Curve-based Planar Graph Routing with Guaranteed Delivery in Multihop Wireless Networks

Transcription

1 Curve-based Planar Graph Routing with Guaranteed Delivery in Multihop Wireless Networks Hannes Frey Universität Paderborn Department of Computer Science Computer Networks Group Pohlweg 47-49, Paderborn, Germany Matthias Hollick, Adrian Loch Technische Universität Darmstadt Department of Computer Science Secure Mobile Networking Lab (SEEMOO) Mornewegstr. 32, Darmstadt, Germany Abstract Localized geographic routing schemes operating on planar graphs promise scalability for use within large multihop wireless networks. Existing schemes base routing path construction on faces defined by the planar graph. Once running on a particular planar graph, none of the existing schemes is flexible enough to adapt the sequence of faces visited by the constructed path. Thus, real-world constraints such as network congestion, limited node energy levels, or noncooperation of nodes might severely impact the performance and the robustness of existing planar graph routing variants. To address this problem, we extend planar graph routing with one further degree of freedom: control over the sequence of visited faces. Basically, our face routing extension now follows a sequence of faces intersected by any curve we can freely adjust. We investigate basic schemes for choosing curves dealing with imperfections in the network, and derive algorithms for routing and forwarding along these curves. We analytically prove that our scheme is loop free and allows for guaranteed delivery in arbitrary planar connected graphs. We implement curve-based routing and show its feasibility by means of a simulation study. As a proof-of-concept scenario, we investigate the case of noncooperating nodes. Our results show that curve-based routing is able to sustain the delivery of packets where traditional schemes fail. Keywords-Algorithms, network protocols, wireless multihop networks, localized routing, delivery guarantees I. INTRODUCTION Geographic routing schemes that operate on local state information have been proposed to meet the challenge of scalable routing within large-scale multihop wireless networks. By utilizing local state, these schemes avoid costly exchange of routing updates, which helps conserving the limited bandwidth of wireless environments. Instead of pinpointing a node s position in the network via an IP address, geolocation serves as a location identifier and allows to forward packets towards the intended destination. In fact, following a trivial approach, nodes without any knowledge about the rest of the network but their direct neighborhood can relay packets to neighboring nodes that present some positive progress towards the destination in terms of geographical distance. This allows for scalable operation within large multihop wireless networks. However, the aforementioned purely localized approach cannot guarantee delivery, since local minima in the graph can lead to dead ends, for example, if a node has no neighbor which is closer to the destination than itself. Purely localized geographic algorithms can be beneficially combined with face routing [1], which uses the direct line connecting source and destination as a reference to escape from local minima. Its key concept is routing packets along faces, which are the polygons formed by the edges of a planar network graph. Face routing is confronted with a number of challenges if operating under realistic topology assumptions, as observed, for example, in realistic wireless mesh/sensor networks: Dynamic optimization such as interference-aware or congestion-aware routing cannot easily be realized. Usage of limited resources such as the energy of battery powered nodes cannot be balanced or optimized. Non-cooperative or misbehaving nodes can fatally affect the resiliency of the routing system. The above issues are particularly critical, since existing face routing schemes are non-adaptive, i.e., if the planar graph is not changing, the sequence of visited faces will always be the same. If by chance the face sequence passes a region that only allows for substandard and inferior service, the quality of the forwarding service is likewise. Even worse, adverse and hostile network conditions in certain parts of the network can disallow the packet forwarding entirely. Without additional mechanisms to control the visited face sequence, face routing is not able to deal with such challenges. In contrast, if the visited face sequence can be adjusted, nodes can adapt to the network conditions and deviate for instance from the direct line trajectory (which is the most popular approach so far) to optimize the routing performance w.r.t. certain network constraints. For example, if most of the traffic in a battery powered network always follows the same face sequence, energy depletion of the nodes on the route might lead to network partitioning. Having the freedom to adjust the sequence of visited faces, a node could send packets over an alternative path to avoid low energy regions /12$31.00 c 2012 IEEE

