Geographic Routing on Improved Coordinates

Transcription

1 Geographic Routing on mproved Coordinates Ulrik Brandes, Daniel Fleischer Department of Computer & nformation Science, University of Konstanz Abstract We consider routing methods for networks when geographic positions of nodes are available. nstead of using the original geographic coordinates, however, we precompute virtual coordinates using a barycentric embedding. Combined with simple geometric routing rules, this greatly reduces the lengths of routes and outperforms algorithms running on the original coordinates. We thoroughly investigate capabilities and incapabilities of these ideas. Along with experimental results we show guaranteed message delivery and give time bounds for the precomputation. Finally, a method with a less consuming precomputation is introduced. Our methods apply to networks where short routes are of great importance, and the one-time-precomputation is affordable. 1. ntroduction Routing in a communication network G = (V, E) denotes the task of sending a message from a source node s V to a target node t V. When no direct connection is available this means to forward the message on a path from s to t using intermediate nodes. While in wired networks the specific path is usually determined by routers, in wireless networks, each node has to decide how to forward the message. Specific for geographic routing is the existence of geographical positions of the nodes. This information can be used for the search of an s-t-path. Among the simplest routing algorithms are, e.g., Greedy Routing (see, e.g., [16]) and Compass Routing [8]. Greedy A preliminary version of parts of this paper appeared in Proc. 11th nt. Conf. nformation Visualisation (V 07), pp EEE Computer Society, address: Ulrik.Brandes@uni-konstanz.de, Daniel.Fleischer@uni-konstanz.de (Ulrik Brandes, Daniel Fleischer) Preprint submitted to Theoretical Computer Science November 3, 2008

2 Routing always forwards the message from a node v to its neighbor w closest to t. Since w must be strictly closer to t than v (by definition), the method can get stuck in a dead-lock. Compass Routing chooses the neighbor with least (absolute) deviation angle, but again message delivery to t is not guaranteed. The first geographic routing algorithm with guaranteed delivery was Face Routing (originally called Compass Routing [8]). Currently, the most efficient algorithm is GOAFR+[10], which we therefore use as a reference. Our methods employ a mixture of Greedy and Compass Routing running not on the original geographic coordinates, but instead on precomputed virtual coordinates. This greatly reduces the lengths of routes and guarantees message delivery. Our choice of virtual coordinates is motivated by a theorem of W. T. Tutte [17] on barycentric embeddings, also known as rubberband embeddings. The improvement of route lengths is mainly due to a decreased number of dead-locks because in planar barycentric embeddings, each face is convex. Hence, Greedy Routing, which heuristically delivers short routes, leads to t more often (dead-locks still remain possible for convex faces, but we will see how to fix that). n [14], Rao et al. also used barycentric embeddings, but not necessarily planar ones, where message delivery is not guaranteed. Recently, in [11], Leighton and Moitra proved the conjecture of Papadimitriou and Ratajczak (see [13]) that every planar, 3-connected graph can be embedded into the Euclidean plane such that Greedy Routing guarantees message delivery. Anyhow, this embedding is not straight-forward computable in a distributed manner, and the edge lengths range from at least 1 to extremely small values, which makes this embedding rather only of (important) theoretical interest. General surveys about geographic routing and virtual coordinates are given, e.g., in chapters [21] and [3]. Our routing methods (presented in full detail in Sect. 3) together with a brief summary of its properties are: BR (Barycentric Routing): very simple routing rules, good in practice for a certain density range of networks, guaranteed delivery, GBR (Greedy BR): very short routes for all densities, outperforming routing methods on geographic coordinates, guaranteed delivery, AGBR (Adaptive GBR): fixes worst-cases of GBR, guaranteed delivery, GBFR (GB Face Routing): less precomputation, guaranteed delivery.

