Proceedings Chapter

Greedy Routing on Virtual Raw Anchor Coordinate (VRAC) System

LEONE, Pierre, SAMARASINGHE, Kasun

Abstract

Geographic routing is an appealing routing strategy that uses the location information of the nodes to route the data. This technique uses only local information of the communication graph topology and does not require computational effort to build routing table or equivalent data structures. A particularly efficient implementation of this paradigm is greedy routing, where along the data path the nodes forward the data to a neighboring node that is closer to the destination. The decreasing distance to the destination implies the success of the routing scheme. A related problem is to consider an abstract graph and decide whether there exists an embedding of the graph in a metric space, called a greedy embedding, such that greedy routing guarantees the delivery of the data. A common approach to assign geographic coordinates is to measure distances (for instance the distances between neighboring nodes) and compute (virtual) coordinates. The rationale of the Virtual raw Anchor Coordinate System (VRAC) is to use the (raw) measured distances as coordinates in order to avoid further computations. More precisely, each node needs [...]

Reference

LEONE, Pierre, SAMARASINGHE, Kasun. Greedy Routing on Virtual Raw Anchor Coordinate (VRAC) System. In: International Conference on Distributed Computing in Sensor Systems. IEEE, 2016.

DOI : 10.1109/DCOSS.2016.21

Available at: http://archive-ouverte.unige.ch/unige:87830

Disclaimer: layout of this document may differ from the published version.

1 / 1 Greedy Routing on Virtual Raw Anchor Coordinate (VRAC) System

Pierre Leone and Kasun Samarasinghe Centre Universitaire d’Informatique University of Geneva Switzerland {pierre.leone,kasun.wijesiriwardana}@unige.ch

