Distributed Construction of an Underlay in Wireless Networks

Transcription

1 Distributed Construction of an Underlay in Wireless Networks Hannes Stratil Institute of Computer Engineering, Vienna University of Technology, Austria Abstract A wireless network consists of a large number of nodes that use wireless communication links to collectively perform certain tasks in various application domains (industry, military, etc.). Since wireless nodes are irregularly scattered over a large application area, a suitable wireless multihop routing protocol is needed to facilitate the communication between arbitrary nodes. Many geographic routing protocols use a planar graph as the underlying network topology. This paper presents an efficient algorithm for the localized computation of such an underlay, namely, the Short Delaunay Triangulation (SDT). The SDT contains all edges of the Delaunay Triangulation that are shorter than the communication range. The great asset of the Short Delaunay Triangulation is its spanning property: It approximates the shortest path of the Unit Disk Graph by a constant factor. Our distributed algorithm for constructing the SDT surpasses alternative underlay construction algorithms ( [1], [2]) and requires point-to-point communication links only. I. INTODUCTION Ad-hoc wireless networks are a collection of nodes that are randomly dispersed over some area of interest. They are considered useful for a wide range of application domains including military applications, surveillance, and wireless sensor networks. Ad hoc wireless networks are made up of nodes that communicate with each other over a wireless medium in the absence of a fixed infrastructure and any centralized control. Any computation in an ad-hoc wireless network hence needs to be carried out in a decentralized manner and should be localized (i.e., use information of its local neighborhood only), self-organized and self-stabilizing. Direct communication between two nodes is guaranteed only if the distance between them is less than the minimum of their respective communication ranges. It is generally not possible (nor desirable) that all nodes are within the communication range of each other. Communication between non-neighboring nodes thus requires a multihop routing protocol. We assume that all nodes are distributed in a twodimensional plane and stipulate the existence of a position service that provides network participants with their location. Location knowledge is an important information for supervision and security features. In wireless networks the position is in fact more important than a specific node ID. For example, tracking applications are more interested in where the target is located than in the ID of the reporting node. We do not concentrate on the implementation of a particular routing protocol in our work, but rather on the computation of an efficient underlay over which different routing protocols can be implemented. Various proximity graphs, e.g., elative Neighborhood Graph [3], Gabriel Graph [4], Yao graph [5] and Delaunay Triangulation [6], have been proposed for this purpose. Efficient and scalable algorithms for wireless networks may only use information about the local neighborhood. Hence, the computation of the underlay should also be done locally. Localized versions of the elative Neighborhood Graph and the Gabriel Graph can easily be computed, because a node only needs information about its single hop neighborhood to decide whether a communication edge exists in the elative Neighborhood Graph (resp. Gabriel Graph) or not. For the computation of a localized version of the Delaunay Triangulation a node needs more information than the single hop neighborhood: If the Delaunay Triangulation was only computed with the single hop neighborhood, the computed graph would neither be implicitly a subgraph of the centralized computed graph nor would the graph be necessarily planar. An important advantage of the Delaunay Triangulation is the spanning property. In computational geometry, the spanning ratio of a graph is defined as the maximum ratio of the length of the shortest path connecting any pair of sites in the graph to their euclidean distance. A graph is called a spanner, if the spanning ratio is bounded by a constant. The elative Neighborhood Graph and the Gabriel Graph are not

2 spanners for the complete graph, because their spanning ratios are unbounded. The Delaunay Triangulation has a constant spanning ratio, however. Another important property for an underlay is planarity, because several routing protocols need a planar graph to guarantee the delivery of messages (e.g., [7] [9]). Whereas the Yao graph is not planar, the Delaunay Triangulation is not only planar but also one of the densest planar graphs. In this paper, we present an algorithm for the construction of a planar subgraph, called the Short Delaunay Triangulation (SDT), which relies upon its local neighborhood only. It guarantees scalability, because the algorithm is independent of the total number of nodes in the wireless network. The SDT contains all edges of the centralized computed Delaunay Triangulation that are shorter than the communication range, but it can also contain some edges that are not present in the centralized computed version. Hence, the Short Delaunay Triangulation is not necessary a subgraph of the Delaunay Triangulation. We will show that the SDT is a graph-spanner of the Unit Disk Graph (UDG). A graph-spanner is a subgraph of the UDG in which the length of the shortest path connecting two nodes is within a constant factor of the length of the shortest path connecting these two nodes in the UDG. The spanning property is important, because it can be shown that an underlay with a constant spanning ratio is also within a constant factor of its energy costs [10]. Another novel feature of our algorithm is its use of dedicated communication links between the nodes, rather than the broadcast approach of [1] and [2], which is advantageous e.g. from the viewpoint of security: Wireless networks may be used in hostile environments, where authentication is necessary to protect the communication against malicious nodes. It is unacceptable in a highrisk environment to reveal the exact location of nodes to anyone within range [11]. Therefore, pairwise cryptographic techniques must be employed to protect the position information exchange between any two nodes in position based routing protocols. Several secure routing protocols based on shared keys have been proposed in the literature (e.g., [12], [13]), which are much easier to maintain if dedicated communication links can be used. The remaining sections are organized as follows: Some assumptions and the basic principles of planar