3 2. Preliminaries We consider wireless networks modeled as unit disk graphs G = (V, E) whose nodes are embedded in the Euclidean plane R 2, and two nodes are adjacent iff their distance is not greater than 1. Note that most of our ideas easily carry over to the more realistic d-quasi unit disk graph model (or are even still applicable as long as we can determine any planar subgraph). We assume that G is connected and there are no two nodes at the very same position. The number of nodes is denoted by n. Geographic routing uses these positions to route a message from a source node s to a target node t under the assumptions that the coordinates of t, the coordinates of all neighbors of v and its own position are known to each node v. The path s, v 1, v 2,... along which a message is routed is called message path. A mapping p: V R 2 is called embedding. Let R 2 be equipped with open and closed sets induced by the Euclidean norm. Each embedding p of a graph G naturally corresponds to a closed set G R 2 (we identify G and G), where edges of G are drawn as straight lines between its incident nodes. We call an embedding p planar if its straight-line embedding G is planar. nstead of using original geographic coordinates, that will be denoted x v, ŷ v, we compute virtual coordinates x v, y v and z v during a precomputation phase. The Euclidean distance of two nodes v, w wrt original coordinates is d(v, w), wrt virtual coordinates d(v, w). (Distances wrt virtual coordinates neglect the z-coordinate.) The angle tvw, the (counter-clockwise oriented) angle at v from t to w, is called deviation angle and is defined to take values in ] π, π]. Note that all nodes w on the (open) right hand side of the (infinite) line vt have negative deviation angle. The angles ϕ t (s), ϕ t (v 1 ), ϕ t (v 2 ),... are called direction angles to the target node t, where s, v 1, v 2,... is a message path. The first direction angle ϕ t (s) is defined to be 0, and for each further node ϕ t (v) is increased or decreased according to the angle at t from the predecessor w on the path to v, i.e. ϕ t (v) = ϕ t (w) + wtv. For a (periodic) message path s, v 1, v 2, v 3, s, v 1,... clockwise around the corners of a square with t in its center, the corresponding direction angles would be 0, π/2, π, 3π/2, 2π, 5π/2,..., see Fig. 1. Thus, actually, ϕ t (v) also depends on the position of v in the message path (but we neglect that) since ϕ t (v) is only unique up to adding multiples of 2π. During the precomputation phase the restricted Gabriel Graph G GG (see [5]), i.e., the intersection of G with its Gabriel Graph, will be used. This graph is planar (embedded), and it is connected if the original unit disk graph is connected. Each node

4 v 1 v 2 β 1 β 2 t s α v 3 Figure 1: Message path s, v 1, v 2, v 3, s, v 1,... with deviation angle α = tsv 1 = π/4 and direction angles β 1 = ϕ t (v 1 ) = π/2, β 2 = ϕ t (v 2 ) = π v G GG maintains an ordered list (counter-clockwise) of its neighbors. To pass a message by the right hand rule denotes to pass the message to the next neighbor in this list after the sender if a sender exists. Otherwise, we will give some direction: to pass a message by the right hand rule directed to vw denotes to take the next neighbor after vw. Sometimes, virtual edges are added to G GG, which possibly do not exist in G and have to be realized as paths. However, by construction, such virtual neighbors can easily be reached by sending the message by the right or left hand rule, which has to be specified for each virtual edge. When adding a virtual edge, the incident nodes simply update their lists. Since virtual edges are always inserted within a face of G GG, planarity is never destroyed (although the straight-line embedding of G GG may temporarily be not planar). For ease of simplicity we call a subdivision (replacing edge by edge-node-edge) of a 3-connected graph also 3-connected. Moreover, we even allow cut pairs on the nodes of the outer face. The following theorem of Tutte also holds for this class of graphs. 3. Barycentric Routing n this section we introduce our routing methods BR, GBR, AGBR and GBFR running on virtual coordinates that are computed in a precomputation phase. The theorem of W. T. Tutte can be formulated as follows. Theorem 1 (Tutte, [17]). Fixing nodes of a face of a planar embedded, 3- connected graph onto the corners of a strictly convex polygon C, and setting

5 the positions of the remaining nodes, such that each node is in the barycenter of its neighbors, yields a planar embedding. We call the embedding obtained from this theorem barycentric embedding p C. After fixing the nodes of C it is unique (G is connected) and can be computed using the Laplacian matrix L of the given graph G = (V, E) defined as follows. deg(v) if v = w, L vw = 1 if {v, w} E, 0 otherwise, where deg(v) is the number of neighbors of v. Embedding p C = (x v, y v ) v V is then given by the unique solutions of Lx = 0 and Ly = 0, where the positions of the nodes of C are fixed and its corresponding lines in the equation systems deleted, see, e.g., [6]. These equations can iteratively be solved by the following Jacobi iteration for all nodes v V \ C, see also [7], x v w N(v) x w deg(v) and y v w N(v) y w deg(v). (1) Note that Jacobi iteration (1) only needs communication between adjacent nodes. An equivalent way to define p C is the following, see, e.g., [6], { } p C = arg min d(v, w) 2 : nodes of C are fixed. (2) p v,w V Note that this formulation directly implies that all nodes v V are within the closed (bounded) set delimited by the strictly convex polygon C Precomputation Phase The precomputation phase for the following routing rules is divided into four steps that once have to be executed to obtain the desired virtual coordinates (x v, y v, z v ) v V. After this precomputation phase messages can be routed easily and on very short paths. Note that the steps 1, 3, 4 can be skipped if G is a planar, 3-connected graph. Step 1 computes the (planar) restricted Gabriel Graph G GG and determines tree children, i.e., nodes that would be deleted if successively removing degree-one-nodes from G GG.