2 In this work we enhance localized routing schemes to follow global forwarding rules, while avoiding the inherent drawbacks of global routing schemes. In particular, we present an approach, which expands the concept of localized geographic routing by allowing curved trajectories between source and destination. Similar to conventional face routing, our scheme maintains its locality, because it only requires nodes to know about their immediate neighbors. At the same time, our scheme allows to exercise global control over the routing/forwarding path, since nodes can define end-toend trajectories that indirectly control which regions in the network to visit or to evade 1. By choosing different enough curves, packets follow different paths. Our contribution is the design of a curve-based, localized, geographic routing scheme for planar graphs 2. In particular, we design a robust algorithm based on face routing and give a rigorous mathematical definition of our scheme, which enables the source of a packet to define an arbitrary curve as an abstract trajectory. We prove that the curve-based planar graph routing is loop free and allows for guaranteed delivery of packets in planar graphs. While our routing mechanism is localized, defining the curve itself might require global knowledge. However, together with a global scheme, we also discuss a distributed strategy to find trajectories. As a proofof-concept, we implement curve-based planar graph routing and study its performance using simulation. In particular, we analyze the scenario of non-cooperative nodes. The remainder of this work is organized as follows. Section II is dedicated to related work and provides a short introduction to face routing. We define and mathematically formulate the basis for our scheme in Section III. The algorithms for curve-based routing are presented in Section IV, while in Section V, we prove that our scheme operates loopfree and can provide delivery guarantees. In Section VI we present our scenario for the proof-of-concept implementation. We further design and implement a system using curvebased routing in a network simulator. We study and analyze its performance in Section VII. Finally, Section VIII presents conclusions and an outlook on future work. II. RELATED WORK Using trajectories or just a sequence of anchor points to guide localized geographic routing has been considered in connection with geographic greedy routing so far [3], [4], [5]. Yet, none of the known approaches supports pure localized forwarding while providing delivery guarantees at the same time. In fact, existing localized schemes require mechanisms to recover from the well known greedy routing failure [6]. 1 In a sense, our curve-based routing can be seen as a form of geo-trafficengineering for multihop wireless networks. In contrast to the control and optimization of traditional traffic engineering, the focus is on controlling the traffic flow with respect to geographical areas that are impacting on the performance of the multihop wireless network. 2 A sketch of the idea has been presented as a poster at MASS 2011 [2]. Face routing is the only known general localized forwarding scheme to recover from greedy routing failure and to guarantee message delivery. It was first mentioned as Compass Routing II in [7] and was picked up and extended in many follow-up publications [1], [8], [9], [10], [11]. The main idea is to send packets along the interior of the faces formed by a planar drawing of the network topology. Packets visit the faces intersected by the straight line connecting source and destination. Traversing a single face is done by left or right hand traversal. The right hand rule is the same as if a person would walk on the interior of the face holding its right hand against its edges. The left hand rule is the same but using the left hand instead. Face routing and trajectory based forwarding has not been considered in combination so far. This is the contribution of this work. In this work we specify in detail a face routing variant to follow arbitrary trajectories, we technically proof its delivery guarantees, and we show an application example of this face routing extension. Deviating from a straight line has been briefly introduced in [12], yet following welldefined trajectories is not discussed. With that we extend and generalize the knowledge about face routing and the delivery guarantees rigorously studied in [6]. For all variants and also for the variant presented in this work, a planar graph is assumed, which means that the nodes of the graph are located on a plane and that the edges of the graph cross at most in their end points. However, usually the graph of a wireless network is not planar. Therefore, a previous step is necessary to planarize it. Arbitrary graphs can be transformed by the non-localized topology control scheme CLDP [13] into a (not necessarily planar) graph where face routing always succeeds. However, a localized topology control scheme to transform an arbitrary graph into one which supports face routing cannot exist. This is an immediate consequence of [14]. Such a negative result renders face routing useless at the first look. However, wireless networks are not arbitrary graphs, they have a local structure which has recently been exploited in [15]. In that work, it has been shown that in wireless networks it is possible to locally construct graphs which are planar with a very high probability. Simulations in realistic wireless graph models based on log-normal shadowing revealed actually a 100% probability. In this work we are not concerned about how a planar graph is constructed but what to make best of such graph once it is constructed. In the following we just assume any planar graph, though, we do not assume any further structural properties. III. NOTATIONS AND ASSUMPTIONS A. Geometric and Graph Notations We consider undirected graphs G = (V, E) with finite node sets. Two nodes s and t are said to be connected if there exists at least one path in G with end points s and t.

