppose that \ A 1 A 2 A 3 is equilateral and that all no des kno w the distances b et w een the anchors: | A 1 A 2 | , | A 1 A 3 | , | A 2 A 3 | . 3 Distributed graph planarization In this section, given an Unit Disk Graph G , we build a planar subgraph e G . W e further extend it to e G 0 by c hanging some of its edges by virtual edges, where a virtual edge represen ts a path of G . Recall that in [8] it is prov en that if we consider a graph G = ( V , E ) and that we ha ve three total order relations, < 1 , < 2 , < 3 , on the set of no des and – the in tersection of the three order relations is empty , – for each edge ( x, y ) ∈ E and for each vertex z 6∈ { x, y } there is at least one order < i suc h that x < i z and y < i z . then the graph admits a planar em b eddings. In this paper, we adapt this result to UDG. It leads to a simple and lo c alize d distribute d algorithm to planarize a communication graph of a wireless netw ork and to a simple description of the comm unication graphs that accept an efficient routing algorithm. Our aim is to define three suitable order relations by using virtual raw anchor co ordinate system. The order relations hav e to satisfy some prop erties to ensure that the resulting communication graph admits a planar em b edding. In this pap er, w e show how to lo c al ly compute the planar embedding by using only the distances to the anchors. W e assume that there are no pair of no des x, y such that for a given k ∈ { 1 , 2 , 3 } , d ( x, A k ) = d ( y , A k ). It do es not restrict the generalit y since the (Leb esgue) measure of these p ositions is zero. Giv en the no des’ co ordinates, w e define three total order relations, < 1 , < 2 , < 3 on the set of no des V in the follo wing wa y: Definition 2. F or k ∈ { 1 , 2 , 3 } , no des x and y with c o or dinates ( x 1 , x 2 , x 3 ) and ( y 1 , y 2 , y 3 ) satisfy the r elation x < k y , if and only if x k < y k . Lemma 1. Given that the thr e e anchors ar e not aligne d, we c onsider the set of no des that ar e inside A , se e Figur e 1. If we c onsider the r estriction of the or der r elations < k on A × A denote d < k | A then their interse ction is empty. 3 \ k =1 < k | A = ∅ . (1) 4 Pr o of. T o prov e that the intersection is empt y is equiv alen t to prov e that given any p oin t x that b elongs to the con vex h ull of the three anchors the triangular area A is cov ered by the three circles cen tered on the anc hors and passing through x . Indeed, if the intersection is not empty there is a p oin t y ∈ A that b elongs outside of the three circles (and recipro cally) , i.e. x < k y , for k ∈ { 1 , 2 , 3 } . Because the area A is the union of the three triangles \ A 1 xA 3 , \ A 1 xA 2 and \ A 2 xA 3 , see Figure 2(b), it is sufficien t to sho w that the three triangles are cov ered by the circles. W e consider \ A 1 xA 3 particularly and the pro of extend to the others triangles. W e decomp ose the triangle \ A 1 xA 3 in to tw o sub-triangles \ A 1 xx 0 and \ A 3 xx 0 , where x 0 is such that the line xx 0 crosses the line A 1 A 3 p erpendicularly . Because the length of the segment A 1 x is larger than the length of the segment A 1 x 0 the sub-triangle \ A 1 xx 0 is cov ered by the circle cen tered in A 1 and passing through x . The same argument apply to the sub-triangle \ A 3 xx 0 and this concludes the pro of. y > 2 x y > 1 x y > 3 x T 3 k =1 < k 6 = ∅ A 1 A 3 x A 2 Fig. 1. If we do not restrict ourselv es to the region A , T 3 k =1 < k 6 = ∅ . T 3 k =1 is represented in gra y R emark 1. Notice that if we do not assume that the no des b elong to the area A then, the intersection (1) ma y not b e empty . Indeed there are p oin t y whose the distances to the three anchors are larger than the distances of x to the three anc hors, i.e. x < k y , ∀ k = 1 , 2 , 3, see Figure 1. Definition 3. We define the thr e e binary r elations e < 1 , e < 2 , e < 3 by ∀ x, y ∈ V , k = 1 , 2 , 3 , x e < k y ⇐ ⇒ x < k y and y < j x for j 6 = k . F rom Lemma 1, we deduce that the graph G S chnyder e < 1 , e < 2 , e < 3 induced by these three total orders is planar. Ho wev er, as we mentioned in the introduction, there are some ma jor issues: 1) G S chnyder e < 1 , e < 2 , e < 3 ma y not be a subgraph of an UDG, and 2) G S chnyder e < 1 , e < 2 , e < 3 can not b e computed locally . W e denote min k the minimal z with resp ect to the order relation < k . The next lemma, gives a lo c al condition to ensure planarit y . Lemma 2. Given an UDG G = ( V , E ) , if ∀ ( x, y ) ∈ E and ∀ z ∈ V \ { x, y } with max {| xz | , | y z |} < √ 5 r / 2 , ther e exists k ∈ { 1 , 2 , 3 } such that x < k z and y < k z then the gr aph is planar. Pr o of. The condition that max {| xz | , | y z } < √ 5 r / 2 is particular to UDG. Indeed, for an edge ( u, v ) ∈ E with either u or v at distance larger than √ 5 r / 2 from x and y , the tw o edges ( xy ) and ( uv ) (whose lengths are b ounded by r ) cannot in tersect. This condition limits the set of no des that p oten tially can b e linked to an edge in tersecting ( xy ) and ensures that the verification can be done lo cally . W e now consider tw o edges ( xy ) and ( uv ), with b oth u and v at distance at most √ 5 r / 2 from x and y . By assumption, there exist k 1 , k 2 , k 3 , k 4 ∈ { 1 , 2 , 3 } such that u, v < k 1 x, u, v < k 2 y , x, y < k 3 u, x, y < k 4 v . 