6 Step 2 determines the nodes of the outer face of G GG (i.e., the perimeter nodes) and sets them equidistantly on a circle enclosing G GG to form the strictly convex polygon C. Step 3 establishes 2-connectedness of G GG by adding virtual edges that do not destroy planarity of G GG. Step 4 establishes 3-connectedness (in the sense given in Sect. 2) of G GG, such that the theorem of Tutte, Theorem 1, can be applied. Algorithm 1: BR step 1 (determine tree nodes) compute restricted Gabriel Graph G GG and denote all nodes non tree-nodes foreach v having exactly one non tree-node neighbor do denote v a tree-node repeat the last step (at most) n rounds foreach non tree-node v having at least one tree-node neighbor do denote v a tree-root foreach non tree-node v do z v 0 foreach tree-root v having children v 1,..., v k do send z = i, z = i + 1 to v i foreach tree-node v receiving z, z, having children v 1,..., v k do set z v z and send z + (z z) i/(k + 1), z + (z z)(i + 1)/(k + 1) to v i repeat the last step (at most) n rounds Starting from degree-one-nodes Algorithm 1 detects all tree-nodes in G GG and assigns values z v that induce a prefix ordering of each tree. This allows easy addressing by Routing Rule 1. The following algorithm steps run on non tree-nodes and tree-roots. After these have obtained their final x, y coordinates, all tree-nodes inherit the x, y values of their tree-root. Algorithm 1 is mainly used to decrease the number of virtual edges inserted in the next three steps. Algorithm 2 has to be extended as follows if the outer face may contain cut nodes. f a message ( x v, ŷ v, c) initiated from node v with coordinates x v, ŷ v reaches a node w more than once (which can only happen if w is a cut node), w does not set its c w value, but inserts a virtual edge between its

7 Algorithm 2: BR step 2 (set perimeter nodes) let M denote all nodes v with minimum ŷ v among their neighbors foreach v M do pass message ( x v, ŷ v, c) containing the coordinates x v, ŷ v and a counter c = 1 by the right hand rule directed (0, 1) foreach v V do if v received message ( x, ŷ, c) and ŷ ŷ v (and x > x v if ŷ = ŷ v ) then set cv c pass message ( x, ŷ, c + 1) by the right hand rule if v received message ( x, ŷ, c) and x = x v, ŷ = ŷ v then // v is the rightmost node on the outer face with minimum ŷ v pass message (r x, r y, r, c) by the right hand rule directed (0, 1), where (r x, r y ) denotes the center of a circle of radius r that encloses all nodes on the outer face (minimum/maximum x, ŷ can be collected on the walk around the outer face to get a reasonable value for r) set x v r x, y v r y r if v received message (r x, r y, r, c) (and x v, y v are not yet set) then pass message (r x, r y, r, c) by the right hand rule set x v r x + r sin(2πc v /c) set y v r y r cos(2πc v /c) repeat the last step at most 2n rounds predecessor and its successor on the outer face and then passes the message ( x v, ŷ v, c) by the right hand rule (neglecting virtual edges). Note that the inserted virtual edges may temporarily destroy planarity of the embedding, but they do not affect planarity of the barycentric embedding afterwards. Parameters ε, κ 1, and ϑ, κ 2 for the next step, together with an appropriate stop criterion have to be adapted depending on the specific scenario. Choosing ε too small and the stop criterion too weak may result in a non-detection of a cut node. Choosing ε rather too great (we used 10 2 while r 100 in our tests) is the better choice. Unnecessarily inserted edges do not affect the correctness of the algorithm. Algorithm 3 terminates (i.e., it stops inserting virtual edges) because of the Euler Formula for planar graphs. n fact, when choosing ε sufficiently small, virtual edges are inserted unnecessarily (i.e.,

8 Algorithm 3: BR step 3 (establish 2-connectedness) repeat foreach v V do if v is not a perimeter node then apply Jacobi iteration (1) let U = {u V : d(v, u) < ε} and W = {w V : d(v, w) ε} if v both had neighbors u U, w W for more than κ 1 rounds then insert virtual edge between succeeding neighbors u U, w W if this edge not already exists until stop criterion is fulfilled if G is already 2-connected) only if the edge is part of a component G 1 that is separated from G by a cut pair. Lemma 2. Let G be a connected planar graph, where the nodes of a face have been fixed onto a strictly convex polygon C. G is 2-connected if Algorithm 3 (ε > 0, κ 1 > 0, no stop criterion) does not insert a virtual edge. Proof. f G is not 2-connected, there exists a cut node v separating G into connected components G 1,..., G k, k 2. W.l.o.g. G 1 does not contain a node of C, and v has a neighbor in G 2 with a position different from the position of v. Minimizing (2) sets all nodes of G 1 to the position of v. An edge between neighbors of v in G 1 and G 2 does not exist since v is a cut node. Hence, a virtual edge is inserted. Lemma 3. Let G be a 2-connected planar graph, where the nodes of a face have been fixed onto a strictly convex polygon C. f Algorithm 4 (ϑ > 0, κ 2 > 0, no stop criterion) does not insert a virtual edge, G is 3-connected (in the sense of Sect. 2). Proof. Assume that v, w is a cut pair separating G into non-trivial (i.e., none of them is a path) connected components G 1,..., G k, k 2, where G 1 does not contain a node of C. W.l.o.g. v has at least two neighbors in G 1, and G 1 is maximum with respect to inclusion (among all G 1 with the previous properties). An embedding, where G 1 is not embedded on a line between v and w, cannot minimize (2). Hence, there are at least two neighbors w 1, w 2 G 1 of v within a sector S of an angle tending to zero.