graphs are explained in Section II. A detailed description of the underlay construction algorithm is presented in Section III. Section IV gives a detailed relation to other existing work. In Section V, we study the performance of our algorithm and compare it with the algorithms of Li et al. [1] and Gao et al. [2]. Section VI outlines some future directions for research and Section VII concludes the paper. II. PELIMINAIES A wireless network consists of a large number of nodes. Every node is a device that integrates at least a processing unit, some memory, a power unit and a wireless communication interface. The communication interface is used to transmit messages to neighbor nodes and to receive messages from neighbor nodes. A. Assumptions An algorithm for the computation of a planar subgraph requires the positions of the nodes: We assume the existence of a position service that allows every node to estimate its own coordinates locally, e.g., GPS. A survey of position services can be found in [14]. Moreover, we assume that the nodes in the network have negligible difference in altitude, so they can be considered roughly in a plane. B. Network Model The above specified network can be modeled as an undirected communication graph G(S, E) in the plane, with a set of sites S and a set of edges E. Each site s i of the set S := {s 0,..., s N 1 }, represent a node of our wireless network. The total number of sites is N = S. An edge (s i, s j ) E, s i, s j S, represents a wireless link of the network in G(S, E). Each node s has a real communication area A(s) which depends on transmission power, noise, interference, and blockages due to physical obstructions. A(s) is different at each node, irregular and generally unknown. We assume a known circumcised communication range (s). However, with the property that the disc disc(s; (s)) with center s and radius (s) satisfies disc(s; (s)) A(s). The neighborhood of a site s S, denoted by N (s), is the set of sites within s s circumcised communication range (s). Any site s i N (s) is called visible to s, and we assume n to be an upper bound on the nodes in the neighborhood n N (s), s S. The following properties for N (s) are required: Completeness: All sites s i within the circumcised communication range (s) are elements in the neighborhood set of site s. s i disc(s; (s)) : s i N (s) (1)

3 Consistency: If a site s i is in the neighborhood set of site s, than site s is in the neighborhood set of s i. s i N (s) : s N (s i ) (2) Both (1) and (2) are implied by the following property: s i disc(s; (s)) : s disc(s i ; (s i )) (3) Note that (3) is fulfilled by setting (s) = for all s, when = max{ : disc(s, ) A(s) for all s} is the minimum of the communication ranges of all nodes. In the remainder of this paper, we will hence assume that (3) with s : (s) = holds. An edge (s i, s j ) is present in G(S, E) if and only if s i, s j is less or equal than the communication range, where s i, s j denotes the euclidean distance between s i and s j. Therefore, G(S, E) is a undirected graph with at most N(N 1) 2 edges. If we normalize to one unit, than the communication graph is called the Unit Disk Graph. Definition 1 (Unit Disk Graph): The Unit Disk Graph UDG(S) of a set S is defined as an undirected graph, where there is an edge between two sites s i, s j if and only if the Euclidean distance between s i and s j is at most 1. The communication graph G(S, E) as well as the Unit Disk Graph are usually not planar. Informally, a graph is planar, if no two distinct edges cross each other. C. Spanning ratio Consider a graph G over a finite non empty set S I 2. For each pair of sites s i, s j, the length of the shortest path connecting s i and s j measured by euclidean distance is denoted by Π G (s i, s j ), while the direct euclidean distance is s i, s j. Definition 2 (Spanner): A graph G is called a spanner, if Π G (s i, s j ) is no more than a constant factor larger than s i, s j, for all s i, s j S. Definition 3 (Spanning atio): The spanning ratio of the graph G is defined by: Π G (s i, s j ) σ G := max s i, s j S (4) (s i,s j) s i, s j The complete graph has σ = 1, but is less interesting because the graph is not planar and the number of edges is quadratic. If the graph is not connected, then σ is infinity, so it is reasonable to focus on connected graphs only. III. SHOT DELAUNAY TIANGULATION The Delaunay Triangulation is a fundamental geometric construction. For a given set of sites, Delaunay Triangulation constructs a triangle tessellation of the plane with the initial sites as vertices. Definition 4 (Delaunay Triangulation): A triangulation of a set S of sites in the plane is called a Delaunay Triangulation DT (S) of S, if the circumcircle of each of its triangles does not contain any site of S in its interior 1. Definition 4 does not consider the case where all sites are colinear. The following Definition 5 is a little bit stronger and more convenient for our purpose: Definition 5 ( empty circle property): An edge (s i, s j ) is a Delaunay Edge if there exists a circle through s i and s j so that no site of S lies properly inside this circle. The Delaunay Triangulation is unique for a given set S of sites. The bound on the spanning ratio in the following Lemma 1 is an upper bound on all possible sets of sites in I 2. It is impossible to find a set S I 2 which has a higher spanning ratio than the presented bound, although there may be sets where the spanning ratio is less. Lemma 1: [15] The spanning ratio of the Delaunay Triangulation has a constant upper bound, which is at π least 2 and at most 2π 3 cos( π ) It is conjectured 6 that the spanning ratio of the Delaunay Triangulation is σ DT = π 2. The optimal worst-case time complexity of computing a Delaunay Triangulation for N sites in two dimensions is O(N log N). See [16] for an overview of available Delaunay triangulation algorithms. Note that the Delaunay Triangulation is not unique if four or more sites are cocircular. Each computation algorithm needs a deterministic rule for breaking ties here, i.e., to decide which of the edges are to be included in the Delaunay Triangulation and which of them are not. For simplicity, we will assume subsequently that no four or more sites are cocircular, and extend our approach in Subsection III-C. We recapitulate in the following lemma the planarity of the Delaunay Triangulation. Lemma 2: DT (S) is a planar graph. Proof: We show that no two edges may cross in the Delaunay Triangulation. Suppose two crossing edges, (, ) and (, ), appear in the Delaunay Triangulation. By Definition 5, there is a circle through and not containing or, and likewise there is 1 We consider that the interior of a circle is an open disc, i.e., the boundary is excluded.