Abstract—Geographic routing is an appealing routing strategy hence alternative mechanisms are required to guarantee the that uses the location information of the nodes to route the data. delivery. This technique uses only local information of the communication graph topology and does not require computational effort to Geometric routing requires an underlying location ser- build routing table or equivalent data structures. A particularly vice for assigning coordinates to nodes. Such a service can efficient implementation of this paradigm is greedy routing, be expensive, especially considering the wireless networks where along the data path the nodes forward the data to a with power constrained wireless devices. As an alternative, neighboring node that is closer to the destination. The decreasing distance to the destination implies the success of the routing in [1] geometric routing based on virtual coordinates instead scheme. A related problem is to consider an abstract graph of physical coordinates has proposed. In the same spirit, and decide whether there exists an embedding of the graph in the more important problem of virtual coordinate assign- a metric space, called a greedy embedding, such that greedy ments such that greedy routing guaranteeing delivery, namely routing guarantees the delivery of the data. A common approach greedy embedding has introduced [2]. In subsequent re- to assign geographic coordinates is to measure distances (for instance the distances between neighboring nodes) and compute search, algorithms for greedy embedding on different geomet- (virtual) coordinates. ric spaces [3] for different classes of graphs were proposed. The rationale of the Virtual raw Anchor Coordinate System Although, computation of greedy embeddings in a distributed (VRAC) is to use the (raw) measured distances as coordinates in environment is expensive in terms of the message complexity. order to avoid further computations. More precisely, each node needs to measure three distances. In this paper, we investigate Virtual Raw Anchor Coordinate System (VRAC) uses raw the existence of greedy routing in the VRAC coordinate system distances (Euclidean distance) from a set of anchor nodes as using a metric free characterization of greedy paths that is more the coordinate of a node [4], [5] (see Figure 1). In this paper, general than in previous works. We show that if the graph is we consider the problem of greedy routing on Virtual Raw saturated (see definition in the text) then the greedy algorithm Anchor Coordinate (VRAC) system. We propose a greedy guarantees delivery. Interestingly, the approach of greediness here applies to Schnyder drawings of planar triangulations. In- routing algorithm, which does not rely on an underlying deed, by choosing the measured distances appropriately Schnyder metric, rather based on a metric free definition of a greedy drawings of planar triangulations are always saturated and hence path. When compared to [4], [5], [6], [7], [8], we provide a our greedy routing algorithm succeeds. more general definition of greedy paths. This new approach The VRAC coordinates have conditions to satisfy to make of greediness can also be used as a building block of geo- greedy routing successful. These conditions can be inferred from geometric considerations. However, we formulate these conditions graphical routing algorithms of the type Greedy-Face-Greedy in an abstract way in order to avoid geometric considerations as we propose for VRAC in [8]. Alternatively, we develop and in order to make possible further derivation of virtual here on the new connexion of VRAC with the problem of VRAC coordinate systems, i.e. using only the abstract graph greedy embedding that makes possible the new approach of description. In particular using only local information would lead greediness. Actually, our greedy routing algorithm guarantees to distributed algorithm. delivery on all Schnyder drawings of a planar triangulation I.I NTRODUCTION (see the discussion at the end of Section IV) and this extents previous results of existential flavor1. Geometric routing is an appealing routing technique that uses geographic positions of nodes for routing data in commu- nication networks. Its most attractive form, greedy routing forwards messages along a distance decreasing path towards 1We refer to [12] for description of Schnyder drawings. What is relevant here is that given a planar triangulation and a set of weights of the triangular the destination. Greedy routing decision is completely based faces we obtain a Schnyder drawing (embedding) on the plane of the graph. on the local neighborhood information (positions of the neigh- Existential result says that there exists a set of weights such that the Schnyder boring nodes) and the position of the destination node, hence drawing is greedy. Application of the greedy routing algorithm shows that every set of weights leads to successful greedy routing. We emphasize that considered a local routing algorithm. Nevertheless, greedy in this second case there is no embedding of the graph. The result is useless routing does not always guarantee the delivery of messages, for but relevant to routing. II.B ACKGROUND In this article, we propose a greedy routing algorithm based on virtual raw anchor coordinate (VRAC) localization A greedy embedding is an embedding of a graph on a metric paradigm [4], [5], see Figure 1. In VRAC, nodes use the geometric space such that, greedy routing always succeeds. distance from three distinguished nodes as the coordinate, see