9 Algorithm 4: BR step 3 (establish 3-connectedness) repeat foreach v V do if v is not a perimeter node then apply Jacobi iteration (1) if v had neighbors that are within a sector S of angle ϑ for more than κ 2 rounds then insert virtual edge between the left-most (right-most) u S and w, i.e., the first node that starting from the neighbor w / S succeeding (preceding) u by the right hand rule (left hand rule) is outside the sector S rotated by π if such a node w exists until stop criterion is fulfilled Because G 1 is maximum, there exists a path from v by the right hand rule (or left hand rule) outside sector S to a node w / G 1, such that the angle at v to w 1 and w differs from π, see Fig. 2. Hence, a virtual edge will be inserted. t remains the case, when all connected components (in this case only two are possible since then v, w C) G 1, G 2 contain nodes of C. Let v, w be such a cut pair that minimizes G 1 with respect to inclusion. Then C G 1 {v, w} (plus an additional edge or a path from v to w that may also already exist in G 1 ) is convex, such that its barycentric embedding is planar and it does not affect the remaining embedding, see Fig. 3. Algorithm 4 terminates because each inserted virtual edge decreases the number of cut pairs (only cut pairs v, w C as mentioned in the proof may remain). Note that, as an alternative, Algorithms 3 and 4 might be replaced by simply triangulating faces. But this may produce lots of unnecessary virtual edges, in particular, if obstacles (i.e., areas without nodes or without edges) are present. The use of virtual edges can cause a delay of position information during the communication of Jacobi iteration (1) if the inserted edges do not exist in the original graph G, i.e., a node v that has a virtual neighbor w at a distance k only receives its position after k rounds instead of after one round. Lemma 4. Jacobi iteration (1) with delayed positions tends to p C.

10 w w v v w* w* w w v v w* w* Figure 2: From left to right, top-down: Original graph G, where the dotted (virtual) edge is inserted first and the dashed edge follows (nodes v, w and w correspond to the first edge). Barycentric embedding of G. nsertion of first edge. nsertion of second edge v v w w Figure 3: A cut pair v, w on C separating a minimum component G 1 whose barycentric embedding (right hand side) does not affect the embedding of the remaining graph

11 Proof. Jacobi iteration (1) can be rewritten as x D 1 A x + D 1 Ãx C, (3) where D is the diagonal matrix that contains the degrees of all inner nodes v / C, A is the adjacency matrix of the subgraph induced by the inner nodes and à is the matrix obtained from the adjacency matrix of G by omitting columns corresponding to inner nodes and omitting rows corresponding to perimeter nodes C. The vectors x, x C denote the x-positions of inner and perimeter nodes. teration (3) converges because the matrix D A is essentially diagonally dominant, see, e.g., [7]. The same holds for the matrix D A defined as follows. Let k be the maximum delay and let A i denote the adjacency matrix of inner nodes that have a distance of i to each other, such that A = k i=1 A i. Let Ãi be defined analogously, and D be the diagonal matrix that contains the degrees of all inner nodes on the first entries and then is filled with ones. Furthermore, let denote an identity matrix and 0 zero matrices of appropriate sizes. Delayed iteration (3) can be written as ( ) ) x (D ) 1 A1 A k 1 A k (Ã1 x + (D ) 1 à k x C, }{{} =: A where the first entries of x are used for the positions of the inner nodes and the limit on these is the same as in iteration (3) BR Rule We now present our routing rules. These assume that virtual coordinates, i.e., a barycentric embedding p C, have been precomputed. The following very simple rule for BR (Barycentric Routing) is sufficient for guaranteed message delivery and already delivers good results in practice for graphs of certain densities (see Fig. 12). For all rules, v always denotes the current node that has to pass the message to one of its neighbors. Routing Rule 1. f d(v, t) = 0, pass message to w with maximum z w z t. f d(v, t) > 0 and z v > 0, pass message to the neighbor w with minimum z w. f d(v, t) > 0 and z v = 0, pass message to w with minimum angle tvw 0.