3 Each node in V is a point in 2D Euclidean space. For two points p and q we use p q to express the Euclidean distance between p and q. We say that G is planar if the edges in E intersect at most in their end points. p left right a q (a) Left and right of ab. b C 2 a b C 1 c (b) Cones of ab and ac. Figure 1: Definitions for straight lines. Given two points a and b, the straight line from a to b is denoted by ab. A point p is said to be located left of ab if the counter clockwise angle between ab and ap is less than 180. Otherwise, it is said to be located right. For example, in Fig. 1a point p is located left of ab and point q is located right. Two straight lines ab and ac with same origin a define two areas, one is the area defined by the clockwise angle from ab to ac and the other one defined by the counter clock wise one. We term both areas the cones defined by ab and ac. For example, in Fig. 1b the straight lines ab and ac define the cones C 1 and C 2. A special case occurs, when one of the angles between ab and ac is 0. In this case we consider however two cones, one being the entire R 2 and the other being just a line originating from a and passing through b, c. B. Continuous Curves A curve is a function f : [0,1] R 2. The value x in the expression is termed parameter. If we write we implicitly assume x [0,1]. A curve is continuous if for each x and ǫ > 0 there exists a δ > 0 such that x y δ implies f(y) ǫ. A continuous curve connects a source node s with a destination node t, if f(0) = s and f(1) = t. A curve intersects a line ab if there exists an x 0 such that f(x 0 ) is located on ab. Continuous curves obey the properties expressed in the following three lemmas. In the following, we take these properties for granted when defining our routing algorithm. Lemma 1: If a continuous curve intersects a straight line ab, then there exists a largest q 1 such that f(q) is located on ab. Proof: Let q = sup{x 1 : is located on ab}. We claim that q is the largest parameter such that f(q) is located on ab. We have to show that f(q) is located on ab and that f(q ) is not located on ab for all q > q. The latter is obvious. Otherwise, we would have the contradiction q < q sup{x 1 : is located on ab} = q. Assume for the sake of contradiction that f(q) is not located on ab. Let ǫ be the minimum distance between f(q) and ab. Since is continuous, there exists a δ > 0 such that f(q) f(y) ǫ/2 for all y with q y δ. Moreover, is not located on ab for all x > q. Thus, f(q δ +x) is not located on ab for all x 0. It follows: q = sup{x 1 : is located on ab} q δ, a contradiction. Lemma 2: Assume a continuous curve intersects a straight line ab. Consider the largest q such that f(q) is located on ab. If f(q) is not an end point of ab, then there exists a δ > 0 such that for all q < x < q +δ, all are located either on the left hand or the right hand side of ab. Proof: Without loss of generality assume that f(q) a f(q) b (the following arguments can be written the same way for the opposite case). Thus, changing the side of ab requires the curve starting at q to advance at least by f(q) a. Define ǫ = f(q) a /2. Since is continuous there exists a δ > 0 such that all x > q with x q δ satisfy f(q) ǫ < f(q) a. Thus, is located on one side of ab all these x. Lemma 3: Assume a continuous curve intersects two straight lines ab and ac which are originating from a. Consider the largest parameter q such that f(q) is located on ab or ac. If f(q) = a, then there exists a δ > 0 such that for all q < x < q+δ, all are located in one of the two cones defined by ab and ac. Proof: Without loss of generality assume that a b a c (the following arguments can be written the same way for the opposite case). Thus, changing the cone defined by ab and ac requires the curve starting at q to advance at least by a b. Define ǫ = a b /2. Since is continuous there exists a δ > 0 such that all x > q with x q δ satisfy a ǫ < a b. Thus, is located in one of the two cones defined by ab and ac for all these x. As a short hand notation of the facts derived with Lemma 2 and Lemma 3, we just speak of the initial part of is located left/right of ab for x > q and the initial part of is located in one cone for x > q, respectively. A. Formal Specification IV. ALGORITHM The main part of the curve assisted planar graph routing algorithm is shown in Alg. 1. It basically works as follows (refer to Fig. 2 for an example): The faces intersected by a continuous curve connecting source with destination node need to be traversed. The edges of a face are traversed until an intersection p of the curve with a face edge sv is found.