4 a circle through and not containing or. If these properties are fulfilled, the two distinct circles must intersect at four points, which is impossible 2. Each planar triangulation (e.g., the Delaunay Triangulation) is denser than any other planar subgraph. Lemma 3: [6] A triangulation of a set S of sites is a maximal planar subdivision: no edge connecting two sites can be added to the triangulation without destroying planarity. The above presented graph is not directly applicable to our network model (Subsection II-B). The concept of the Delaunay Triangulation is originated in Computational Geometry one unit computes a planar subgraph of the complete graph. However, algorithms for wireless networks must be localized. It is therefore impossible to compute anything in a centralized manner. Moreover, the edges in our network model must be shorter than the communication range, whereas the length of edges in the Delaunay Triangulation are only depending on the geographic positions and are hence not bounded by any communication range. We hence assume that each node 3 s i S computes a local planar subgraph only with the nodes of its neighborhood N (s i ), denoted by DT (N (s i )). We denote by DT (s i ) all edges of this planar subgraphs which are originating at s i ; DT (s i ) = {(s i, s j ) DT (N (s i )) : s j N (s i )}. The term localized computed is used for the union of these graphs. Definition 6 (localized computed graph): The localized computed version of the Delaunay Triangulation is defined as: locdt (S) := DT (s i ) (5) s i S The Delaunay Triangulation of the whole set S will be denoted with DT (S) and we use the term centralized computed. The localized computed graph is important for our further work, although they are only abstract. In general, no device in the network is aware of the whole localized computed graph. Nevertheless, this graph must be connected and planar to be applicable for routing in wireless networks. In contrast to the elative Neighborhood Graph and the Gabriel Graph, the positions of the single hop neighbors are not sufficient to compute a localized version 2 No four or more sites are cocircular. 3 We denote nodes with the letter s to stay uniform with the graph definitions and to avoid confusions with the number of nodes. emember: n is the number of nodes in the neighborhood and N is the total number of nodes in the network. of the Delaunay Triangulation. We present a distributed algorithm for the computation of a planar subgraph of locdt (S), which we call Short Delaunay Triangulation SDT (S). The Short Delaunay Triangulation fulfills the Delaunay property of Definition 4 locally, but it is not a subgraph of the centralized computed Delaunay Triangulation. It contains all edges of the Delaunay Triangulation that are shorter than the communication range and some additional edges which are not present in the centralized computed Delaunay Triangulation. First of all, we will show in Lemma 5 that locdt (S), the union of the DT (s) s, s S, is connected, and then Lemma 6 will show that locdt (S) might not be planar although the DT (s) s are planar individually. The local Delaunay Triangulation DT (N (s)), s S, at each node is only computed by considering the single hop neighborhood. The local Delaunay Triangulations at different nodes might be inconsistent, i.e. a Delaunay Edge may be in the local Delaunay Triangulation DT (N (s i )) of node s i but not in DT (N (s j )). This could happen, since each node s uses only the nodes within its communication range for the computation of DT (N (s)) and ignores the nodes outside of the communication range. Fig. 1. The shaded region is outside of s communication range. Figure 1 explain this issue: If the triangle (,, ) is a Delaunay Triangle, then no other node is inside the circumcircle of this triangle by Definition 4. Node cannot communicate with a node in the shaded region, so for node the edge (, ) is a Delaunay Edge. If a node was located in the shaded region, node would compute a Delaunay Edge to this node and omits the edge to (because of Definition 5). Hence, the local Delaunay Triangulations would be inconsistent. The localized computed Delaunay Triangulation is by Definition 6 the union of the DT (s) for all s. For each edge (s i, s j ) of the Unit Disk Graph that is not in locdt (S), the edge is neither in DT (s i ) nor in DT (s j ). Before we proof the connectivity of locdt (S) we need

5 a lemma to verify the conditions when an edge is not a Delaunay Edge. Lemma 4: An edge (s i, s j ) of the Unit Disk Graph is not in DT (s i ), if and only if (s i, s j ) is intersected by an Delaunay Edge (s k, s l ). s i, s j, s k, and s l N (s i ). Proof: By Lemma 2, the local Delaunay Triangulation DT (s i ) is planar. It is by Lemma 3 impossible to add an edge to a triangulation without destroying the planarity. Hence, each non Delaunay Edge is intersected by at least one Delaunay Edge. The localized computed Delaunay Triangulation locdt (S) is obviously a subgraph of the Unit Disk Graph. If U DG(S) is connected, locdt (S) is also connected. We proof in the following the connectivity in the neighborhood of node s i, the proof is the same for the neighborhood of node s j. Lemma 5: locdt is connected. Proof: Suppose U DG(S) is connected and there are disconnected components C 1, C 2,..., C M in locdt (S). Let s i and s j be two nodes with s i, s j is minimal such that (s i, s j ) UDG(S) but (s i, s j ) / locdt (S) and s i C i, s j C j. Since edge (s i, s j ) UDG(S), there must be some node s l inside the circumcircle of the triangle (s i, s j, s k ) (Figure 2, Definition 4). Edge (s i, s j ) divides the circle through s i and s j in two circle segments, whereas node s k lies in the first segment and node s l lies in the other segment (Lemma 4). In at least one segment, w.l.o.g. the segment with node s k, s i, s k < s i, s j and s j, s k < s i, s j, (s i, s k ) UDG(S) and (s j, s k ) UDG(S). There are three cases: (s i, s k ), (s j, s k ) / locdt (S) s k with s k C k and s i, s k < s i, s j and s j, s k < s i, s j, (s i, s j ) violates minimality assumption. (s i, s k ) or (s j, s k ) / locdt (S) since s i, s k < s i, s j and s j, s k < s i, s j, (s i, s j ) violates minimality assumption. Both (s i, s k ),(s j, s k ) locdt (S) contradiction to the demanded disconnectivity of the components. locdt (S) is connected but not necessarily planar. Lemma 6: The union of local planar Delaunay Triangulations is not necessarily planar. S I 2 such that locdt (S) DT (S) (6) Proof: The computation of the local Delaunay Triangulations only with the single hop neighbors does not prevent two edges from intersecting. Figure 3 shows an example of 4 nodes (S := {,,, )}) where 4i=1 DT (s i ) is not a planar graph. Figure 3(a) shows the Unit Disk Graph of the network. The distance between node and node is longer than one unit (resp. longer as the communication range if we use the communication graph instead of the Unit Disk Graph), the nodes and are therefore twohop neighbors. The local Delaunay Triangulation of node (DT (N ( ))) and the edges originating at node (DT ( )) are shown in Figure 3(d). Figure 3(e,f) shows DT (N ( )) and DT (N ( )), which are equal because both nodes can communicate directly with the same set of nodes. However, DT ( ) and DT ( ) are obviously different. Finally, DT (N ( )) and DT ( ) are shown in Figure 3(g). The localized computed Delaunay Triangulation locdt (S) = 4 i=1 DT (s i ) is not a planar graph because of the intersection of edge (, ) with (, ) (Figure 3(b)). Figure 3(c) shows that the edge (, ) violates the conditions of a Delaunay Triangle, because the circumcircle of the triangle (,, ) contains node (Definition 4). All local Delaunay Triangulations DT (N (s i )), i = 1,..., 4 are planar individually the localized computed Delaunay Triangulation (Figure 3(b)) is non planar, however, because DT ( ) is inconsistent with DT ( ): The edge (, ) belongs to DT ( ), but not to DT ( ). The computation of a generalized local Delaunay Triangulation at node s with the positions of all single and all double hop neighbors DT ( N (s i )), s i N (s) would guarantee that all edges DT (s) from node s to its single hop neighbors are Delaunay Edges [2]. The drawback of this algorithm is that the time complexity could be as large as O(n 3 log n). Fig. 2. s j s k s i sl Edge (s i,s j) is no Delaunay Edge A. Algorithm We now describe our distributed computation algorithm for the Short Delaunay Triangulation. Each node s computes its local Delaunay Triangulation DT (N (s)) to eliminate the communication edges which are originating at node s and which are definitely not Delaunay Edges. The remaining edges originating at node s are denoted by DT (s). The union locdt (S) of the DT (s) s s S

6 (a) UDG(S) (b) locdt (S) (c) non Delaunay Edge (, ) (d) DT (N ()), DT () (e) DT (N ()), DT () (f) DT (N ()), DT () (g) DT (N ()), DT () Fig. 3. local planar vs. global non planar is non planar (Lemma 6), but would become planar, if each node ignored the edges which are not Delaunay Edges in at least one local Delaunay Triangulation. The removal of those non Delaunay Edges is the main task of our algorithm. Algorithm 1 (Short Delaunay Triangulation): Each node s: Step 1: computes DT (N (s)) Step 2: generates edge-ignore instructions, when s i, s j N (s) and edge (s i, s j ) / DT (N (s)). Step 3: unifies the edge-ignore instructions for every node s i N (s) to a message and sends this messages to the corresponding nodes in its neighborhood. Step 4: omits an edge (s, s i ) in its DT (s), if just one message from a single hop neighbor node contains an edge-ignore instruction for (s, s i ). We prove the planarity of the Short Delaunay Triangulation and show that the SDT is connected if the Unit Disk Graph is connected. Moreover, we show that the spanning ratio of the Short Delaunay Triangulation with respect to the Unit Disk Graph is the same as the spanning ratio of the Delaunay Triangulation with respect to the complete graph. For our major theorems, we need a number of properties of the SDT and the UDG, which will be established below. The following Lemma 7 shows that a communication edge (s i, s j ) cannot be an edge of the Short Delaunay Triangulation, if it is ignored in just one local Delaunay Triangulation DT (S 1 ), s i, s j S 1. Lemma 7: If an edge (s i, s j ) does not exist in just one local Delaunay Triangulation DT (S 1 ), s i, s j S 1, this edge does not exist in any larger Delaunay Triangulation DT (S 2 ) with S 1 S 2. Proof: An edge (s i, s j ) does not exist in a Delaunay Triangulation DT (S 1 ) if the empty circle property (Definition 5) is violated. This property is also obviously violated in DT (S 2 ), S 1 S 2. It is impossible for a node s to decide locally which of the edges of DT (s) (the edges originating at node s) are non Delaunay Edges in any other local Delaunay Triangulation, but the following Lemma 8 (from the work of Gao et al. [2]) gives us a useful tool to solve this problem.