12 3.3. GBR Rule For dense graphs BR suffers from its restriction to the Gabriel Graph. GBR (Greedy BR) prevents this drawback. Whenever possible, GBR advances in greedy mode to t using the original graph G instead of G GG. This leads to an algorithm with overall good performance on the whole density spectrum, outperforming routing algorithms on geographic coordinates (see Fig. 12). Routing Rule 2. Pass message to a G-neighbor w with minimum distance d(w, t) < d(v, t) if such a neighbor exists. Otherwise, pass the message according to Rule 1 until reaching a node w with d(w, t) < d(v, t) and then again apply Rule AGBR Rules AGBR (Adapted GBR) needs an additional step 0 during the precomputation phase to prove Theorem 11 that is analogous to a theorem concerning the message path lengths of GOAFR+, see [10]. Hence, AGBR fixes the unbounded message paths of GBR in the worst-case (see Theorem 10), while still being good in practice. Step 0 computes a backbone graph G BG (see, e.g., [19]), i.e., a subgraph of the original unit disk graph G forming a dominating set of G. The remaining steps afterwards run on the backbone graph, and routing between backbone graph G BG and G is straight-forward. nitially, set ρ = 2. Routing Rule 3. Pass message according to Rule 2 if it will thereby be passed to a node w with d(w, t) ρ d(s, t). Otherwise, walk by the right hand rule along the outer face of the graph G ρ formed by all nodes u with d(u, t) ρ d(s, t) until returning to v. f t was not surrounded by this walk, set ρ 2ρ and again apply Rule 3. Otherwise, set ϕ ϕ t (v), v v and apply Rule 4. Routing Rule 4. Pass message according to Rule 1 if it will thereby be passed to a node w with d(w, t) ρ d(s, t). Otherwise, walk by the right hand rule around the outer face of G ρ until a node w with ϕ t (w) ϕ, w v is reached. Set ϕ ϕ t (w), v w and apply Rule 4.

13 3.5. GBFR Rules Finally, GBFR (GB Face Routing) uses Rule 1 and Rule 2, but with a less consuming precomputation phase. Jacobi iteration (1) may be applied just a few times. Thus, the exact coordinates of p C are not obtained and message delivery may fail because of two reasons: there is no node w with tvw 0 in Rule 1, or, a node is visited more than once on a message path, see Fig. 5. n both cases (note that the latter does not require that a node remembers that it already received the message before) GBFR then applies Face Routing [8] on the original graph G. Since convergence of Jacobi iteration (1) is geometric, we might already be very close to p C after a few iterations. The good performance of GBFR is due to this fact, see Fig. 10. Note that we used the following additional heuristic: degree-2-nodes pass the message to the other neighbor. 4. Theoretical Results n this section we consider provable properties of the routing methods just introduced. Experimental results follow in the next section. Routing Rule 1 directly implies the following lemma for tree nodes. Lemma 5. Routing Rule 1 guarantees message delivery if s and t are in the same tree. Otherwise, the message will be passed from s to its tree-root if s is a tree node. f t is a tree node, the message is passed from its root to t. Theorem 6. BR, GBR, AGBR and GBFR guarantee message delivery. Theorem 6 implies that the application of Rule 1 guarantees message delivery on 3-connected graphs, which is also proved in [12]. Anyhow, we present a simpler proof relying only on (2). Proof. Consider Fig. 4. The message has to be passed from source node s to target node t. Assume that both nodes are not tree-nodes, otherwise Lemma 5 could first be applied. The (infinite) line st divides the plane into two half-planes. Because either s is in the barycenter of its neighbors or s is a node on the convex polygon C, there exists a neighbor v of s in the closed right hand half-plane with minimum deviation angle. f the direction angle from s to v does not increase, the distance to t decreases. Hence, if the message is not delivered to t the message path finally leads to a node w with a direction angle of ϕ 2π. To prove Theorem 6 it is sufficient to show

14 that w has to be within the star-shaped (open) domain D, delimited by the message path and the (dashed) line ts. Assume the contrary, as depicted in Fig. 4. Let G D be the subgraph induced by all nodes within D. All the edges connecting a node from the message path to a node in G D have negative deviation angle. All the other edges connecting G D to G GG \G D have to pass the dashed line between s and the upper intersection point with the edge to w. Since D is star-shaped with t as center, there exists a sufficiently small angle α > 0, such that the rotation of G D by α reduces the lengths of all edges leaving G D, in contradiction to (2). This proves message delivery for BR. Since the greedy mode of GBR (GBFR) will only be restarted after BR mode reaches a node strictly closer to t, GBR (GBFR) also guarantees delivery. The case when AGBR reaches a node v such that Rule 2 would lead to a node w with d(w, t) > ρ d(s, t) remains to be shown. First note that the Theorem of Whitney [20] implies the existence of a homeomorphism g : R 2 R 2 that sends the original graph G GG (with non-straight-line virtual edges that do not cross) to its barycentric embedding. Hence, v is a node on the outer face f ρ of the graph G ρ formed by all nodes u with d(u, t) ρ d(s, t). f t was not surrounded by the walk along f ρ then t is not reachable within G ρ and ρ has to be increased. f t was surrounded then t is reachable. There is a BR-path from v to t that has to intersect f ρ. f not reaching t earlier, Rule 4 leads to the BR-path from the last intersection point of the BR-path to t. w v s D t Figure 4: BR and GBR guarantee message delivery. The message cannot be passed to a node w such that the message path crosses the dashed line further distant from t than s