4 Algorithm 1 Curve-based planar graph routing. Require: start node s, destination node t. Ensure: message delivery if t reachable from s, message drop, otherwise. 1: set p = 0 and v = s. 2: repeat 3: find the largest q 1 such that f(q) is located on an edge vw E. 4: if f(q) = w for an edge vw E then 5: set node_rhr and node_lhr to w 6: else if f(q) = v then 7: find node_rhr and node_lhr with Alg. 3. 8: set p = q 9: else 10: find node_rhr and node_lhr with Alg : set p = q 12: end if 13: right/left hand traversal until one of the conditions hold destination node t is reached. visited node has outgoing link intersecting with for p < x 1. traversal start edge visited again in the same direction. 14: set v to the currently visited node. 15: until no more intersections found on current traversed face Three cases need to be considered: the intersection might regularly intersect sv at a point p s (see Fig. 3a), it might irregularly intersect at a point p = s (see Fig. 3b), or it might intersect the end point w of an outgoing link vw. For the latter case, the algorithm just forwards to w. For the other two cases, the next hop node and traversal rule selection are decided by Alg. 2 and 3, respectively. In both cases, traversal changes into the face intersected by the remaining part of the curve connecting the intersection f(p) with the destination t. Face traversal is determined by the values stored in node_rhr and node_lhr. They determine the next hop to be used when right hand or left hand rule is applied. The whole algorithm stops if the destination is visited or if no more intersections are found. B. Execution Example Consider Fig. 2 for an example of Alg. 1. Node s has a message to be sent to node t. The face sequence to follow is s v 3 q 3 v 1 q 1 q 2 q v 4 v 4 2 F 1 F 2 F 3 F 4 v 5 q 5 F 5 Figure 2: Curve-based face traversal. t Algorithm 2 Forwarding decision for regular intersections (see Fig. 3a). Require: forwarding node s, destination node t. Ensure: next hop nodes node_rhr and node_lhr for left or right hand traversal. 1: find largest q 1 such that f(q) is located on an edge sv E. 2: if is initially located left of sv for x > q then 3: let sw be the edge in counter clockwise direction of sv. 4: set node_rhr to v. 5: set node_lhr to w. 6: else 7: let sw be the edge in clockwise direction of sv. 8: set node_rhr to w. 9: set node_lhr to v. 10: end if determined by, depicted by the dashed curve. Node s sees only intersecting irregular with its outgoing links. Thus, Alg. 3 is used to determine the start edges for left or right hand traversal. Say left hand traversal is used. In this case, the message will be forwarded to v 1. At nodev 1 an intersection with is found again. Node v 1 finds intersecting regularly with its outgoing edges. It determines the largest parameter q 1 which satisfies that f(q 1 ) is located on an outgoing edge, v 1 v 2 in this case. The curve is initially located right of v 1 v 2 for x > q 1. Thus, with Alg. 2 v 1 v 2 becomes the start edge for left hand traversal. Edge v 1 s becomes the edge for right hand traversal. For the following assume that with each face change at regular intersections, we always follow the link which was intersected by. In this example we then follow the link v 1 v 2. At node v 2 we find intersecting with outgoing edges again. The latest intersection is at q 2, where f(q 2 ) lies on edge v 2 v 3. Alg. 2 determines that edge v 2 v 3 is to be followed with the right hand rule. At node v 3 face F 3 will then be followed by left hand rule until reaching node v 4. Here again an intersection of an edge with at q 4 > q 3 is found. Face F 4 is then Algorithm 3 Forwarding decision for irregular intersections (see Fig. 3b). Require: forwarding node s, destination node t. Ensure: next hop nodes node_rhr and node_lhr for left or right hand traversal. 1: find largest q 1 such that f(q) is located on s. 2: find edges sv and sw which define the smallest cone where is initially located in. 3: if counter clockwise angle from sv to sw is less than 180 then 4: set node_rhr to v. 5: set node_lhr to w. 6: else 7: set node_rhr to w. 8: set node_lhr to v. 9: end if