5 It is clear that k 1 6 = k 3 , k 4 and k 2 6 = k 3 , k 4 and we can assume that k 1 = k 2 and then u, v < k 1 min( x, y ). Indeed, if k 3 6 = k 4 w e hav e k 1 = k 2 b ecause k i = 1 , 2 , 3. If k 3 = k 4 w e apply the same argument to u, v instead of x, y . W e conclude that ( uv ) do not cross ( xy ) b ecause each p oin t of ( uv ) are < k 1 smaller than x and y , see Figure 2(a). 00 00 00 00 00 00 00 00 00 00 00 00 11 11 11 11 11 11 11 11 11 11 11 11 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 A k 1 v u x y (a) u, v < k 1 min( x, y ) implies that ( uv ) and ( xy ) do not in tersect. y > 3 x | \ A 1 xA 3 y > 1 x | \ A 1 xA 3 x A 2 A 1 A 3 (b) T 3 k =1 < k | A = ∅ . Fig. 2. Lemma 3. Given a gr aph G = ( V , E ) and thr e e anchors A 1 , A 2 , A 3 . We define the sub gr aph e G =  V , e E  of G by ∀ x, y ∈ V , ( xy ) ∈ e E ⇐ ⇒ ( xy ) ∈ E and ∃ k ∈ { 1 , 2 , 3 } such that y = min k { z | x e < k z } or x = min k { z | y e < k z } . If the lengths of al l the e dges in the r esulting gr aph e G =  e V , e E  ar e smal ler than 2 r / √ 5 ≈ 0 . 8944 then the gr aph e G is planar. Pr o of. Let ( xy ) b e an edge of e G . Since by hypothesis, all edges of e G are of length at most 2 r / √ 5, it can b e seen as a subgraph of a Disk Graph with radius 2 r / √ 5. T o apply Lemma 2, it is sufficient to chec k that ∀ z 6∈ { x, y } with max( | xz | , | y z | ≤ √ 5 / 2 ∗ 2 r / √ 5 = r , there exists k ∈ { 1 , 2 , 3 } such that x, y < k z . Only , if suc h a z exist, in our construction of e G , w e would not ha ve the edge ( xy ), but instead, we w ould hav e an edge ( xz ), a con tradiction. R emark 2. The selection pro cedure of the edges naturally induces an orientation. Hence the obtained graphs ma y b e seen as digraphs. W e will use this remark in Section 4. e G is not a subgraph of G S chnyder e < 1 , e < 2 , e < 3 , how ever, Lemma 3 provides a sufficien t condition ensuring that the subgraph e G of the communication graph G is a planar graph, see Figure 3. The adv antage of e G is that it needs each no de to kno w only its neighbors. It means that it is sufficient that eac h no de broadcasts its iden tifier and its VRA C coordinates, so that its neighbors kno w them. It induces a comm unication complexity of O (log ( n )) bits. When the densit y is high enough, the prop osed condition is sufficien t. How ev er, when the densit y is low, we observe that there are situations in which considering only the edges of length at most 2 r √ 5 leads to disconnect the graph. T o a void this, a solution is to reconstruct G S chnyder e < 1 , e < 2 , e < 3 , but tw o questions arise: can we still do it lo cally ? If it is not a subgraph of the communication graph, how can we detect the missing edges ? F or this, the solution that w e propose is to in tro duce virtual links . If we are in the situation where x e < k y and x e < k z , z < k y but z is out of range of communication of x , from Schn yder’s theory , x would rather be connected to z than y . Only , due to the limited range of x , it do es not o ccur. Then, the edge ( xy ) can p oten tially cross an edge from z if | z y | < r . T o av oid this, we replace ( xy ) b y a virtual edge ( xz ). F or 6 this, the no de y that knows its neighborho od informs x and a virtual edge b et ween x and z (through y ) replaces the edges ( xy ). In turn, z also chec ks if it is in the same situation as y . Ultimately , we would lik e that the computed graph ˜ G 0 = ( V , ˜ E 0 ) is a subgraph of the graph induced by the three total orders. The algorithm to compute the ˜ G 0 go es as follo w: – (As in Lemma 3) Each node x kno ws its neighboring no des and compute the no des y k , k = 1 , 2 , 3 such that y k = min { z | x e < k z } . – (Virtual edge) Each no de x chec ks with its neighboring no des y k , k = 1 , 2 , 3 that there do es not exist a no de y k 0 in its second neighborho od such that y k 0 < k y k , x e < k y 0 ( d ( y 0 , y ) < r ) and y k 0 is out of the comm unication range of x . • If suc h a no de do es not exist the edge ( xy ) b ecomes active. • If such a node exists, y k c heck with y k 0 that there do es not exist a similar no de that