15 Figure 5: Example, where routing suffers from finite precision arithmetic Note that Theorem 6 actually requires the exact coordinates of the barycentric embedding p C that may not be representable with finite precision arithmetic. Consider Fig. 5, where a message from the uppermost node has to be sent to the center node and assume that perimeter nodes w 1, w 2 at 3 and 9 o clock have virtual coordinates (30,1) and (-30,-1), respectively. The exact y-coordinates of the inner nodes adjacent to w 1, w 2 are 2/3, 2/3. Thus, for finite precision arithmetic the deviation angles at w 1 and w 2 remain negative after each application of Jacobi iteration (1), even in the (finite precision) limit. Rounding the exact coordinates (that may have been obtained somehow) to finite precision yields the same negative result. A heuristical workaround that works well in practice is to modify Rule 1 from tvw 0 to tvw ε for some small ε > 0. Another workaround is a rational representation of coordinates as follows. Theorem 7. Let the virtual coordinates of nodes in C be small integer numbers, representable by q 2 log C bits ( x v 2 q for all v C). Then the coordinates of p C are rational numbers with at most η = 2.416n bits for denominators and η+q bits for numerators. Furthermore, using finite precision arithmetic (2(η + q) bits) the rational coordinates of p C can be reconstructed after finitely many steps of Jacobi iteration (1). Proof. Let x and x C again denote x-positions of inner and perimeter nodes, respectively (we will use this notation also for other variables, e.g., V, V C ). Then x = L 1 ( L)x C, where L is the submatrix of L induced by inner nodes, and L is implicitly defined by Lx = 0. Hence, denominators (positive integers) of x are bounded by det(l ) # spanning trees of G GG (16/3) n, where the first inequality is Kirchhoff s matrix tree theorem together

16 with the fact, that merging the nodes of (the circle) C does not increase the number of spanning trees, and the last inequality is from [15]. Note that this yields a bound very much better than the Hadamard bound n n/2 n, where is the maximum degree of G GG. The size of the numerators then follows from x v 2 q for all v V. Since we use 2(η + q) bits, Jacobi iteration (1) will approximate the exact coordinates of each node up to a deviation smaller than 2 2η 1, which is sufficient to reconstruct the finite continued fraction of the exact rational coordinates simply by the Euclidean algorithm applied to the approximate coordinates, see, e.g., [18], for a similar application. A strictly convex polygon C with integer coordinates of bitlength at most 2 log C may be obtained for nodes i = 0,..., c := C /8 by coordinates ( i j=1 (c j + 2), (c + 5)c/2 i) and defining the remaining positions symmetrically. Actually, a strictly convex polygon can also be defined within a square of width in O( C 3/2 ), see [1], instead of O( C 2 ). Note that arithmetic with 2(η + q) Θ(n) bits maybe replaced by arithmetic with a fixed number of bits and stepwise storing highest bits that already remain fixed by Jacobi iteration (1), see also, e.g., [18]. Also note that reconstruction of the exact coordinates should be attempted repeatedly from time to time. A reconstruction may be possible very early and it can easily be checked, whether the exact coordinates are found since only these set each node to the barycenter of its neighbors. Bounds when the reconstruction is possible for sure can be obtained as follows. Let f denote the mapping on inner nodes implicitly defined by Jacobi iteration (1). Furthermore, let x (and ŷ ) denote some initial positions of the inner nodes within C, e.g., (scaled) original positions, and let x (and y ) denote the exact rational coordinates. We first consider a special case. Let d = max v GGG dist(v, V C ), where dist(v, V C ) denotes the minimum hop-distance from v to a node in C. Let G GG be a quadratic grid with n = 4(d + 1) 2 nodes (d > 0), where the 4( n 1) perimeter nodes are set to the integer positions of the rectangle (0, 0), ( n 1, 0), ( n 1, n 1), (0, n 1) (thus C is not strictly convex, but, anyhow, p C is planar). The bound for denominators for the grid given by (16/3) n det(l ) 2 n 1 /n (see [9]) can be replaced by 1 since the exact coordinates are integers. Since the matrix D 1 A is 1/4 times the (symmetric) adjacency matrix of a grid with 4d 2 nodes, its spectral radius R is cos(π/(d + 1)), see, e.g., [2]. The worst-case (for 2-norm convergence) initial positions x of the inner nodes are given when x x is an eigenvector