5 followed by right hand rule until arriving at node v 5. Here the last intersection is found at q 5 > q 4. Face F 5 is then traversed by left hand rule until arriving at the destination node t. w s f(q) v (a) Regular intersection. t w v s f(q) (b) Irregular intersection. Figure 3: Possible face traversal start cases. A. Loop Free Operation V. ANALYSIS Theorem 1: Curve-based planar graph routing is loop free. This holds even if it is applied in arbitrary not necessary planar graphs. Proof: Whenever in Alg. 1 an intersection of the remaining curve for x > p with an edge e is encountered, parameter p is advanced to a value q > p such that does not intersect with e for x > q. Thus, at each edge a traversal of a new face is started at most once. It follows, since G is finite, only a finite number of times a new face will be explored. Thus, a routing loop may occur only if the last face F is traversed infinitely. However, if F is traversed without further face changes, with Theorem 9 in [6], the traversal start edge e will eventually be traversed twice in the same direction again. Thus, Alg. 1 will stop at e and drop the message. B. Guaranteed Delivery s w f(q) F v (a) Initial curve located left. f(q) s v F w (b) Initial curve located right. Figure 4: Considering the initial part of the curve. Lemma 4: Let G be a planar graph and let t be reachable fromsing. Assume there exists a largestq such that f(q) is located on an edge sv. Assume f(q) s,v. When starting in s, face exploration according to the rule and first edge determined by Alg. 2 will always encounter an edge e which satisfies: there exists a q with q < q 1 such that f(q ) is located on e. t Proof: Assume for the sake of contradiction that no such intersection exists. Consider the case that is initially located left of sv for x > q (see Fig. 4a). With Alg. 2 face traversal will follow sv with right hand rule or sw with left hand rule. In both cases, the same face F is traversed. Moreover, is initially located inside F for x > q. When the boundary of the traversed face does not intersect the curve for x > p, the curve is located completely inside F for p < x 1. This follows immediately from the fact that is continuous. If we consider the case that is initially located right of sv for x > q (see Fig. 4b), it follows the same way that a single face F is traversed and that the curve is located completely in F for p < x 1. With t = f(1) we get for both cases that t is located inside the traversed face F. With Lemma 2 in [6] follows that there exists no path in G which connects t with a node on the boundary of F. Thus, t is not reachable from s, since s is on the boundary of F, a contradiction. Lemma 5: Let G be a planar graph and let t be reachable from s in G. Assume there exists a largest q such that f(q) is located on an edge sv. Assume f(q) = s. When starting in s, face exploration according to the rule and first edge determined by Alg. 3 will always encounter an edge e which satisfies: there exists a q with q < q 1 such that f(q ) is located on e. Proof: Assume for the sake of contradiction that no such intersection exists. For x > q, the initial part of is located inside the cone of the outgoing edges selected by Alg. 3. Let F be the face traversed according to the outgoing edge and rule determined by Alg. 3. When the boundary of the traversed face does not intersect the curve for x > p, the curve is located completely inside F for p < x 1. This follows immediately from the fact that is continuous. With f(1) = t follows that t is located in the interior of F. By Lemma 2 of [6], s and t can not reach each other, a contradiction. Theorem 2: Curve-based planar graph routing has guaranteed delivery as long it is applied on planar graphs. Proof: Let G be an arbitrary planar graph. Let s and t be source and destination nodes which are reachable in G. With Theorem 1 the message is either dropped or delivered after a finite number of forwarding steps. Assume for the sake of contradiction that the message is dropped. If the last visited face was selected due to the fact that intersected the end point w of an edge vw originating at the current forwarding node v, then Alg. 1 advances to w and selects a next face at w. If the last visited facef was selected due to Alg. 2, then by Lemma 4 a further face will be selected afterf. IfF was selected due to Alg. 3, then a further face will be selected due to Lemma 5. Thus,

6 in all cases we have a contradiction since the last face was already selected. VI. APPLICATION SCENARIO AND SYSTEM DESIGN We next demonstrate the feasibility of our scheme in a concrete application scenario and describe the design and implementation of a proof-of-concept system. A. Application Scenario Curve-based planar graph routing departs from the trivial line trajectory that existing face routing schemes employ. As a result, it introduces an additional degree of freedom, which allows to either circumvent, or force the visiting of, defined geographic zones in the network. Applications we envision include (1) to mitigate interference or congestion in the wireless network, (2) to balance load within the network to achieve fairness w.r.t. utilization of resources such as energy or bandwidth, and (3) to counteract non-cooperative or misbehaving nodes by avoiding the affected zones. For the proof-of-concept of our scheme, we have chosen scenario (3), i.e., the avoidance of misbehaving nodes in the network. Related work on robust operation of classical multihop wireless networks