In other words, between every node pair u, v there is another Figure 1, i.e. the usual scheme for localizing the nodes is 1) node w adjacent to u, such that d(u, v) > d(w, v), where d(.) measure distances and, 2) compute the coordinates. In this is the underlying metric on the geometric space. Papadimitrou work our goal is to avoid using geometry. Our algorithm uses & Ratajczak [2] were the first to study the existence of three order conditions <∗ to perform routing. Note that the greedy embeddings. In particular, they constructively proved i order relations come in an abstract way, although they can be the existence of a greedy embedding of a 3-connected graph derived from geometric properties as well. in 3. Moreover, it is conjectured that any 3-connected planar R Our contribution In this paper we; graph admits a greedy embedding in R2. The conjecture was proved affirmatively for different classes • Provide a metric-free definition of greedy paths. of graphs. In [9] it is proved for 3-connected graphs, in [10] • Design a greedy routing algorithm that guarantees deliv- for Delaunay triangulations, in [11] for graphs that satisfy ery if the communication graph is saturated, see definition conditions with respect to the power diagram. In [3], a greedy 5. embedding in the hyperbolic plane of a connected finite graphs • Provide a greedy routing algorithm for all schnyder is constructed. More related to our approach, in [12] the drawings of a planar triangulations. conjecture is proven for planar triangulations (maximal planar The rationale that motivate this work is to extend the ca- graphs), see also [13], [14], [15]. Another direction of greedy pabilities of the VRAC coordinate system by proposing a embedding research considers the efficient representation of definition of greediness as general as possible. We hope that coordinates. In [16] authors proposed a greedy embedding on further investigations will lead to the design of distributed a hyperbolic space with Olog(n) bit complexity. Such a algorithm that compute the VRAC coordinates using only coordinate representation is called succinct, which is important the abstract form of the communication graph, i.e. the graph for the design of scalable routing schemes. In subsequent G = (V,E). In a sense, by avoiding the computation of the research, succinct greedy routing schemes are proposed in greedy embedding, our approach decouples the problems of [17], [14]. greedy and the design of successful greedy Common approach in above proposals is to compute an algorithms. embedding given a graph and to use the underlying metric The algorithm is numerically validated with a simple Java of the respective space to perform greedy routing. However, simulator. Results are not reported here since the simulations our approach is to avoid the definition of a metric and the bring nothing more than confirmation that the algorithm is computation of the planar embedding, see for instance [13] successful. The simulator can be obtained under request to for an algorithm to compute the greedy embedding of planar the authors. triangulations. We rely on the metric-free definition of greedy In sections IV and III, we present Schnyder’s characterization paths in [18] - without embedding the graph. Moreover, the of planar maximal graphs. In section V we prove the properties computation of coordinate systems used in [12], [14], [15] that we need to build a greedy path between any two nodes. requires global topology knowledge, as opposed to the local Finally, in section VII we state the main result of the paper approach in our work. In [19], a local routing algorithm is about the existence of greedy paths. proposed for half − θ6 graphs, which is a special case of a Delaunay triangulation. It utilizes the geometric properties of III.V IRTUAL RAW ANCHOR COORDINATES (VRAC) a half −θ6 graph and proves a strict upper bound on the path stretch. This approach does not compute a greedy embedding, Virtual Raw Anchor Coordinate System (VRAC), is a cost yet based on the underlying geometry on the plane. effective localization scheme, which uses raw distances (Eu- In Schnyder’s work [20], planar graphs are characterized clidean distance) from a set of anchor nodes as the coordinate by the existence of three total order relations on the vertex of a node [4], [5]. The motivation for using VRAC is to set of the graph (and extra conditions). Using these order avoid the expensive computations required to compute the relations Schnyder builds three spanning2 directed trees called geographic coordinates. Indeed, most localization schemes the realizer and the coordinates are computed using these measure the distances from a set of anchors and embed the trees. This coordinate system has relevant properties, for graph communication graph in a metric space. In VRAC coordinate drawing, see for instance [21], that we do not need for system such an embedding is avoided and raw distances are routing3. In [12], [14], [15], Schnyder’s characterization of maintained as coordinates. (maximal) planar graphs [20] is used. This characterization is Despite being computationally efficient, VRAC does not discussed in section III and is also used in this work. have any underlying geometric properties to perform geomet- ric routing. Although, we can define three order relations on 2Actually, spanning internal nodes of the triangulation the set of nodes V . 3Although it is relevant to ask if these properties are necessary for constructing a greedy embedding Definition 1. The three order relations