is out the range of comm unication of y 0 . This op eration is rep eated recursiv ely un til no no de satisfying this property is found and a virtual edge is created b etw een x and the last node found. The original edge ( xy k ) is remo ved. R emark 3. In the next section, w e will see that, using the mo dified VRA C coordinates, a virtual edge represen ts a path of length 2. In this case, the recurrence is useless. Ho wev er, using the original VRAC co ordinates, a virtual edge can represen t a longer path. In the middle of Figure 3 we plot a comm unication graph resulting from the selection described in Lemma 3 without restricting the lengths of the edges to b e smaller than 2 r √ 5 . W e observe that tw o edges cross. In the right of the Figure 3 the virtual links mechanism is used. W e observ e that the crossing is remov ed and the graph is planar. By comparing with the left side of the figure, we observe that the connectivity of the graph is better with the virtual links. Ho wev er, when the no de densit y is high enough, the selection of edges of length less than 2 r √ 5 is sufficient. F urthermore, when the density increases, the num b er of virtual edges tends to zero (cf Section 6. Fig. 3. On the left the planar graph obtained by considering only edges of length at most 2 r √ 5 . On the middle the graph obtained by considering all the edges, we observe t w o edges crossing. On the righ t the planar graph obtained with virtual links (in green), the crossing is remov ed. 7 ˜ G 0 may not be a subgraph of G , whic h means that some of its edges ma y ha ve length greater than r . F or this, w e can not use Lemma 2. Ho wev er, if we can pro ve that it is a subgraph of G S , w e would obtain from [8], that it is planar. In the next section, we sligh tly change the VRA C co ordinate system in order to pro ve that it is the case. It also allows us to give guaran tees on the stretch of e G 0 . 4 Prop erties of the planar em b edding In this section we discuss a simple extension of the VRAC co ordinate system. Using this new co ordinate system, we prov e that in e G 0 , the distance b et ween tw o no des is at most t wice the distance in the original graph G . F urthermore, we sho w that a virtual edge e e corresp onds to a path of at most tw o edges in G , and that the length of suc h an edge is upp er b ounded b y 2 r / √ 3. In this section, w e make the follo wing hypothesis: – There are three anchors A 1 , A 2 , A 3 , the nodes b elong to the conv ex hull A of the anchors and they kno w their distances to all three anc hors. – \ A 1 A 2 A 3 is equilateral. – the no des kno w the distances b et w een the anchors ( | A 1 A 2 | , | A 1 A 3 | , | A 2 A 3 | ). With respect to the first part of the paper, the tw o last hypothesis are new. By using the distances b et ween the anchors, each node x can compute the heights of the triangles \ A 2 xA 3 , \ A 1 xA 3 and, \ A 1 xA 2 . As \ A 1 A 2 A 3 is equilateral, it is equiv alen t to compute the surface of these triangles or the heights and it is then easy to see that their sum is constant. W e denote these v alues ( x 1 , x 2 , x 3 ), see Figure 4(a). One adv antage of this co ordinate system with resp ect to just using the distances to the anchors is that the sum of the three triangle areas is constant, so w e can normalize the co ordinates suc h that x 1 + x 2 + x 3 = 1. A reason for doing this is that b ecause of the measurement errors on the physical location of the no des, it is likely that the distances to the anchors do not corresp ond to co ordinates inside a same plane. With the normalization, we pro ject the co ordinate on a same plane. 4.1 Adapting results of Section 3 and further Results of Section 3 Using the co ordinates defined ab o ve, we define the order relations < 1 , < 2 , < 3 and e < 1 , e < 2 , e < 3 the same w ay we did in Section 3. In Section 3, given a no de x , the no des satisfying y e > k x were outside the circle centered at A k of radius | xA k | . With the new definition of the distance function d (c.f. Section 2.2), the no des y satisfying y e > k x are contained on the half plane containing A k defined by the line parallel to ( A k mod 3+1 A k mod 3+1 ) going through x , as illustrated in Figure 4(b). Using this observ ation, it is easy to see that the intersection of the three order relations is empt y , so Lemma 1 is still v alid in this co ordinate system. Similarly , Lemma 2 remains also true and we can adapt the proofs of Lemma 3. T o summarize, all the results we ha ve pro ved previously are v alid with the new co ordinate system . Connectivit y results and stretc h Definition (Figure 4(d)): Given a no de x , we call the greedy regions of x the three regions A x i = { z | x e < i z } , for i ∈ { 1 , 2 , 3 } . Definition (Figure 4(d)): Given a no de x , for i ∈ { 1 , 2 , 3 } , we denote the region b et w een the tw o regions A x i and A x i mod 3+1 b y ¯ A x i . R emark 4. A node x has at most one outgoing edge to wards a no de in eac h of its greedy regions. It has no outgoing edge to w ards no de not in its greedy regions, ho wev er, it ma y ha ve an ingoing edge from an y no de. 8 0 0 0 0 0 0 0 1 1 1 1 1 1 1 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 00000000000000000000 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 11111111111111111111 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 00000000000000000 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 11111111111111111 0000000 0000000 0000000 1111111 1111111 1111111 A 1 A 3 A 2 x 3 x 2 x 1 x = ( x 1 , x 2 , x 3 ) (a) The new coordinate system: x = ( x 1 , x 2 , x 3 ) where x 1 , x 2 and x 3 are respectively the heigh ts of the triangles \ A 2 xA 3 , \ A 1 xA 3 and, \ A 1 xA 2 . A 3 x ˜ < 3 y A 1 x A 2 (b) y with x e < 3 y . A k 1 A k 2 A k 3 u v y x (c) New pro of of Lemma 2 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 00000000000 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 11111111111 A 1 ¯ A x 2 A 3 A 2 A x 2 A x 1 ¯ A x 3 A x 3 ¯ A x 1 x (d) Definition of the greedy re- gions Fig. 4. Adapting pro ofs to the new settings. Lemma 4. Given an e dge ( x, y ) ∈ E , ther e is a p ath P fr om x to y in e G 0 . The p ath is c ontaine d in either { z ∈ A x i | z ≤ i y } or { z ∈ A y i | z ≤ i x } for some i ∈ { 1 , 2 , 3 } , and it verifies: P e ∈ P | e | ≤ 2 | xy | . Pr o of. Given an edge ( x, y ) ∈ E , without loss of generalit y , we can supp ose y ∈ A x 1 . ( x, y ) ∈ E . By h yp othesis, | xy | ≤ r . W e prov e the lemma by induction on the length of xy 2 . By our definition and Lemma 3, ( x, y ) ∈ e E 0 iff y = min 1 { z | x e < 1 z } . If y = min 1 { z | x e < 1 z } , then ( x, y ) ∈ e E 0 and there is a (direct) path b et ween x and y . If not, there is x 0 with x 0 e < 1 y and x 0 = min 1 { z | x e < 1 z } . Notice that x 0 ∈ { z ∈ A x i | z ≤ i y } . W e no w prov e that | x 0 y | < | xy | . Using the notation of Figure 5(a), we obtain that | x 0 y | is maximum for x 0 = x , x 0 = c 1 or x 0 = c 2 . W e hence ha ve to pro ve | c 2 y | ≤ | xy | . The case | c 1 y | ≤ | xy | is symmetric. In the purple triangle of Figure 5(a), w e ha ve | c 2 y | sin (2 π / 3 − β ) = | xy | sin ( π / 3) . So | c 2 y | | xy | = sin (2 π / 3 − β ) sin ( π / 3) . As β ≥ π / 3, w e hav e | c 2 y | | xy | ≤ 1. So | c 2 y | ≤ | xy | . Recall that w e supposed that no t w o no des in the net work ha ve the same coordinates compared to a giv e n anc hor. It means that x 0 can neither b e c 2 nor c 1 3 . W e hence ha ve | x 0 y | < max( | xy | , | c 1 y | , | c 2 y | ) ≤ | xy | ≤ r . So we can apply the induction on the edge ( x 0 y ) which is in E and strictly shorter than ( xy ) and this pro ves that there is a path betw een x and y through x 0 . 2 The length of an edge tak es v alue in I R , how ever there are a finite num b er of edges (upp er b ounded b y n 2 ), hence our induction will terminate. 3 Because d ( x, A 2 ) = d ( c 1 , A 2 ) and d ( x, A 3 ) = d ( c 2 , A 3 ) 9 A 2 A 3 A 1 c 2 x 0 c 1 π / 3 y x β (a) 2 √ 3 | x 2 − y 2 | − 1 √ 3 | x 1 − y 1 | x y 2 √ 3 | x 1 − y 1 | | xy | | x 2 − y 2 | | x 1 − y 1 | (b) Fig. 5. | x 0 y | < | xy | and P e ∈ P | e ≤ 2 | xy || Let us now lo ok at the stretch factor. W e hav e x < 1 x 0 , y > 1 x 0 and there is a path y 1 = x 0 , y 2 , . . . , y l = y in the graph e G that satisfies y 1 < 1 y 2 < 1 . . . < 1 y l = y by construction of the virtual edge. Because the co ordinate with resp ect to A 1 increases(monotonically) along the path w e hav e that | x 1 − y 1 1 | + | y 2 1 − y 3 1 | + . . . + | y l − 1 1 − y l 1 | = | x 1 − y 1 | (the subscript indicates the co ordinate with respect to A 1 ). If x 0 ∈ A y i for i ∈ { 2 , 3 } , we hav e y < i x 0 and y > 1 x 0 . By induction, if such a i exists, the rest of the path will b e in A y i and the i th-co ordinate decreases (monotonically) along the path. If not, x 0 ∈ ¯ A y 1 and y > 1 y 0 , y < 2 x 0 and y < 3 x 0 and b oth coordinates with resp ect to A 2 and A 3 decreases along the path. This prov es that in all cases, w e ha ve that there exists i ∈ { 2 , 3 } suc h that | x i − y 1 i | + | y 1 i − y 2 i | + . . . + | y l − 1 i − y l i | = | x i − y i | . In summary , along the path P , for z ∈ P , z 1 increases from x 1 to y 1 and there is an i ∈ { 2 , 3 } such that z i decreases from x i to y i . The distance cov ered in A x 1 to go from a no de with i th co ordinate x i to a no de with i th co ordinate y i is upp er-bounded by 2 √ 3 | x i − y i | , see Lemma 5, since we supp