7 Lemma 8: If two edges (, ) and (, ) cross in UDG, at least one of the four nodes has communication edges to the three other nodes. Proof: Both distances, and, are shorter than the communication range. Assume that they intersect at point p. By the triangle inequality,, p +, p, and, p +, p,. Summing these two inequalities, we get, +,, +,. (7) Therefore,,,,, or both have length shorter than the communication range. Similarly,, p +, p,,, p +, p, and by summation, +,, +,. (8) By this inequality,,,,, or both have length shorter than the communication range. No matter in which case, the node shared by the two short edges has communication edges to all three other nodes. The following Definition 7 gives us a useful notation for local Delaunay Triangulations: Definition 7 (correctly computed): We call a local Delaunay Triangulation DT (N (s)) computed correctly with respect to the final Short Delaunay Triangulation SDT (S), if DT (s) SDT (S), s S. The intersection of communication edges, which must be resolved in the course of planarization, can appear in three different cases: Case 1: Each node has communication edges to all three other nodes. In this case the local Delaunay Triangulation is computed correctly at each node. One of the intersecting communication edges must be a non Delaunay Edge, however. Both corresponding nodes of the non Delaunay Edge omit this communication edge in their local Delaunay Triangulation. Nevertheless, both corresponding nodes s i, s j of the non Delaunay Edge (s i, s j ) generate an edge-ignore instruction for (s i, s j ), because the nodes use only local information and do not know whether the corresponding node computed the local Delaunay Triangulation correctly or not. Case 2: Two edges intersect and two nodes have communication edges to the three other nodes. This case makes only sense, if the two nodes with the communication edges to the three other nodes are not connected by one of the intersecting edges. Otherwise no intersection may occur. In Figure 3, and are such nodes. In this example, the communication edge (, ) is a Delaunay Edge and edge (, ) is a communication edge which is not a Delaunay Edge. Node computes its local Delaunay Triangulation DT (N ( )) correctly, but node computes a non-correct Delaunay Triangulation DT (N ( )). The different local Delaunay Triangulations became inconsistent. Only node generates an edge-ignore instruction for edge (, ) in this case. Case 3: In the last case only one node has communication edges to all of the three other nodes. This case is a little bit harder to deal with because this node must inform two single hop neighbors that the edge between them must be ignored. Node is the only node in Figure 4 which has communication edges to all other nodes and is therefore the only node which computes its local Delaunay Triangulation correctly. The nodes and have no communication edges to node, so edge (, ) is locally a valid Delaunay Edge for both nodes. Node generates edge-ignore instructions for both nodes, and. Fig. 4. (, ) (, ) (s1, s3) α 1 α 2 Only node has communication edges to all other nodes. Other cases of two intersecting communication edges cannot occur, because the neighborhoods of the corresponding nodes must always be complete and consistent (Subsection II-B). Each node collects the edge-ignore instructions for the neighbor nodes and sends to each neighbor node at most one message with all edge-ignore instructions relevant for this node. Hence, the message complexity per node of our algorithm is at most O(n). The complexity for Case 3 is not optimal, however. Just generating an edge-ignore instruction when two single hop neighbors in local Delaunay Triangulation have no Delaunay Edge between them, would unnecessarily increase the time complexity and the message length. Consequently, a test is needed for Case 3 to verify whether a node must really generate edge-ignore instructions for the other nodes.