exists and mainly concentrates on cryptographically secure routing protocols as well as schemes to increase the robustness of the forwarding of data packets (see [16] for a survey on this topic). The avoidance of insecure areas in the network using geographic approaches has been studied in [17]; the proposed scheme GeoSec introduces so-called quarantine zones, which are excluded from the routing. However, GeoSec does not use the abstraction of curves to specify routing trajectories, but has been designed as an extension to traditional reactive routing protocols for multihop networks such as AODV. Within our application scenario, we assume the network to contain a number of malicious/non-cooperative nodes. For the special case of face routing, nodes could manipulate the graph topology, or they could simply drop all packets instead of forwarding them to the next node. For our discussion, we assume that the faces which contain at least one misbehaving node are potentially malfunctioning and need to be avoided by our routing. We assume that information about the areas to avoid is available in the network, for example, by means of an intrusion detection system. B. System and Protocol Design As described, our curve-based routing approach extends the original face routing variant [1] by allowing the trajectory to be an in principle arbitrary curve connecting source and destination. The curve is defined by the source according to a given policy and included in the header of each packet. Policies are derived from the application scenario; an exhaustive definition of possible policies is out of the scope of this work. For the chosen scenario, the policy could be prohibitive, i.e., specifying to evade areas with misbehaving nodes from any network traffic, or permitting, i.e., specifying to visit areas with trustworthy nodes. Trajectories can be coded by any parametric function from source to destination. In practice, this can be simplified by approximating the curve with a sequence of straight lines and including their end points in the header. The overhead depends on how detailed the curve is represented. To implement our curve-based routing, we investigate two main design alternatives: distributed and centralized. 1) Distributed Scheme: The first design alternative is a fully distributed scheme. Since no global information about the network status is available, this approach has to use a trial and error strategy to find a working curve. We implement a scheme that deviates from the straight line by choosing curves with stepwise increasing diameter as shown in Fig. 5a. For practical implementation, the source node needs some end-to-end feedback such as an acknowledgment (ACK) to know whether a packet has arrived or not. If ACKs are not received after a set interval of time, a retransmission is triggered. Note that such a packet resent on the original straight line would allow to deal with transient failures of the routing system, even without our curve-based extension. The latter is absolutely required in case of misbehaving nodes, though. Advantages of this distributed scheme are that it can operate as long as the source node itself is able to operate. Since it does not rely on any centralized components, it does not possess a single point of failure. 2) Centralized Scheme using Location Service: The second design alternative takes advantage of a central location service. Such centralized location services have been proposed for existing planar graph routing schemes to resolve the mapping between geographic coordinates and logical addresses of the destination nodes. In static networks, this information might be distributed in advance, but mobile environments require a reliable and dynamic location service 3. Access to the location service is necessary before sending a packet. We build on this and propose to piggyback information about the network status (trusted zones, insecure zones, etc.) when the service answers a location request. Thus, nodes obtain up-to-date information about challenged areas prior to sending a packet and can adapt the curve for the message accordingly, as shown in Fig. 5b. In our scheme, the location service instruments the network by collecting the network status gathered by the nodes. E.g., if a node detects a misbehaving neighbor in its direct vicinity, it sends an alert to the location service, which maintains a geo-referenced database of such information. 3) Implementation and Protocol Operation: We implemented both, the distributed and centralized approach. To allow for flexibility, we have chosen to include curve construction methods based on line segments, parabolic segments and Bézier segments for both approaches. 3 A location service can be seen as equivalent to a DNS service in an IP-based network, but translates names into geo-locations.