ose that the anchors form an equilateral triangle. F rom this we deduce that the length of the path P is upp er bounded by 2 √ 3 | x 1 − y 1 | + 2 √ 3 | x i − y i | , this longest path is obtained b y moving along the path where the i th co ordinate is constan t and the first co ordinate goes from x 1 to y 1 and then along the path where the i th co ordinate go es from x i to y i and the first one is constan t. W e now express | xy | in terms of | x 1 − y 1 | and | x i − y i | . By the configuration of the different triangles, c.f. Figure 5(b), w e ha ve | xy | 2 = | x 1 − y 1 | 2 + ( 2 √ 3 | x i − y i | − 1 √ 3 | x 1 − y 1 | ) 2 = 4 3 ( | x i − y i | 2 + | x 1 − y 1 | 2 − | x i − y i || x 1 − y 1 | ). Hence the stretc h factor c verifies: c 2 = ( 2 √ 3 | x 1 − y 1 | + 2 √ 3 | x i − y i | ) 2 4 3 ( | x i − y i | 2 + | x 1 − y 1 | 2 −| x i − y i || x 1 − y 1 | ) ≤ 4. So w e hav e c ≤ 2 as claimed. Corollary 1. e G 0 is a sub gr aph of G S which is e qual to the half- θ 6 gr aph. Pr o of. The definition of the half- θ 6 graph giv es the same graph as G S when using the three total orders using the modified VRAC coordinates. F rom the previous lemma, we get that if there is an edge ( xy ) ∈ ˜ E 0 , with y in a greedy region A x k , then y is minimum according to ˜ < k . Hence, ( xy ) is an edge of G S . How ever, there are examples in whic h e G 0 6 = G S F rom this corollary , we immediately obtain that e G 0 is planar. When ˜ G 0 = G S , we also deduce that it is geometric 2-spanner. Ho wev er, in the general case, we can not deduce any information on its stretc h, which is the ob ject of the next theorem, which is implied b y Lemma 4. Theorem 1. Given a c onne cte d gr aph G , the gr aph e G 0 is planar, and for any two no des x and y , if ther e is a p ath of length ` fr om x to y in G , ther e is one of length at most 2 ` in e G 0 . 10 Ho wev er, notice that the previous theorem applies to e G 0 which may contains virtual edges that are not edges of the Unit Disk Graph G , instead, a virtual edge represen ts a path in G . The next lemma sa ys that suc h a path has length tw o. Notice further that the virtual edges are edges of the Unit Hexagonal Graph (c.f. [16]). Finally , simulations in Section 6 show that the virtual edges are rare and disapp ear as the no de densit y increases. Lemma 5. A virtual link ( xy ) ∈ e E 0 has length at most 2 r / √ 3 when the anchors form an e quilater al triangle. Pr o of. Without loss of generality , we can suppose that the link is oriented from x to y and with resp ect to minimizing co ordinate y 1 . The virtual links represents a path x, z , . . . , y where z is inside the communication range of x , so d ( x, z ) ≤ r . W e know by construction that considering the first coordinates, we ha ve y 1 ≤ z 1 . It means that y is in the triangle T 1 delimited b y the three lines { u | u 1 = z 1 } , { v | v 2 = x 2 } and { w | w 3 = x 3 } as depicted in Figure 6. The furthest points of this triangle are the t wo summits other than x . This triangle is equilateral and the edges are of length 2 r / √ 3, so | xy | ≤ 2 / √ 3 r . F rom this lemma, w e obtain that Algorithm 1.1 constructs correctly ˜ G 0 . Input: A Unit Disk Graph G . Output: ˜ G 0 . for all x ∈ V do for k ∈ 1 , 2 , 3, y k = min k { y ∈ A x k , | y k x | < r } do x broadcast ”activ ate ( xy k )” y 0 k = min k { y ∈ A x k , | y k y | < r } if y k 6 = y 0 k then y k broadcast ”disable ( xy k ) and activ ate ( xy 0 k )” end if end for end for Algorithm 1.1. Distributed construction of ˜ G 0 . Corollary 2. Given a c onne cte d gr aph G , if the density is high enough, e G 0 is a planar 2-sp anner of G . Pr o of. When the density increases, the num b er of virtual edges decreases, and if the densit y is high enough, with high probabilit y , there remains none of them. In this case, e G 0 is a subgraph of G , and the result follo ws from Theorem 1. T o summarize, e G 0 is a planar graph such that 1) the length of a shortest path in this graph is at most t wice the length of a shortest path in the communication graph, and 2) virtual edges corresp ond to a path of length tw o in the comm unication graph. F urthermore, to c onstruct e G 0 (Algorithm 1.1), the communication complexit y in terms of bits at a no de x is as follow: eac h no de broadcast once its Id and co ordinates, then, eac h no de broadcast the Id of its three neighbors minimizing the orders, and finally , each of this neighbors ma y send to x an other no de Id whic h is the extremit y of a virtual edges starting from x . Hence, eac h node induces an exchange of at most 6 no des (excluding the neighborho o d discov ery), which gives a total of at most 6 n no des’ ids that are broadcast. The computational complexit y is O ( n∆ ), with ∆ the maxim um degree of the communication graph: each no de x computes which of its neighbors minimizes each order, plus, x requires eac h of the selected neighbors y 1 , y 2 , y 3 to v erify if there should b e a virtual edge, whic h also consists in computing a minim um of a set of at most ∆ elements. No w, if instead of using VRA C co ordinates, we use the Euclidean co ordinates, each no de y can computes on its o wn if it is minimum for one of the three orders for one of its neigh b ors. It means that w e can av oid the statement ”broadcast ”activ ate ( xy k )”” in Algorithm 1.1. The mo dified version of Algorithm 1.1 needs the broadcasts of at most 3 n nodes iden tifiers that can all b e p erformed in a single round of communication. 11 In case there are no virtual edges, the resulting graph is a 2-spanner of the Unit Disk Graph, and in this case no messages are exc hanged. It answers the open question 22 of [16] under the h yp othesis that the density id high enough. Recall that ˜ G 0 is alw ays a subgraph of a Unit Hexagonal Graph (c.f. [16]), and hence a planar spanner for these graphs. 5 A lo cal routing algorithm W e now prop ose a lo cal routing algorithm. This algorithm has tw o modes depending on if the destination is in a greedy region of the sender or not. Lemma 6. R e c al l that we supp ose the triangle \ A 1 A 2 A 3 e quilater al. L et x b e a no de with a message for a destination y . If y b elongs to a gr e e dy r e gion of x , and that x has an out-neighb or in this gr e e dy r e gion, the algorithm pr o c e e ds as fol low: – (Data delivery) | xy | ≤ r in which c ase x tr ansmits the data dir e ctly to y . – (Greedy routing) | xy | > r and x tr ansmits to its neighb or x 0 that b elongs to the same gr e e dy r e gion as y . We have | xy | ≥ | x 0 y | . Pr o of. If x transmits directly to y there is nothing to prov e. F or the other case, let x 0 b e the node that receiv es the message. W e note the v ector xy = a e iα 1 and xx 0 = b e iα 2 ( | xy | = a and a ≥ b ). The v ector x 0 y = a e iα 1 − b e iα 2 , | x 0 y | = a 2 + b 2 − 2 ab cos( α 1 − α 2 ) and, 0 ≤ α 1 , α 2 ≤ α b ecause x 0 and y b elong to the same greedy region. Then, − α ≤ α 1 − α 2 ≤ α and cos( α 1 − α 2 ) > 1 / 2. This is sufficien t to conclude that | x 0 y | ≤ | xy | = a . Lemma 7. We assume that the anchors form an e quilater al triangle. L et x b e a no de that wants to send a message to a destination y using the gr aph e G 0 or dir e ct tr ansmission. We assume that y do es not b elong to a gr e e dy r e gion of x . Without loss of gener ality, y is in ¯ A x 2 . We further assume that x has thr e e out- neighb ors and so do any no de in the e quilater al triangle T with b ase the se gment p ar al lel to ( A 2 A 3 ) , c enter e d at x , of length 4 / √ 3 r and with the other summit in ¯ A x 2 (c.f. Figur e 6). Under those hyp othesis, we have two p aths (without c onsider ation on the orientation of the e dges) P 1 = x, u 0 , P 0 1 , u 1 , ..., u k − 1 , P k − 1 1 , u k , z and P 2 = x, v 0 , P 0 2 , v 1 , ..., v l − 1 , P l − 1 2 , v l , z , without loss of gener ality u 0 > 1 v 0 (when P 2 6 = x, z , as if P 2 = x, z , ther e ar e no v 0 ), and we have: – k − 1 ≤ l ≤ k . – the P j i , for i ∈ { 1 , 2 } and j ≤ k ar e monotone p aths with r esp e ct to > 1 , p otential ly of length 0. – ∀ u ∈ P 1 \ { z } , u ∈ A x 2 . – ∀ v ∈ P 2 \ { z } , v ∈ A x 3 . – ∀ 0 ≤ i ≤ l , ther e is an oriente d e dge fr om u i to v i and, ∀ u ∈ { u i , P i 1 } , u > 1 v i . – ∀ 0 ≤ i ≤ l − 1 , ther e is an oriente d e dge fr om v i to u i +1 and, ∀ v ∈ { v i , P i 2 } , v > 1 v i +1 . – z is in ¯ A x 2 . Given these two p aths, either a no de fr om { x, u 0 , ...u k , v 0 , ...v l , z } has y within its c ommunic ation r ange, or | z y | < | xy | . Pr o of. By hypothesis, x has a neighbor (in e G 0 ) u 0 in A x 2 and a neighbor v 0 in A x 3 , without loss of generalit y u 0 > 1 v 0 . By Lemma 5, which b ounds the length of an edge, u 0 is in T , so u 0 has three out-neighbors. In particular it has an out-neighbor u 0 in A u 0 3 . u 0 can not be in A u 0 3 ∩ A x 2 as otherwise w e w ould hav e a virtual link ( xu 0 ) instead of the link ( xu 0 ). If u 0 is in ¯ A x 2 , if w e can pro ve that ( xu 0 ) ∈ e E 0 setting z = u 0 , the theorem would b e verified with P 1 = x, u 0 , z and P 2 = x, z . But, we ma y not hav e an edge b et ween x and u 0 , ho wev er we pro v e that u 0 can b e replaced b y a z ” satisfying all the desired prop erties. u 0 is in T , so it has an out-neighbor u ” in A u 0 1 . If u ” 6 = x , u ” has an out-neigh b or in the greedy region oriented tow ards A u ” 2 that in turn has a neighbor in its 12 ¯ A x 3 ¯ A x 2 ¯ A x 1 x A 1 A 3 A 2 y u v w T T 3 T 1 T 2 