Deterministic Allocation by Oriented Edge Coloring for Wireless Sensor Networks

Deterministic Allocation by Oriented Edge Coloring for Wireless Sensor Networks Lilia Lassouaoui 1, Stephane Rovedakis 1, Linqing Gui 2, Anne Wei 1 1 CEDRIC Laboratory, Conservatoire National des Arts et Métiers, 292 rue Saint-Martin, Paris, France {lilia.lassouaoui, stephane.rovedakis, anne.wei}@cnam.fr 2 Department of Electronic and Optical Engineering, Nanjing University of Science and Technology, 200 Xiaolingwei Street, Nanjing, China guilinqing@gmail.com Abstract: In wireless sensor networks, network lifetime is among the most important criteria. Network lifetime mainly depends on the link scheduling established at the Medium Access Control layer. Indeed, the avoidance of transmission conflicts enable energy savings since there are no message retransmissions. We are interested in deterministic allocation of the wireless medium for data collection in tree based sensor networks. In this paper, we consider a generalization of the distance 2-edge coloring problem, in which transmission and interference edges are taken into account. We propose a distributed algorithm for this problem, called D2EC, which ensures that conflicts are avoided. We also carry out simulations to compare D2EC with a random allocation strategy of the wireless medium. The simulation results show that there is a significant reduction on packet loss by using D2EC. Moreover, D2EC extends the lifetime of 250% in the best case regarding the random allocation of the wireless medium. Keywords Wireless Sensor Networks, deterministic link scheduling, distance-2 edge coloring. I. INTRODUCTION In the last decade, a large number of applications, such as security surveillance, health care, or monitoring, have emerged thanks to the deployment of sensor networks. One of the main tasks that a sensor network has to fulfill is the collection of sensed data at a particular node, called sink node. The data collected by the sink are analyzed in order to make decisions, e.g., fire detection in a forest or animal tracking [1]. Most of the time, sensor nodes are battery powered and it is either difficult or impossible to recharge batteries of nodes since they can be spread in inaccessible areas. Therefore, an important issue is to reduce the energy consumption in order to increase the network lifetime. Among actions performed by sensors, wireless communication is the most energy consuming. As a result, an effective allocation of the wireless medium allows reducing energy consumptions. Spanning tree is a well used routing structure for data collection, which avoids loops. However, in wireless networks message collisions can arise at a node when several messages sent simultaneously by nodes in the same communication range are received. When messages collide, the messages have to be transmitted again which involve additional energy consumption. The Medium Access Control (MAC) layer is in charge to avoid collisions and to turn off the transceiver node as soon as possible to improve the energy efficiency for data transmission. A lot of works has been devoted to the design of efficient MAC protocols for multi-hop wireless networks [2,3,4,5,7]. It exists three categories of MAC protocols: (a) contention based protocols, such as CSMA/CA (Carrier Sense Multiple Access with collision avoidance), (b) schedule based protocols, such as TDMA (Time Division Media Access), and (c) hybrid protocols. A TDMA approach achieves high energy efficiency since the transceiver of a node is switched off when it is neither in transmission nor in receiving mode. In this case, a TDMA protocol computes an allocation of time slots to sensor nodes, by taking into account possible communication interferences in the network. We consider that each sensor node is equipped with a half-duplex transceiver using a single frequency. Two concurrent transmissions are interfering with each other if and only if (cf. Figure 1): (i) they have the same receiver, (ii) the transmitter of one transmission is also the receiver of the other transmission, or (iii) at least one of the receivers is within the communication range of the non-intended transmitter. Interferences of types (i) and (ii) are known as primary conflicts, while interferences of type (iii) are secondary conflicts [6]. To avoid collisions, two interfering transmissions cannot occur during the same time slot. An assignment of time slots to links is referred to as link scheduling. Figure 1: primary and secondary conflicts. In order to increase the network lifetime, we are interesting in TDMA approach. In this context, such a resource allocation can be modeled as a vertex or edge coloring problem, where each color corresponds to a unique time slot. A solution for the minimum vertex (or edge) coloring problem is to find a color allocation to vertices (or edges) of minimum cardinality such that two neighboring vertices (or edges) have distinct colors. It is known that the vertex and edge coloring problems are NP-complete to solve for the class of general graphs [9]. A solution to the minimum edge coloring problem cannot avoid