17 corresponding to R, say, with -norm 1 ( x is within the perimeter nodes). Hence, f t ( x ) x = R t (4) (neglecting the fact that we actually may have a rounding error smaller than 2 2η q for each iteration), such that t = 1 2 log sec(π/(d + 1)) = ln 2 n + Θ(1) = n + Θ(1), (5) π2 where the constant in Θ(1) is negative, is the sharp bound, when the reconstruction is possible for sure in this case. Note that worse initial coordinates, where x x = c, we still have t n(1 + log c) ln 2/π 2 and did not lose very much. Also note that if we had the worst-case denominators as large as n (removal of a few inner edges of the grid increases denominators, while R remains nearly constant), we still have t n( n) ln 2/π 2 n 2 /3. The bound of O(n 2 ) may also be used for networks in practice. Sufficient conditions may be given as follows. Let deg j (v) denote the number of neighbors w of v with dist(w, V C ) = j. Theorem 8. Let d = max v GGG dist(v, V C ) O( n) and V C O( n). For each v V with dist(v, V C ) = j let deg j 1 (v) deg j+1 (v). Then the exact coordinates can be reconstructed after t O(n 2 ) iterations. Proof. Since V C is in O( n), any initial positions x that are within C fulfill x x c 1 n for a constant c 1. Thus ( π ( x x ) v c 1 nd sin 2 dist(v, V ) C). (6) d Hence by Jacobi iteration (1) and the sum formula for the sine ( ( π )) t f t ( x ) x c 1 nd cos, (7) 2d such that t = (4.831n+1+log c 1 nd) 0.562d 2 O(n 2 ) iterations are sufficient for reconstruction of the exact coordinates with worst-case denominators and worst-case initial positions. However, for the general case one can also construct instances, where convergence is worse. Consider the graph of Fig. 6, suitable for our application, which needs the worst-case number of iterations t Θ(n 3 ) because d Θ(n).

18 Figure 6: Example for slow convergence to the barycentric embedding Let G GG be a rectangular grid with dimension 7 (d + 2), where some nodes and edges are removed according to Fig. 6. Let the left lower and upper perimeter nodes have coordinates (0, 0) and (0, 6), respectively (again, p C is planar although C is not strictly convex). Furthermore, let x have integer coordinates according to Fig. 6. Then analogous to the last proof ( ( π )) t. f t ( x ) x cos (8) 2d The determinant of L is the number of spanning trees of a 3 (d + 1) grid (see also [9]) and the exact coordinates of the uppermost inner node is given for d = 1, 2 by 4/5, 19/24 and for d 3 by a d /b d, both sequences a d, b d with characteristic polynomial T 2 5T + 1. All a d, b d are coprime, such that the denominators grow exponentially with respect to d (and n). Hence, t Ω(n 3 ). The following theorem states that this is already the worst-case in O-notation. Theorem 9. Exact coordinates can be reconstructed after O(n 3 ) iterations. Proof. Let G GG be any graph as considered above. n general, D 1 A is not symmetric, so we define the matrix norm D = D 1/2 D 1/2 2 (where 2 is the matrix norm that corresponds to the vector 2-norm) that coincides with the spectral radius R on D 1 A. First, R(D 1 1 A ) = 1 R(L 1 D ) n, (9) 2 where the equation holds because the spectrum of D 1 A is in ]0, 1[ (L is essentially diagonally dominant) and the inequality uses a lower bound for

19 the smallest eigenvalue λ 1 of D 1 L given by 2λ1 U E min, (10) =U V U E see [4], where U E is the number of edges with nodes both in U and V \ U, and U E denotes the sum of degrees of nodes in U. Since G GG is planar and 2-connected, the minimum is greater than 2/(6n). Any initial positions x within C fulfill x x n 2 /4. Hence, f t ( x ) x n 3 D 1 A t D = n 3 R(D 1 A ) t. (11) This yields a bound of t = (4.831n log n) n 2 worst-case denominators and worst-case initial positions. O(n 3 ) for The last theorems together have shown that both memory and time consumption to compute exact coordinates may exceed memory and time needed for complete routing tables, which is obviously in O(n log n) and O(n 2 ) (e.g., n-times application of BFS), respectively. Hence, for practical purposes, clearly, GBFR is the better choice. Now we state results about the lengths of the message paths that are computed by our routing methods. Theorem 10. There exists a sequence of unit disk graphs with a designated target node t, such that the expected quotient of the length of the BR-path (and GBR-path) between a random source node s and t divided by the length of a shortest s-t-path is unbounded. Proof. Consider Fig. 7, where the left hand side displays a restricted Gabriel Graph G GG of a sequence of unit disk graphs. The number n 1 of nodes on the half circle can be varied. Let the lowermost node be the target node t. The right hand side shows G in barycentric embedding, which is trivial in this case because there are no inner nodes. For all source nodes s different from t and its neighbors the length of a shortest path is 2, whereas the BR-path (and GBR-path) has an expected length of about n/4. Theorem 11. The length of an AGBR-path is in O(l 2 ), where l is the length of a shortest s-t-path.