8 Figure 5 illustrates an example of Case 3 and shows the critical regions. The critical regions for node s are the shaded sections A and B. Only two nodes, one from section A and one form section B are endangered to allow an edge in their local Delaunay Triangulations that is not in the SDT. Node s has only to generate edgeignore instructions for all edges where one node lies in the first section and the other node lies in the second section. π/3 π/3 α 1 α 2 s s 5 Section A Section B Fig. 5. egions with endangered nodes. In the example of Figure 5, node s is the only node which has communication edges to two nodes, and. The number of these nodes is non relevant, it only matters that this nodes are not within the communication range of the nodes in the critical sections (specification of Case 3). A simple but rough test to verify whether a node has to generate the edge-ignore instruction is based upon elementary properties of triangles. Lemma 9: If an angle in a triangle is less or equal than π/3, then the edge opposite to this angle is smaller or equal than the longest edge in the triangle. Lemma 9 implies that two single hop neighbors have a common communication edge if the angle between the communication edges to those single hop neighbors is less or equal than π/3. Therefore, for the existence of Case 3 the following two properties are essential: There is an angle between two successive communication edges (called α 1 ), which is larger than π/3. There is a second angle between two successive communication edges (called α 2 ), which is larger than π/3. The simple test generates an edge-ignore instruction for an edge (s i, s j ), if the angle between s i and s j is smaller than π and if it contains α 1 and α 2 as subangles. However, the simple test considers larger regions than the critical regions, i.e., the regions between the dashed lines in Figure 5. Node s generates edge-ignore instruction for (, ) and (, s 5 ). An additional and more precise test can be used to reduce the number of edge-ignore instruction further by eliminating the small circle segments between the dashed lines and the critical regions (Figure 5). Consider a network of four nodes that are placed like that in Figure 4. If the inequalities of Lemma 10 below are true, then node is the only node with communication edges to the other nodes and edgeignore instructions must be generated. Lemma 10: If and, > 2, cos α 1 (9) (10), > 2, cos α 2 (11) then, >, and, >,. Proof: The law of cosinus provide us with,, 2 =, 2 +, 2 2,, cos α 1 (12) where α 1 is the angle between, and,., <, (13), 2 <, 2 +, 2 2,, cos α 1 (14) 2,, cos α 1 <, 2 (15) 2, cos α 1 <, (16) The proof for node is the same as for node. A node can exactly recognize via Lemma 10 whether two single hop neighbor nodes possibly allow a wrong Delaunay Edge. Therefore, a node s has only to generate an edge-ignore instruction for a specific edge (s i, s j ) if the corresponding nodes s i and s j lie within the critical sections and if the inequalities of Lemma 10 are true and if edge (s i, s j ) is not a Delaunay Edge in the local Delaunay Triangulation of node s (DT (N (s))). We now show the planarity of the Short Delaunay Triangulation. Theorem 1: The Short Delaunay Triangulation (SDT ) is planar. Proof: The local Delaunay Triangulations are planar due to Lemma 2. By Lemma 8, the three cases above are the only situations how intersections of edges can

9 occur. If we remove the non Delaunay Edges with the algorithm presented above, the resulted graph is planar. Corollary 1: SDT (S) locdt (S) Theorem 1 showed that removing edges form the graph makes the graph planar. We still have to prove that the resulting graph, i.e., the (SDT ) is connected. Theorem 2: The Short Delaunay Triangulation is connected. Proof: We prove the theorem by showing that the removal of edges from the localized computed Delaunay Triangulation, locdt (S), done by our algorithm does not disconnect the network. Note that locdt (S) is connected if the Unit Disk Graph is connected (Lemma 5). The three cases are the same as in Theorem 1: Case 1: No additional edge is removed from the locdt (S). Both corresponding nodes have the edge already removed by the computation of their local Delaunay Triangulations. Case 2: The second case clarifies the inconsistent views of the local Delaunay Triangulations. An edge (s i, s j ) is removed in this case, if this edge is a non Delaunay Edge in the local Delaunay Triangulation of either s i or s j. The node, that computes its local Delaunay Triangulation correctly (assume, w.l.o.g., node s i ), sends a message with an edge-ignore instruction to the corresponding node s j. Node s i has also an alternative path to the node s j (Lemma 5). Case 3: In this case, an edge (s i, s j ) is eliminated from the graph only when there is a third node s k which sends the edge-ignore instructions for this edge to s i and s j. Node s k has communication edges to s i and s j and establishes therefore an alternative path (Figure 4). The communication edge to node s i (resp. node s j ) is either a Delaunay Edge or there exists an alternative path to this node in the local Delaunay Triangulation of node s k. B. Four or more cocircular nodes If four or more nodes in a wireless network are cocircular, then is the Delaunay Triangulation not unique. Computing the Delaunay Triangulation needs a deterministic rule to decide which of the edges shall be included in the Delaunay Triangulation and which of them shall be omitted. Four or more cocircular nodes are a problem for our SDT-algorithm, if it is possible that only one node computes its local Delaunay Triangulation correctly (Case 3). In such a case, this node must first recognize that four or more nodes are cocircular, and second, this node must inform the other cocircular node about the non Delaunay Edges. However, we prove in Lemma 11 that only Case 1 and Case 2 can appear, if the nodes of intersecting communication edges are cocircular. At least one node of a crossed edge has communication edges to the nodes of the crossing edge and computes its local Delaunay Triangulation correctly. These nodes inform the corresponding nodes of whether the communication edge is a Delaunay Edge or not. The computation algorithm of the Short Delaunay Triangulation therefore needs no additional time and communication. Figure 6 gives an example of four cocircular nodes. Fig. 6. Four cocircular nodes Lemma 11: If two edges intersect and the four nodes are cocircular, than at least two nodes have communication edges to all other cocircular nodes. Proof: If two edges (, ) and (, ) cross, then there is at least one node with communication edges to the three other nodes (Lemma 8). We assume w.l.o.g. that node is the node with communication edges to, and. Edge (, ) divides the joining circle of the four nodes in two circle segments, whereas node lies on the perimeter of the first segment and node lies on the perimeter of the other segment. In at least one segment, the edge (, ) is longer than all other chord in this segment. Hence, at least one node of the intersecting edge (, ) lies within the communication range of node. If we assume w.l.o.g. that node is this node, then has communication edges to, and. C. Spanning Property of Short Delaunay Triangulation The spanning ratio of the Delaunay Triangulation (Lemma 1) does not apply to the Short Delaunay Triangulation, because the SDT does not contain arbitrarily 4 A straight line going through two points of a circle is called a secant. Its segment lying inside the circle is called a chord.