all the possible collisions in the network, since only primary conflicts are taken into account. Indeed, an edge coloring can allocate the same color to two distinct links e 1 and e 2 which are not neighboring but are both neighboring of a third link e 3. If the common extremity node x of e 1 and e 3 is a transmitter and the common extremity node y of e 2 and e 3 is a receiver, then x corresponds to a non-intended transmitter for y considered in secondary conflicts. In such a case a collision occurs at node y. To take into account primary and secondary conflicts, it is necessary to compute distance-2 edge coloring. It means that every pair of edges with neighboring extremity nodes has distinct colors [10]. In the example described above, edges e 1, e 2 and e 3 form a path of length three and distinct colors are allocated. Faudree and Schelp [11] show that every arbitrary graph with maximum degree has a distance-2 edge coloring that uses at most 2 2-2 +1 colors, and it can be computed in polynomial time. Moreover, for the class of tree graphs a distance-2 edge coloring uses 2-1 colors. Our contributions. We propose a distributed algorithm to establish a deterministic allocation of wireless medium by computing a distance-2 edge coloring. We perform simulation to evaluate our algorithm with random allocation strategy. The obtain results show that our approach allows a significant reduction of the packet loss and an increase of the network life time. The reminder of the paper is organized as follows. In Section II, we introduce some related work on edge coloring. Section III defines the model considered in this paper. We present the oriented edge coloring with conflicts problem and our algorithm D2EC in Section VI. Simulation results are presented in Section V. Finally, we conclude the paper in last section. II. RELATED WORK A high number of MAC allocation algorithms have been proposed to handle the characteristics of wireless sensor networks. Some related works joint graph coloring and TDMA scheduling in wireless multi-hop networks. To efficiently use the time slots, vertex coloring has been used for node scheduling in [12]. The objective is to enable the maximum nodes in transmission mode without generating communication interferences in each time slot. Edge coloring problems have been used for link scheduling in wireless networks. Gandham et al. proposed a distributed distance-2 edge coloring algorithm based on links orientation (transmitter or receiver) using at most 2 ( + 1) time slots [8]. An edge coloring is computed in a first phase. Then, in a second phase a mapping of each color to a unique time slot is performed based on an edge orientation following a depth first traversal of the network. The algorithm proposed in [8] takes into account the hidden and exposed terminal problems, but it requires an even number of edges in any cycle. The best distributed approximation algorithm for edge coloring is obtained by Grable et al. in [14], which generates a valid edge coloring with using (1+ε) colors in O(polylog(n)) time. Herman et al. [15] expanded the approach of Grable et al. [14] to oriented (distance-2) edge coloring. In their distributed algorithm, an edge coloring is computed in a first step by using the algorithm of Grable et al. [14], and then edges are randomly re-colored with probability p. Finally, an orientation is assigned to each edge to obtain an oriented (distance-2) edge coloring. Compared with [8], Herman et al. improved the time complexity from linear to O(polylog(n)), but the problem with odd cycles remains. In [13] the authors proposed a centralized algorithm to assign time slots to directional links by coloring the edges of a directed graph. The proposed algorithm colors the edges selected following a Breadth First traversal of the network and considering interfering edges at distance 2. It tries to assign the smallest free color to non interfering edges. The problem with odd cycles is solved by the approach used to compute an edge coloring. All the works mentioned above color all the edges of the network, and thus allocate them a time slot, even if they are not used for data transmission. Ghosh et al. are interested in the minimization of the maximum delay for data collection in tree-based sensor networks [6]. They proposed a centralized approach to compute a Multi-Channel scheduling. Secondary conflicts and a part of primary conflicts are taken into account by assigning different frequencies, while the other primary conflicts are avoided using distinct time slots. They show that multiple frequencies can be used to give approximation guarantees on the minimization of the link scheduling problem. The frequencies and slots assignment can be considered as a distance-2 edge coloring, however contrary to previous approaches only the tree edges are colored, reducing the link schedule. However, the approach given in [6] is a centralized algorithm, which cannot be easily spread out on a real sensor networks. Contrary to the existing works, we propose a deterministic algorithm for oriented (distance-2) edge coloring problem with conflicts. Our algorithm computes a solution to this problem in a distributed manner (only information at distance two are needed). III. MODEL AND ASSUMPTIONS We model the sensor network as a graph G = (V, E), where V is the set of nodes and E is the set of edges that represent communication links. Each node v V is identified by a unique identifier. Let r ϵ V be the sink node and T = (V, E T ) be a directed spanning tree of G rooted at r. We assume a static network topology and each node is equipped with a single half-duplex radio that can be turned either to transmission or receiving mode. We consider an asynchronous system, i.e., nodes do not have access to a global clock. The time complexity of a distributed algorithm can be defined as the maximum time consumed by any computation. It is assumed that the processing time of an event is zero time units and the transmission time (time between sending and receiving a message) is at most one time unit. Moreover, information are exchanged using local broadcast, i.e., a message sent by each node is received by all its neighbours.