A spond to sectors s1, s3 and s5 respectively. Based on these 1 d(u; A ) 1 u X partial orders, a node can distinguish its neighboring nodes. S 2 1 x1

u u u S s S 6 1 d(u; A ) 2 su su 3 d(u; A ) 2 6 2 u Su Su 3 5 Su 4 su su 5 3 CHNYDER HARACTERIZATION AND ATURATED su A X A IV.S C S A 4 A 2 3 3 2 3 COMMUNICATION GRAPH

Fig. 1: Two realistic ways of constructing the VRAC. Coordinate assignment with raw distances from anchors We consider an ad-hoc network with VRAC localization on the left and perpendicular distances from heights of infrastructure. Let G = (V,E) represent the communication the triangles on the right. The coordinates of u are graph, where V,E are the vertex and edge sets respectively. (d(u, A1), d(u, A2), d(u, A3)) on the left and (x1, x2, x3) on We assume that three anchor nodes Ai, i = 1, 2, 3 are placed the right. such that the network lies entirely in the interior of the triangle A1A2A3. This ensures that the property a) of (1) is satisfied. Each node u is able to measure the distances V × V are defined by d(u, Ai), i = 1, 2, 3, and uses (d(u(A1), d(u, A2), d(u, A3)) as coordinate and the order relation < , i = 1, 2, 3 are naturally ∀u, v ∈ V u < v ⇐⇒ d(u, A ) > d(v, A ) ⇐⇒ u > v . i i i i 1 i defined using these coordinates. It is important to note that these order relations are to- We concentrate on the special class of graphs, which satis- tal4(see [5] for details). Due to the totality of orders, a fies certain connectivity conditions. These conditions are due minimum element with respect to each total order can be to the schnyder characterization of planar graphs based on defined, which is denoted as mini for i = 1, 2, 3. Furthermore, order relations. Given a G = (V,E), it is proven based on the total orders, we define sectors associated with a in [20] that there exist three total order relations on V × V , node u as follows. denoted <1, <2, <3 such that Definition 2. (Sectors) We define the following sectors asso- T ciated to a node u ∈ V , see Figure 1. Note that the reference a) i=1,2,3

u s.t. (x, z) ∈2 v, u >3 v}. u s2 = {v | u <1 v, u <2 v, u >3 v}. u This is called a (3-dimensional) representation of a planar s3 = {v | u >1 v, u <2 v, u >3 v}. graph. Such representation of a planar graph does not use a su = {v | u > v, u < v, u < v}. 4 1 2 3 (planar) embedding and applies to an abstract graph. Using this u s5 = {v | u >1 v, u >2 v, u <3 v}. representation we get a characterization of a greedy path that u s6 = {v | u <1 v, u >2 v, u <3 v}. applies to abstract graphs as well and ignore the embedding. We also use the notation x