10 long edges. The spanning ratio of the Short Delaunay Triangulation, as defined in Definition 3, is in fact not bounded by any constant. However, we can guarantee the same spanning ratio for another more appropriate type of spanner the graph-spanner. Definition 8 (graph-spanner): A subgraph is called a graph-spanner, if the length of the shortest path connecting any two sites in the subgraph is no more than a constant factor larger than the length of the shortest path connecting the two sites in the original graph. We denote in the following the spanning ratio of the Short Delaunay Triangulation with respect to the Unit Disk Graph as the graph-spanning ratio. Definition 9 (Graph-spanning atio): The graphspanning ratio is defined by: Π SDT (s i, s j ) σ SDT := max s i, s j S (17) (s i,s j) Π UDG (s i, s j ) For the graph-spanner, we replace each communication edge by the shortest path of our Short Delaunay Triangulation. So we have to show, that this path exists, i.e., there is no edge in the path that is longer than the communication range. Lemma 12: The shortest path connecting two sites in the Short Delaunay Triangulation exists. Proof: Suppose a communication edge (s i, s j ). Let C be a circle whose boundary passes through s i and s j, with the diameter s i, s j (Figure 7). If this edge is not in the SDT, then at least on site s k lies by Definition 5 within circle C. (s i, s k ) UDG(S) and (s j, s k ) UDG(S), because s i, s k < s i, s j and s j, s k < s i, s j. For each new edge (e.g. (s j, s k )) exists a circle C 1 whose boundary passes through s j and s k which lies completely inside the circle C. If circle C 1 is empty, then edge (s j, s k ) is part of the SDT. Otherwise, there must be at least one site s l inside circle C 1. Again, (s j, s l ) UDG(S) and (s k, s l ) UDG(S), because s j, s l < s i, s j and s k, s l < s i, s j. The shortest path connecting two sites (s i, s j ) in the Short Delaunay Triangulation lies in any case completely within a circle with diameter s i, s j and therefore is no edge of the shortest path longer than edge (s i, s j ). The following Theorem 3 shows that the Short Delaunay Triangulation is a graph-spanner of the Unit Disk Graph. Additional, we prove that the upper bound of the graph-spanning ratio is the same as the upper bound of the Delaunay Triangulation spanning ratio. Theorem 3: The Short Delaunay Triangulation (SDT ) is a graph-spanner of the UDG. Fig. 7. s i s k C C 1 s l All edges of the shortest path are shorter than edge (s i, s j). The spanning ratio of the SDT is: σ SDT σ DT (18) Proof: The shortest path connecting s i and s j in the Unit Disk Graph consists of m edges: Π UDG (s i, s j ) = s i = t 0, t 1 + t 1, t s j... + t m 1, t m = s j (19) Due to Lemma 12 and Definition 3, t l, t l+1 Π SDT (t l,t l+1) σ DT, 0 l < m. Π UDG (s i, s j ) Π SDT (s i = t 0, t 1 ) σ DT + Π SDT (t 1, t 2 ) σ DT Π SDT (t m 1, t m = s j ) σ DT (20) Π SDT (s i, s j ) = Π SDT (s i = t 0, t 1 ) + Π SDT (t 1, t 2 ) Π SDT (t m 1, t m = s j ) (21) Π SDT (s i, s j ) Π UDG (s i, s j ) σ DT (22) σ SDT σ DT (23) IV. ELATED WOK Planar graphs are frequently used as underlays for position-based routing algorithms: Karp and Kung [7] as well as Bose et al. [8] proposed similar reliable routing algorithms. Both use a localized computed version of the Gabriel Graph as underlay. In the work of Bose and Morin [9] the underlying topology is the centralized computed Delaunay Triangulation. However, the centralized computed Delaunay Triangulation is not applicable to wireless networks, since it can contain arbitrarily long edges. Two methods for the localized computation of planar graphs that are related to the Delaunay Triangulation are published in [1] and [2]. The major difference between these algorithms and our algorithm is that

11 both algorithms use broadcasts to communicate with the neighbour nodes whereas our algorithm uses dedicated communication channels only. The algorithm of Gao et al. [2] constructs a planar graph called estricted Delaunay Graph (DG). First, each node s S computes the local Delaunay Triangulation DT (s) with its single hop neighbors and sends its local Delaunay Triangulation to all of its single hop neighbors. Second, each node s deletes an edge (s, t) from DT (s) if the edge does not exists in the DT (N (u)) (s, t N (u)). However, the time complexity of this algorithm is very high. Li et al. [1] propose the PLDel(S) as planar subgraph. Again, the nodes compute the local Delaunay Triangulations of their single hop neighborhood. Then each node proposes its local Delaunay Triangles and waits for reply messages whether the proposed triangles are also valid in the Delaunay Triangulations of the neighbourhood or not. V. PEFOMANCE ANALYSIS In this subsection we analyze the memory, message and time complexity for the computation of the Short Delaunay Triangulation accordingly to our algorithm. Moreover, we compare the results with two other distributed planarization algorithms; the algorithm of Li et al. [1] and the algorithm of Gao et al. [2]. Both algorithms assume, in contrast to our algorithm, that a node can broadcast a message to all nodes