7 Table I: Overview on Experiments Section Experiment Objective Experiment Results VII-D Show if disallowed areas are evaded Areas are successfully avoided. Exp.(1) of traffic. A significant reduction of packet loss is observed. VII-E Compare the efficiency of distributed Centralized scheme has constant overhead, while overhead of Exp.(2) vs. centralized approach to define curve. distributed scheme increases with # of malicious nodes. For small # of malicious nodes, distributed scheme is more efficient. S (a) Distributed scheme. D S (b) Centralized scheme. Figure 5: Examples for curved paths. For the distributed scheme, the source node initiates the routing process using the direct line to the destination, it sends the packet and starts a timer. If it fails to receive an acknowledgment for a given timeout value, the source calculates a curved path with a predefined step size that marks the distance of the parabola from the center of the reference line. We alternate the curves from the left to the right of the reference line and increase the step size linearly, if still unsuccessful. Thus we explore alternate paths through the network (see Fig. 5a). For the centralized scheme, the source queries the location service and obtains directly the disallowed areas. For our proof-of-concept implementation, we chose a basic variant to implement this scheme. The disallowed areas are distributed arbitrarily in the simulation area and, thus, might overlap. In a first step, our algorithm aggregates all areas that overlap, which results in an independent set of area groups. In a second step, for the trajectory calculation, all area groups intersecting the straight line between both nodes are determined. Next, we calculate the convex hulls enclosing these area groups. Last, the trajectory is planned as straight line segments, which border the convex hulls. The design and evaluation of more advanced combinations of calculating curves under realistic assumptions is highly scenario dependent, and thus outside the scope of this paper. VII. PERFORMANCE EVALUATION In this section, by means of simulation, we analyze our curve-based routing scheme. A. Goal of the Evaluation The first goal of our study is to demonstrate the correct working of curve-based routing and its effectiveness in counteracting non-cooperative nodes in the network. The second goal of our study is to analyze the efficiency of our scheme and its control overhead by comparing the D distributed vs. the centralized approach. Table I shows the corresponding two sets of experiments carried out, and gives a summary of our findings. B. Evaluation Technique and System under Test We perform a simulation study to be able to investigate sufficiently large networks. We have extended the NetSim 2 network simulator developed by Leong et al. 4 In particular, we implemented curve-based routing as described in Sec. VI as well as actual node misbehavior and misbehavior detection/reaction. Benign nodes adhere to the protocol at all times, while misbehaving nodes are dropping packets. C. Experimental Design, Metrics, Parameters, and Factors The effectiveness of our scheme can be measured using the degree by which our scheme is able to bypass malicious areas. As an application level metric, the packet loss ratio, which is defined as the number of successfully received packets divided by the number of sent packets, reflects this. Parameters to be studied are the number of attackers, type of routing protocol (face routing vs. curve-based routing) and the different modes of operation of curve-based routing. To measure the efficiency of the distributed vs. the centralized scheme, we measure the instantaneous overhead of both schemes. We measure overhead by counting how many tries in average have to be performed, until a packet is successfully received. We treat each packet independently and do not apply any reference-line caching. For the case of the approach using a location service, we count the querying of the location service as one try, but we omit the overhead to update the location service. For the distributed scheme, the overhead is measured by the average number of packets that need to be generated until a feasible trajectory is found. Table II gives an overview on all important parameters and factors of the simulation. We present the results for the selected network size of 150 nodes, since our results for different network sizes and network densities did not yield significant qualitative differences. We generate packets with an average packet generation period of 1 packet per second between random sourcedestination pairs. We kept the workload as simple as possible, since more complex workload does not yield any 4 NetSim 2 has been specifically developed for face routing. It abstracts from the medium access and physical layers to focus on routing. It is available at