Property 2. A node u has at most one edge (u, v) ∈ E such u 4 that v ∈ s2i−1. Moreover, such a node v satisfies that v

A of our algorithm. These combinatorial properties are derived 1 v from geometric properties of a saturated graph, yet the greedy path construction is also valid if we assume that the order

u relation 1 v. (2) z <3 u the computation of Schnyder drawing coordinates, where as  u z <1 u our algorithm is not restricted to this setting. If v ∈ s3 , z 6= v we have ⇒ z >2 v. (3) z <3 u  V.G REEDY ROUTING u z <1 u If v ∈ s5 , z 6= v we have ⇒ z >3 v. (4) z <2 u A. Overview of Routing Strategy Property 2 is from [20] and the proof uses part b) of the graph If the (planarized) graph is saturated then each internal node u u u representation (1). There is a nice geometric void condition u, has exactly one edge in each sector s1 , s3 , s5 and an associated to this property (see Figure 2). Indeed, if we assume indeterminate (0, 1, 2,...) number in the remaining sectors u u u u that the edge (u, v) belongs to s1 then the existence of a s2 , s4 , s6 . It is helpful to look at the geometric visualization node u <1 w <1 v violates the second condition of (1), see in the right of Figure 2. u u Figure 2 and Figure 3c of [12]. It is interesting to compare this For routing from a node u to a destination D ∈ s1 (or s3 or u void region with the corresponding ones of the planar Relative s5 ) the natural option is to follow the existing edge (u, v) such u u u u Neighborhood Graphs (RNG) or Gabriel Graphs (GG) [22]. that v ∈ sD (= s1 or s3 or s5 respectively). Next, from v, if v v v In order to construct a planarized subgraph of a given D ∈ s1 (or s3 or s5) we repeat the same strategy. However, v v v graph, each node u decides to keep or remove the edges it may happen that D 6∈ s1 (or 6∈ s3 or 6∈ s5). In this case v u v v v u shared with neighboring nodes in the following way. For D ∈ s2 ∪ s6 (or s2 ∪ s4 or s4 ∪ s6 , see Proposition 3) and the v a neighboring node v, i.e. (u, v) ∈ E, u keeps the edge existence of an edge in the sector sD is not provided by the ∗ if ∃i ∈ {1, 2, 3} such that (u, v) i u . (5) For a greedy path there is a coordinate that changes monotonically. In the following, we build greedy paths from u to D such k u u (a) Destination D ∈ s2i and (D ∈(b) Destination D ∈ s2i and that u 2 u >2 w, and D >3 u >3 v (using (6, 7, 8)) v w these nodes do not cause any trouble because there is a path and D 6∈ s4 and D 6∈ s4 imply from any internal nodes to them with increasing coordinate D > v D < v D > v or  < respectively [20]. Hence, in order to make our best to D 6∈ sv ⇒ 1 2 3 ⇒ D < v i 4 D < v D < v D > v 2 simplify the exposition we no longer make any reference to 1 2 3 (9) these particular nodes in the proofs.  0 w D >1 w D >2 w D <3 w or Proposition 1. If D0 ∈ sD and D00 ∈ sD then D00 ∈ sD D 6∈ s4 ⇒ ⇒ D <3 w i i i D <1 w D >2 w D <3 w Proof. This property follows directly from the transitivity of (10) the inequalities in the definition of the sectors (2). u D Next, because D ∈ s4 ⇒ u ∈ s1 and the assumption of D Proposition 2. We assume that the graph G is saturated. Then saturation, there exists an edge (D,D0) with D0 ∈ s1 . If u u u provided that the destination D belongs to s2 (or s4 , or s6 ) D0 = u we are done. i 0 i+1 D then there is a path {u } in G with u = u such that u ∈ Otherwise, by the property (2) and u ∈ s1 we have that ui i+1 ui i+1 ui s2 (u ∈ s4 or u ∈ s6 respectively), and the path D0 <1 u . converges to D. D By gathering the inequalities corresponding to u ∈ s1 with Along the path, the order <3 (<1, <2) decreases monoton- the ones deduced from (9),(10) we obtain D <1 D0, v >2 u u u ically if D ∈ s2 (D ∈ s4 , D ∈ s6 respectively). D >2 D0, w >3 D >3 D0. Using D0 <1 u, D0 <2 v with u property (3) we obtain D0 >3 u . Proof. For concreteness we consider D ∈ s4 . If u is connected to D we define u1 = D and the proposition is true. Otherwise, Last from D0 <1 u, D0 <3 w and property (4) (with we prove below that there exists a neighboring node of u, u1 edge (u, w) instead of (u, v)) we obtain D0 >2 u . Finally, u1 1 we have proved that D0 ∈ su with the boxes equations and such that D ∈ s4 and D <1 u <1 u. Hence, by applying 4 the construction iteratively we construct the sequence of points D <1 D0 <1 u. The node D0 plays the same role as D in i+1 ui the statement of the proposition but with an increasing < that satisfy u ∈ s4 , lower bounded by D and decreases 1 i+1 i 0 with respect to <1, i.e. D <1 u <1 u . Such a sequence order position. Because of the bound D <1 u we see that converges to D. by applying iteratively the construction we obtain a sequence u D0,D00,... that converges to u and such that all the points Let us prove that given u such that D ∈ s4 there exists x belong to su. Moreover, along the sequence we have D0 ∈ sD, x such that (u, x) ∈ E, D ∈ s4 and D <1 x <1 u. u 4 1 D0 is internal, by the assumption on saturation, there exists two D00 ∈ s1 ,... and Proposition 1 implies that all the points in u u the sequence belong to sD. In particular, for the point x that neighboring nodes of u such that v ∈ s3 and w ∈ s5 . we then 1 D x have is connected to u x ∈ s1 ⇔ D ∈ s4 . We have then proved u x the existence of a point x ∈ s4 that satisfies D ∈ s4 and such D < u, D > u, D > u ⇔ D ∈ su (6) 1 2 3 4 that D <1 x <1 u. u v <1 u, v >2 u, v <3 u ⇔ v ∈ s (7) 3 u u Remark 1. Construction of the greedy path if D ∈ s2i w <1 u, w <2 u, w >3 u ⇔ w ∈ s5 (8) u In order to route from u to D ∈ s2i the node u must first check v w u u If v (or w) is such that D ∈ s4 (or D ∈ s4 ) the next whether for v ∈ s2i+1 and w ∈ s2i−1 one of the condition 1 1 v w point on the path is u = v (or u = w) and (7) shows that D ∈ s2i or D ∈ s2i is satisfied and if yes sends the message 1 v v = u <1 u, and D ∈ s4 ⇒ v >1 D (or (8) shows that accordingly, see the left of Figure 3. Otherwise, the message is 1 w w = u <1 u , and D ∈ s4 ⇒ w >1 D). forwarded to (the existing by Proposition 2) neighboring node u x in x ∈ s2i such that D ∈ s2i, see the right of Figure 3. This • D belongs to an ’odd’ sector of u (Line 5), by the routing scheme converges because the coordinate i decreases saturation assumption, Definition 5, there exists a unique along the path and the path doesn’t step over D, as all the neighboring node of u, called v in the same sector of u points in the path are >i D. than D. v is the next hop. This corresponds to Part 1 of the construction of greedy paths in Remark 2. Proposition 3. Let us assume that (u, v) ∈ E and D, v ∈ su 1 • D belongs to an ’even’ sector of u (Line 7), in this (or su or su). Then, D 6∈ sv ∪ sv ∪ sv (or sv ∪ sv ∪ sv or 3 5 3 4 5 1 5 6 case saturation alone does not ensures the existence of sv ∪ sv ∪ sv). 1 2 3 a neighboring node of u in this sector. The algorithm u Proof. Let us consider v, D ∈1 the other cases are proved considers both neighboring nodes of v and w of u similarly by a permutation of the indices. We have that belong to the ’odd’ sector of u by saturation. If destination D belongs to the same sector of v or w than u v ∈ s1 ⇔ u <1 v u >2 v u >3 v D the node is the next hop (Line 8 and 11). If not (Line u D ∈ s1 ⇔ u <1 D u >2 D u >3 D 14) Proposition 2 proves the existence of a neighboring node of u in th same sector as D. This corresponds to (1) D Part b) of the Schnyder’s conditions implies that must Remark 1 and Part 2 of Remark 2. be larger than u and v for one order and we see on the two inequalities above that it can only be <1. The condition D ∈ VII.R OUTINGIN MAXIMAL PLANAR GRAPH v v v s3 ∪ s4 ∪ s5 implies that v >1 D and hence there is no i ∈ In [12], it is proven using Schnyder’s characterization of planar 1, 2, 3 such that u, v