IV. ORIENTED EDGE COLORING WITH CONFLICTS In this section, we define the problem considered in this paper and we present the distributed algorithm named D2EC that we propose for this problem. A. Oriented Edge Coloring with conflicts problem The objective is to find a TDMA schedule of minimum length in which each node is activated at least once to send data. This is equivalent to computing an oriented edge coloration enabling a maximum of concurrent transmissions at each time. To this end, it is not necessary to color all the edges of the graph. Indeed, data transmissions only occur along the edges of the spanning tree, from each node to the sink node. Therefore, we must compute an oriented edge coloring only for the edges of the directed spanning tree. However, we have to take into account the conflicts between concurrent transmissions. We consider two types of edges in the graph G: communication links are the edges of the directed spanning tree T of G (i.e., E T ), and conflict links are all the edges not in T (i.e., E-E T ). Given a directed spanning tree T rooted at the sink r of a graph G, the Oriented Edge Coloring with conflicts problem is an edge coloring of T such that any two edges e 1 and e 2 of T neighbours in G have the same color if the following conditions are satisfied: (a) Either x ϵ e 1 and y ϵ e 2 are neighbours in G and both head (i.e., receiver) of e 1 and e 2 respectively, (b) or x ϵ e 1 and y ϵ e 2 are neighbours in G and both tail (i.e., transmitter) of e 1 and e 2 respectively. B. Distributed algorithm for Oriented Edge coloring with Conflicts 1) Overview of the algorithm Each node has in input the same static set of colors that is used for the edge coloring, noted C. We can define a total order of the edges of the directed tree following the distances and the identifiers of their extremity nodes, i.e., following the pairs ((d u, u), (d v, v)) with d x the distance of x to the sink r, u the head and v the tail of edge (u, v). The tree edges are colored in a top-down fashion in the tree, and we follow the total order defined above to establish a coloration priority among the interfering transmissions (or edges) at same level in the spanning tree T. Each node has two roles (transmitter and receiver) for data collection, except the sink and the leaves of T which are only receiver and transmitters respectively. In each role, each node cannot use the same color. Therefore, each node maintains a set of authorized colors for each role, i.e., colors which are not used by interfering edges of highest priority for each role. When a new tree edge has been colored, the assigned color is broadcasted in its neighbourhood. This information is used by the nodes of interfering edges with a lowest priority to update their sets of authorized colors. 2) Detailed description In this subsection, we give more details on Algorithm D2EC executed by every node of the network; a formal description is given in Figures 2 and 3. Each node p ϵ V has input information: its neighbors in the network (Neig p ), its parent and children in the spanning tree T (par p and Child p respectively) and its distance in T to the sink (dist p ). Three types of messages are used by our algorithm: <PropCE, q, ece> is used by a node q to send to its parent its set of authorized colors ece in transmission mode, <Diff, q, ecr> is used by each node q to send the ecr set containing the colors in transmission mode for its children, and <DiffAck, q, ce> is used by a node q to inform its neighbors that the color ce has been assigned for transmissions with its parent. Each parent (receiver node) p is waiting for the message PropCE from its children q with their authorized colors in transmission mode. Each child q can send a message PropCE when it has received a message Diff from all its priority neighbors given by PrioE(q). When receiving a message Diff, a node q updates its ace q set by removing the colors contained in the message. By using the edge priority (defined by PrioE(p) and PrioR(p) at p), D2EC ensure a determinist oriented (distance-2) edge coloring, avoiding the hidden and exposed terminal problems. Figure 2: First part of Algorithm D2EC.