8 Table II: Simulation Parameters Parameter Factors # of nodes 150 Node placement Randomly (uniform distribution) Simulation area units Simulation time 100 seconds Planarization method Gabriel Graph Routing scheme {Face, curve-based} Curve-based variants {Centralized, distributed} step size in units {25, 50, 100} (distributed only) # of attackers {1, 2,..., 20} Attacker placement Randomly (uniform distribution) Workload 1 pkt/s, random src-dst pairs # of iterations 40 in Exp. (1), 80 in Exp. (2) Loss ratio Comparison with and without curved-based routing With curved paths Without curved paths Number of attackers / bad areas Figure 6: Effectiveness. Exp. (1), loss ratio. additional insights for our analysis. We perform a sufficiently large number of iterations to obtain statistically significant results. For each iteration, we generate a new random topology. We show the 95% confidence levels for all results. D. Analysis of Effectiveness and Correct Operation Fig. 6 shows the results of the first set of experiments. As expected, the loss ratio using curve-based routing is significantly lower than using plain face routing. Up to five malicious nodes, the loss rate is close to zero (equal to zero for the majority of replications), but for a higher number of malicious nodes, we observe non-zero loss rates. These can be explained by network partitioning. Since entire faces have to be evaded from traffic, even a relatively small number of attackers might isolate regions of the network. If source and destination are not in the same network partition, loss occurs. We conclude that curve-based routing is effective. An interesting observation is the non-linear and asymptotic increase in loss ratio for face routing. By analyzing the packet traces, we could attribute this asymptotic behavior to the fact that the contribution of additional misbehaving nodes gets smaller with each node. For few attackers, the probability of intersecting a source-destination line is roughly the same for each additional attacker (nearly linear increase in Fig. 6). For many attackers, covered areas start to overlap and multiple attackers will intersect the same lines, thus the contribution of each additional attacker gets smaller and smaller. Number of tries Distributed approach Distributed approach (Step 25) Distributed approach (Step 50) Distributed approach (Step 100) Number of attackers / bad areas Figure 7: Efficiency. Exp. (2), overhead (tries until success). E. Analysis of Efficiency and Control Overhead The results of the second set of experiments are shown in Fig. 7. The overhead of the distributed approach increases with the number of attackers in the network, since it gets more difficult to find a valid path if more disallowed areas exist. Yet, for small numbers of attackers, the distributed approach can outperform the centralized one, if no query to the location service is required. In contrast, the centralized approach carries a constant overhead, which stems from a query to the location server. If the location server information is accurate, the first try is going to be successful. We can conclude that for a scenario with only a limited amount of disallowed areas, both approaches are feasible and perform well. The bigger the number of attackers, the bigger the potential benefit of the centralized approach. For different step sizes, the overhead of the distributed scheme varies significantly. For a small step size, a higher amount of tries can be observed, which is due to the fact that multiple tries are required, until areas can be circumvented. F. Summary and Discussion In summary, our simulations show that curve-based routing is feasible. It can significantly improve the performance of planar graph routing in challenging environments. This has been shown for the case of misbehaving nodes, where a remarkable reduction of the loss ratio could be observed. Both suggested practical variants of curve-based routing work according to specification, yet they have pros and cons. The distributed scheme can outperform the centralized w.r.t. to overhead in scenarios with very few attackers and can be implemented in a fully localized fashion. It can further benefit from existing caching techniques to reduce the number of location queries by estimating the position of the destination based on previous message exchange. For large number of attackers, it reaches its limits, though, if the trial-and-error strategy cannot deal with the environment.

9 The centralized scheme has a moderate to low, but constant overhead in terms of number of control messages. This overhead is negligible for routing protocols that access the location service before sending each packet, since the network state can be piggybacked to the reply. In general, it will also yield a smaller delay, since it can directly calculate a valid path. Most important, it is independent from the number of bad areas in the network. The optimal design of the protocol and curve parameters depends on the scenario, and is subject of future work. VIII. CONCLUSIONS We have motivated and presented the curve-based routing paradigm, which is an enhancement to traditional planar graph routing. Curve-based routing enables source nodes to define an arbitrary curve in order to influence the path followed by packets to reach the destination, while traditional schemes utilize straight lines as trajectories, only. We have shown by mathematical proof that the curve-based routing scheme is loop free and guarantees delivery of packets in a planar graph. To the best of our knowledge, this work marks the first technical proof of delivery guarantees for a curve-based face routing variant. We have implemented curve-based routing with two curve-selection variants, and have shown its feasibility by means of a simulation study. As a proof-of-concept scenario, we have investigated the case of non-cooperating nodes. Yet, our scheme is similarly applicable to deal with other realworld constraints such as network congestion or limited node energy levels. Our results show that curve-based routing is able to sustain the delivery of packets where traditional schemes fail. The investigated variants have pros and cons, which need to be traded-off depending on the scenario. 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