within its communication range. For fair comparison, we replace each broadcast by multiple transmissions. Message Complexity: At the beginning, each node sends its position to the nodes within its neighborhood. The number of nodes in the neighborhood is at most n, so each node sends at most n 1 messages 5. After the computation of DT (N (s)), each node s generates the required edge-ignore instructions for the edges in their local neighborhood, unifies the instructions to messages and sends this messages to the corresponding neighbor nodes. To sum up, a node sends at most two messages to a specific neighbor node and the message complexity of a node is therefore O(n). Additionally to the number of messages, we must also consider the message length. The total number of nodes in the network is N and we assume a unique node identifier can be represented by log N bits. Therefore, an edge-ignore instructions has O(log N) bits. How many edge-ignore instructions can, in the worst case, be unified to an edge-ignore message? The number of edge-ignore instructions is one for Case 1 and Case 2, however. The message length for Case 1 and Case 2 is therefore C +log N bits 6. A little bit more complicated is Case 3 with the critical sections. In the worst case, only one node lies in the first section and the remaining nodes lies in the corresponding section. The single node in the first section receive a message with C + (n 3) log N bits. The worst case message length is therefore O(n log N) bits. However, the remaining nodes in the corresponding critical section receives messages with C + log N bits (resp. C + 2 log N bits if Case 1 or Case 2 also appears). More bits appear, if the nodes are evenly distributed in two corresponding critical sections. If the number of nodes is even, each node in a critical sections receives a message with C+( n 2 2 ) log N bits (resp. C +( n 2 ) log N bits). If the number of nodes is odd, the nodes in one section receives messages with C +( n 1 n+1 ) log N bits (resp. C +( ) log N bits) 2 and the nodes in the other section receives messages with C +( n 3 n 1 2 ) log N bits (resp. C +( 2 ) log N bits). Hence, the total number of sent bits for O(n) messages is in the worst case O(n 2 log N) bits. Memory Complexity: Each node has only to store the position information of at most n 1 neighbor nodes. Hence, the memory complexity is O(n). Time Complexity: Each node s computes the local Delaunay Triangulation DT (N (s)) using the nodes of its local Neighborhood N (s). The number of nodes in the neighborhood is bounded by n and therefore the computational complexity at each node is O(n log n). Additionally, each node computes the inequalities of Lemma 10 at most n 1 times in order to verify whether a neighbor node lies within a critical section. Message Memory Time SDT O(n) O(n) O(n log n) Gao et al. [2] O(n) O(n) O(n 3 log n) Li et al. [1] O(n) O(n) O(n log n) The complexities of our work compared with the work of Li et al. [1] are very similar, although the constant factor of the message complexity is lower in our work. The number of messages is important in the context of energy requirements, because wireless devices 2 5 The node itself is also in the neighborhood. 6 C is some constant.

12 are normally equipped with batteries. Therefore a lower number of sending messages increases the lifetime of the devices. The main difference appears in the construction idea, however: Li et al. [1] accepts only a subset of edges at the beginning, the Gabriel edges, and then adds the remaining Delaunay Edges. Our algorithm starts with all edges and then removes the non-delaunay Edges. VI. DYNAMIC NETWOKS AND FUTUE WOK An important aspect for the performance of a distributed algorithm is the behavior of the algorithm in dynamic environments. The mentioned algorithms of Li et al. [1] and Gao et al. [2] are inefficient in the context of dynamic environments, because the algorithms must restart on each change in the topology. Whereas our computation algorithm is suitable for the usage in dynamic environments: By Lemma 7, a communication edge is not part of the Short Delaunay Triangulation, if at least one node sends an edge-ignore instruction for this edge. After changes in the local neighborhood of a node s and the required re-computation of the local Delaunay Triangulation DT (s), node s has only to send the changes in the edge states (from non-delaunay Edge to Delaunay Edge and visa versa) to the neighboring nodes and not the whole information about all non-delaunay Edge. The memory requirements for such a dynamic algorithm are higher than the requirements for the proposed algorithm, because the nodes must store the transmitted and received edge-ignore instructions. A detailed description of the dynamic algorithm and an extensive performance analysis is recommended to our future work. VII. CONCLUSION A position-based routing protocol needs an underlay for reliably delivering of messages from a source node to a destination node. We propose in this paper an efficient computation algorithm for the Short Delaunay Triangulation (SDT). 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