V. EVALUATION BETWEEN D2EC AND RANDOM ALLOCATION In this section, we present simulation results to evalute the path loss performance and energy efficiency of Algorithm D2EC in compraison with a random allocation strategy. In the random allocation that we consider each color has the same probability to be allocated for each transmission in the network. According to [16], in a wireless network the smallest number of possible colors follows O( ), with the maximum degree in the network. Table 2: parameters of algorithm. Symbol of Physical meaning of parameter parameter n Total number of sensor nodes Maxium degree Total number of levels in the network num_col Total number of colors We have considered four different scenarios in our simulation to evaluate and to compare the performance of algorithms. These scenarios are defined by the value of the simulation parameters listed in the following table. Table 3: value for parameters in each scenario. num_col n Scenario 1 3 10 Scenario 2 2 4 4 13 Scenario 3 3 3 4 22 Scenario 4 3 4 4 29 In a first step, we consider network topologies without conflict links (i.e., tree networks). This is a subset of topologies in which solutions for oriented edge coloring and oriented edge coloring with conflicts are identical. In this case, according to Gandham et al. [8] +1 colors are needed. Figure 4 shows the topologies considered in each scenario. (a) Scenario 1 (b) Scenario 2 Figure 3: Second part of Algorithm D2EC. A. Scenarios of Simulation The main parameters used for our simulations of Algorithm D2EC and the random allocation strategy are listed in the following table. (c) Scenario 3 (d) Scenario 4 Figure 4: Network topology of each scenario. B. Evaluation based on Simulation Results Upon the above four scenarios, we simulate both Algorithm D2EC and the random allocation strategy and considering three metrics are used for performance evaluation: average end-toend delay, average packet loss rate, and network lifetime. Here, end-to-end delay is the delay of data transmission from every node to the sink, so average end-to-end delay is the average of all end-to-end transmissions. The average packet loss rate is calculated as the number of collided packets divided by the number of all packets sent in the network. Network lifetime is inversely proportional to the energy used for transmitting all packets. Note that for comparison convenience the network lifetime of D2EC in the first scenario is normalized to 1. Simulation results are shown in Figure 5, 6 and 7. From simulation results, we can find that D2EC algorithm has much better performance than the random allocation strategy.

average end-to-end delay average packet loss rate network lifetime (normalized) 20 15 10 4 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 D2EC, scenario 2 D2EC, scenario 1 Figure 5: Simulation results of average end-to-end delay. &4 Figure 6: Simulation results of average packet loss rate. D2EC, scenario 1&2 D2EC, scenario 1 D2EC, scenario 2 D2EC, scenario 3&4 D2EC, scenario 3 D2EC, scenario 4 Figure 7: Simulation results of network lifetime. As shown in Figure 5, the average end-to-end delay achieved by D2EC in networks with two levels is only 4 slots (in Scenario 1) and 5.17 slots (in Scenario 2), while the delay of the random allocation is 9.22 slots (in Scenario 1) and 10.42 slots (in Scenario 2). The difference on delay performance between D2EC and the random allocation is more obvious in Scenarios 3 and 4 when the network has three levels. The average end-to-end delay increases with the number of levels, since it takes more time to fulfill the end-to-end transmission from the lowest level to the sink. This indicates that when the number of levels in tree networks increases D2EC algorithm has a more obvious advantage on end-to-end delay than the random allocation. As shown in Figure 6, the average packet loss rate of D2EC is equal to zero since the computed oriented edge coloring is collision free. Although the random allocation is simpler than D2EC, it has a much higher packet loss rate because of collisions. When the network has two levels, the average packet loss rate of the random allocation is as high as 46.7% (in Scenario 1) and 54.2% (in Scenario 2). The loss rate increases with the number of levels since when colors are randomly allocated there will be a higher probability for two D2EC, scenario 4 D2EC, scenario 3 nodes to have the same color. As shown in Figure 7, the network lifetime obtained with D2EC is higher in every scenario (extended up to 250% in Scenario 1 compared with the random allocation). VI. CONCLUSION AND FUTURE WORK In this paper, we consider a variation of the oriented edge coloring problem to perform the maximum number of concurrent non interfering transmissions. We propose a distributed algorithm D2EC for this problem. Moreover, we evaluate the performances of our algorithm in comparison with a random allocation strategy through simulations. The obtained results show that D2EC reduces the packet loss considerably. Moreover, D2EC allows to extend the network lifetime up to 250%. Our future work focus on fault-tolerance guarantees, as the fault-tolerance is also an important problem related in wireless sensor networks. VII. REFERENCES [1] P. Rawat, K.D. Singh, H. Chaouchi, J.M. 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