20 Figure 7: A unit disk graph and its barycentric embedding for Theorem 10 Proof. The proof is analogous to the corresponding for GOAFR+ [10] and is thus only sketched here. Backbone graph G BG is a dominating set for the nodes of G. The length of a shortest s-t-path using only intermediate nodes of G BG is only a constant factor longer than a shortest s-t-path in G. Furthermore, G BG has bounded degree, such that the maximum number of nodes in a circle of radius r is in O(r 2 ). This together with the fact that the number of nodes visited during application of Rule 4 in the circle of radius r = ρ d(s, t) l is a constant times the number of nodes in the circle concludes the proof. Note that analogous to the extension from GBR to AGBR we can also extend the rules of GBFR (together with GOAFR+ instead of Face Routing) to obtain an algorithm that has the same upper bound of O(l 2 ). Theorem 12. There exists a series of unit disk graphs with a designated target node t, such that the expected length of an AGBR-path between a random source node s and t is in Ω(l 2 ), where l is the length of a shortest s-t-path. Proof. n the following example, the distance of the nodes on the AGBRpath to t will never increase, such that we can restrict ourselves to GBR. The left hand side of Fig. 8 displays an example of a sequence of unit disk graphs that are constructed as follows. Choose the number of windings of the double spiral, here it is 3. Draw the inner windings like in Fig. 8. For each further quarter of a winding choose the number of nodes to be used for that quarter, such that the outmost nodes are still connected. Finally add the outer circle and the nodes forming the crosshair. The resulting unit disk graph is a planar, 3-connected graph, and it is equal to its restricted Gabriel Graph. Applying barycentric embedding, fixing the outer circle, yields the

21 right hand side of Fig. 8. Let t be the center node that will remain in the center of the circle by symmetry. Now all nodes on the crosshair except the two upper and lowermost edges have negative deviation angle wrt t. Hence, each BR message path, starting from a source s, will avoid these edges and instead be a full spiral path to t. Thus, the length of the message path is roughly proportional to the area of the circle around t, whereas the length of a shortest s-t-path is only proportional to the radius. The very same example also holds for GBR if s is not a node of the crosshair. Greedy mode will be either stopped on the circle or roughly at the center of a quarter of a winding. The following BR phase will again avoid edges of the crosshair and will only stop at some node of the next quarter of the winding. Figure 8: A unit disk graph and its barycentric embedding for Theorem Experimental Results The mean algorithm cost is the quotient of the length of the message path divided by the length of a shortest path. Fig. 12 shows the results of tests on random unit disk graphs with 1000 nodes. Nodes were placed on a circular area uniformly at random. For node densities from 1.0 to 20.0 nodes per unit disk in steps of 0.25 we created 100 networks, and sent messages between 1000 s-t-pairs. For disconnected networks we only considered the largest connected component, which explains the typical non-monotone behavior of the mean algorithm cost. Fig. 9 shows the improvement of the greedy success rate, i.e., the ratio of messages delivered only in greedy mode, due to convex faces in barycentric embedding that make dead-locks improbable. Furthermore, the numbers of tree nodes and inserted virtual edges are shown. Figs. 10 and 11 finally show mean algorithm costs and greedy success rates of GBFR with various maximum numbers of applications of Jacobi iteration (1).

22 60 Tree nodes nserted edges (2-con.) nserted edges (3-con+) Number of tree nodes/inserted edges Node Density [nodes per unit disk] Greedy Success Rate Node Density [nodes per unit disk] Greedy Barycentric Routing GOAFR+ Figure 9: Number of tree nodes and number of virtual edges that were inserted is displayed. Furthermore, success rates of greedy routing for GBR and GOAFR+

23 3 2.5 GBFR 2 iterations GBFR 5 iterations GBFR 10 iterations GBFR 20 iterations GBFR 50 iterations 2 Mean Algorithm Cost Node Density [nodes per unit disk] Greedy Success Rate Node Density [nodes per unit disk] GBFR 2 iterations GBFR 5 iterations GBFR 10 iterations GBFR 20 iterations GBFR 50 iterations Figure 10: Mean algorithm costs and greedy success rates of GBFR (and call of Face Routing if pure GBR failed) at maximum number of iterations 2,5,10,20,50,100

24 1 0.8 Delivery Ratio Node Density [nodes per unit disk] GBR 2 iterations GBR 5 iterations GBR 10 iterations GBR 20 iterations GBR 50 iterations Figure 11: Delivery ratio of pure GBR (without call of Face Routing) Barycentric Routing Greedy Barycentric Routing GOAFR Mean Algorithm Cost Node Density [nodes per unit disk] Figure 12: Mean algorithm costs of BR, GBR and GOAFR+ are compared on random unit disk graphs with 1000 nodes

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