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 1111111111 1111111111 1111111111 1111111111 1111111111 1111111111 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 0000000000000000000000000000 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 1111111111111111111111111111 ¯ A x 1 A x 1 ¯ A x 3 d ( x, A 1 ) d ( v 0 , A 1 ) d ( u 1 , A 1 ) v 0 y z x A x 2 A x 3 ¯ A x 2 u 0 P 0 1 P 0 2 u k v l Fig. 6. On the left side, the three greedy regions asso ciated with x and the edges to the three no des u = min 2 { z | x e < 2 z } , v = min 3 { z | x e < 3 z } and, w = min 1 { z | x e < 1 z } . On the right, a path leading to z . greedy region oriented to wards A 2 , ... By planarit y , since we cannot cross the edge u 0 z this path has to go through v 0 . So we hav e a path from u ” to u 0 . Similarly , u ” has an out-neighbor u ” in A u ” 1 whic h is either x or has a path leading to u 0 . Ultimately , we will hav e a no de z 0 with out-neighbor x and a path to u 0 . Using the same argument and considering neigh b ors in A z 0 2 , we ha ve that z 0 has u 0 has neighbor in A z 0 2 , or it has w whose out-neigh b or in A w 1 is x , and that has a path leading to u 0 in A w 2 . Going along this path, we reac h z ” whose out-neighbor in A z ” 1 is x , and out-neighbor in A z ” 2 is u 0 . Hence the theorem is verified for P 1 = x, u 0 , z ” and P 2 = x, z ” W e now suppose that u 0 is in A x 3 and we w ant to prov e u 0 = v 0 . By contradiction supp ose that u 0 6 = v 0 . Recall that w e hav e u 0 < 1 u 0 , b y construction of e G 0 as w ell as u 0 < 3 v 0 . By planarit y , we ha v e v 0 > 1 u 0 . The no de u 0 has a neighbor in its greedy region A u 0 1 . This neighbor must b e v 0 . Indeed, if not, using u 0 < 3 v 0 and using the same argumen t as b efore, this neigh b or u ” has a neigh b or in the greedy region A u ” 2 that in turn has a neigh b or in its greedy region oriented tow ards A 2 , ... But, since w e cannot cross the edge u 0 u 0 b y planarity , there is an infinity of such no des which is imp ossible. So u 0 has v 0 has neighbor in its greedy region A u 0 1 . But then v 0 has a neighbor in its greedy region A v 0 2 . Looking for a sequence of neigh b ors in the greedy region orien ted tow ards A 2 , will ultimately lead to cross edge ( u 0 u 0 ) whic h con tradict the planarit y of e G 0 . So u 0 = v 0 Since v 0 is in T , it has a neigh b or v 0 in its greedy region orien ted tow ards A 2 . v 0 cannot b e in A u 0 3 , the same region as v 0 b ecause if it was true, b y construction of e G 0 , there would b e an edge b et w een u 0 and v 0 instead of the edge ( u 0 v 0 ). First supp ose that v 0 ∈ ¯ A x 2 . W e would like to set z = v 0 , but, as for the v ery first case considered in the pro of, it ma y not b e appropriate. Indeed, there ma y not b e an edge ( z v 0 ). So w e pro ceed as follow: w e consider the out-neigh b or v ” of v 0 in A v 0 1 . If v ” = v 0 , then we are done by setting z = v 0 . Else, we can pro ve that v ” has u 0 as out-neighbor in A v ” 3 . W e pro ceed recursiv ely un til we reach v ” 0 whic h has u 0 as out-neigh b or in A v ” 0 3 and v 0 as out-neighbor in A v ” 0 1 (Notice that u 0 can not ha v e v ” 0 as out-neighbor in A u 0 3 as it already has v 0 ). W e set z = v ” 0 and the theorem is pro ved. 13 W e now study the case v 0 ∈ A x 2 . Let call v ” the out-neigh b or of v 0 that is in the greedy region orien ted to wards A 1 . If v ” = v 0 , all is fine. Else, we must hav e v ” > 0 u 0 (otherwise the neighbors directed tow ards A 3 will cross the edge v 0 v 0 ). Considering the follo wing out-neighbors in the greedy region oriented tow ards A 1 , we will reach a no de v ” 0 whose out-neighbor in A v ” 0 1 is u 0 , or which is the out-neigh b or of x in A x 2 . W e call this path from v ” to v ” 0 , P 0 1 and w e set v 0 = v 1 . W e con tinue similarly b y considering the out-neigh b or of v 1 in A v 1 2 . T o see that this pro cess terminates, notice that the first coordinate of the v ertices on the paths we construct decrease as we get closer to the line ( A 2 A 3 ), plus the distance b etw een a no de in A x 2 and a no de in A x 3 is low er b ounded by a b ound which increases when we get closer to the line ( A 2 A 3 ), plus the length of an edge is upp er bounded by Lemma 5. Hence the process has to finish on a vertex z in ¯ A x 2 . W e now prov e the second part of the Lemma. The maxim um length of a virtual edge is 2 / √ 3 r , hence w e ha ve that the p olygon formed b y x, u 0 , ...u k , z , v l , ...v 0 is composed of triangles with tw o side of length at most 2 / √ 3. So it is co vered b y the union of the disk of radius r centered on { x, u 0 , ...u k , v 0 , ...v l , z } . If y is inside the p olygon formed by x, u 0 , ...u k , z , v l , ...v 0 , it is within the communication radius of one of these vertices. If not, y is b ello w the p olygon x, u 0 , ...u k , z , v l , ...v 0 and out of range from b oth v l and u k in particular, and then | z y | < | xy | . Theorem 2 (Zig-Zag : an extended greedy routing). We assume that the anchors form an e quilater al triangle. L et x b e a no de that ne e ds to tr ansmit a message to a destination no de y . If any no de at distanc e less than 4 / √ 3 r of x has thr e e out-going neighb ors, then the fol lowing str ate gy delivers the data either to y or to a no de z closer to y than x . – If y is in the c ommunic ation r ange of x , x sends the message to y . – If y is in a gr e e dy r e gion of x , x sends the message to its out-neighb or which is in the same gr e e dy r e gion. – Otherwise, use the r estricte d gr e e dy r outing pr o c ess starting at x . Wlo g y ∈ ¯ A x 2 . • x sends the message to its out-neighb or in A x 2 ∪ A x 3 which has the highest first c o or dinate. • A no de u ∈ A x 2 sends the message to y if p ossible or to its out-neighb or v verifying u > 1 v and v > 3 x . If v > 3 x and v > 2 x (i.e. u ∈ ¯ A x 2 ), end the r estricte d gr e e dy r outing pr o c ess. • A no de v ∈ A x 3 sends the message to y if p ossible or to its out-neighb or u verifying v > 1 u and u > 2 x . If u > 3 x and u > 2 x (i.e. u ∈ ¯ A x 2 ), end the r estricte d gr e e dy r outing pr o c ess. Pr o of. By applying Lemmas 6 and 7 we see that the routing strategy leads to y or to a no de that is closer to y than x ( z in Lemma 7). Indeed, Lemma 7 ensures that a restricted greedy routing pro cess starting at x follo ws the path u 0 , v 0 , u 1 , ..., z using the same notations. 6 Sim ulations W e implemen ted both the planarization algorithm and Zig-Zag, the routing algorithm, and w e present b ello w the results of the simulations. W e considered a net work comp osed of 300 sensors spread in a square area [0; 1] × [0; 1] with three anc hors at position (0 . 5 , 3 . 5), ( − 5 √ 3 + 0 . 5 , − 1 . 5) and ( 5 √ 3 + 0 . 5 , − 1 . 5). W e considered a comm unication radius r for the sensors whic h ranges from 0 . 11 to 0 . 225. F or eac h v alue of the comm unication radius, w e p erformed the av erage o ver 1000 net works and successfully routed messages. Notice that the v alue are plotted with resp ect to the av erage degree of the no des in the UDG, as in ˜ G 0 , the av erage degree is upp er b ounded b y six since it is planar. 14 W e first plot the n umber of virtual edges in Figure 7(a). The simulations indicate that the num b er of virtual edges tends to 0 when the no de density increase. Indeed, the a v erage n um b er of virtual edges decreases from 1.6 when the net work is sparse, to 0.03 when it is dense. Then, in Figure 7(b), we plot the a verage stretch of the path computed in ˜ G 0 by Zig-Zag, where the stretc h of a path is the length of the computed path divided by the euclidean distance b et ween the source and the destination. W e observe that we hav e a stretch which is b et ween 1.3 and 1.4 which is b etter than the theoretical stretch factor of 2, and we compare it to the stretc h of the greedy algorithm (in ˜ G 0 ), which is sligh tly better and of appro ximately 1.3. This apparent gain of efficiency is to b e mitigated by the results sho wn on Figure 7(c) which indicates the success rate of the greedy algorithm and of Zig-Zag. The success rate of Zig-Zag tends to 100% as the densit y increases, which back en the theoretical results of Theorem 2. Indeed, when the density increases, the hypothesis of Theorem 2 are verified, whereas they may not b e v erified at some nodes when it is lo w, th us reducing the success rate. The success rate of the greedy algorithm also increases with the no de densit y , ho wev er, it remains b ello w 80%. 5 1 0 1 5 2 0 2 5 0 0 .5 1 1 .5 2 A v e ra g e de g ree N umb e r of V ir tua l e dg e s (a) Av erage n umber of virtual edges 5 1 0 1 5 2 0 2 5 1 1 .1 1 .2 1 .3 1 .4 1 .5 1 .6 Z i g -Z a g G re ed y A v e ra g e de g ree P a th stretc h (b) Av erage path stretch 5 1 0 1 5 2 0 2 5 0 0 .2 0 .4 0 .6 0 .8 1 Z i g -Z a g G re ed y A v e ra g e de g ree S u c c es s ra te (c) Success rates Fig. 7. References 1. P . Bose, P . Morin, I. Sto jmenovi ´ c, and J. Urrutia, “Routing with guaran teed deliv ery in ad ho c wireless net w orks,” Wir eless Networks , vol. 7, no. 6, pp. 609–616, 2001. 2. B. Karp and H. T. Kung, “GPSR: greedy p erimeter stateless routing for wireless netw orks,” in Pr o c e edings of the 6th annual international c onfer enc e on Mobile c omputing and networking . ACM New Y ork, NY, USA, 2000, pp. 243–254. 15 3. F. Kuhn, R. W attenhofer, and A. 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