Time-efficient Algorithms for the Outdegree Limited Bluetooth Scatternet Formation Problem

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1 Time-efficient Algorithms for the Outdegree Limited Bluetooth Scatternet Formation Problem Ahmed Jedda, Guy-Vincent Jourdan and Hussein T. Mouftah School of Electrical Engineering and Computer Science University of Ottawa {ajedd077, gvj, Abstract We present in this paper three Bluetooth Scatternet Formation (BSF) algorithms which forms deterministically connected and outdegree limited scatternets. Our algorithms improve the execution time of algorithm BlueMIS, which consists of two consecutive phases called BlueMIS I and BlueMIS II. The first phase, BlueMIS I, is a BSF algorithm that forms connected and outdegree limited scatternets in a time-efficient manner. Our algorithms improve BlueMIS by improving BlueMIS I. First, we introduce a time-efficient implementation of communication rounds, called OrderedExchange, that is more suitable for Bluetooth networks, where in a communication round each node sends and receives a message to and from all its neighbors. Using OrderedExchange, we introduce algorithms ComputeMIS I and ComputeMIS II which form the same scatternet as those formed by BlueMIS I but with less execution time. We also introduce algorithm Eliminate. Instead of letting each node u forms a maximal independent set of all its neighbors and then considers them as slaves, as in the case of BlueMIS I, Eliminate let each node u forms a maximal independent set of all its neighbors that have smaller identifiers. Eliminate forms scatternets that are similiar to those formed by BlueMIS I, and more efficient with respect to some performance metrics. Eliminate improves the execution time of ComputeMIS I and ComputeMIS II by about 60%. I. INTRODUCTION Bluetooth is a short-range wireless technology that can be considered as a strong enabler of wireless personal area networks. The technology offers devices that have low cost, low energy consumption, high robustness to interference and noise, and ad hoc networking capabilities. Moreover, Bluetooth devices are widely available in today s laptops, cell phones and tablets. These features and others motivate our work on Bluetooth. According to the Bluetooth specifications, a device can be either master or slave. A node that is a master to one or more slaves forms a star topology called a piconet. The communication of any two nodes in the piconet must be via the master. The master uses an intra-piconet scheduling algorithm to control the flow of packets in the piconets. A scatternet is an interconnection of more than one piconet through bridge nodes, where a bridge is either an M/S bridge (that is, a node that is master to one piconet and slave to one or more other piconets) or an S/S bridge (that is, a node that is slave to more than one piconet). The way a scatternet is formed affects its performance, energy consumption and throughput. There are some criteria that must be considered when forming a scatternet. For instance, it is preferred to have piconets with few slaves (at most 7) since a master can keep only 7 of its slaves active. The other slaves would be parked (that is, inactivated). This imposes a penalty on the scatternet performance. It is also preferred that a node have few roles, where a node belonging to k piconets has k roles and has to share its time among all these piconets. It is preferred thus to decrease the average role per node in the scatternet. Moreover, the number of masters (piconets) in the scatternet is preferred to be as minimum as possible, since masters consume more energy. We focus in this paper on two performance metrics, which are the connectivity and the size of piconets. We aim at introducing fast algorithms (in term of execution time) to form connected and outdegree limited scatternets deterministically. We say a scatternet is connected if there is at least one path for each pair of nodes in the scatternet. Such path may consist of masters and slaves. We say a scatternet is outdegree limited if each of its piconets has at most 7 slaves. Our work is motivated by the BlueMIS algorithm [9]. The approach followed in [9] is to form in a time-efficient manner scatternets that do not necessarily have good properties (in term of other performance metrics). The scatternets are later improved with simple local rules. BlueMIS consists of two phases; BlueMIS I and BlueMIS II. The idea of BlueMIS I is to let each node u constructs a maximal independent set MIS(u) of its neighbors; where a subset of the neighbors of u is independent if no pair of nodes in are neighbors, and it is maximal if there it was not a subset of any other independent set. A node u considers each neighbor v in MIS(v) as a slave except if u v and v MIS(u), where comparison between nodes is in term of identifiers. BlueMIS II improves BlueMIS I scatternets by implying simple local rules. We should not however that the description of BlueMIS II lacks details of implementation that may significantly change the behavior of the algorithm. Thus, we focus mostly on BlueMIS I in this paper. We present in this paper three time-efficient algorithms that form connected outdegree limited scatternets. The algorithms improve the execution time of BlueMIS I. The first two, ComputeMIS I and ComputeMIS II, form the same scatternets that BlueMIS I form but with an improved execution time. The third algorithm, Eliminate, form scatternets that outperform those formed by BlueMIS I in term of average role per node and number of SS bridges, and has an execution time /12/$ IEEE

2 that is about 0.6 the execution time of ComputeMIS I and ComputeMIS II. Our approach is to study the link establishment procedures in Bluetooth networks, which obviously affect the execution time of distributed algorithms in Bluetooth networks. Our idea is to efficiently implement communication rounds in Bluetooth networks, where in a communication round each node in the network sends and receives a message to and from all its neighbors. Many distributed algorithms in the literature consists of such communication rounds [5], among them is BlueMIS I. We study two algorithms, RandomExchange and OrderedExchange, that implement communication rounds in Bluetooth networks. The challenge of these algorithms is mainly due to the inability of a Bluetooth node to broadcast messages to its neighbors. RandomExchange has O( ) time complexity whereas OrderedExchange run in O(n), where is the maximum degree of the graph. Simulation experiments however show that OrderedExchange outperforms RandomExchange with respect to time in relatively small networks (110 nodes and less). This is a practical size for Bluetooth networks and personal area networks in general. Thus, we use these results to design time-efficient distributed algorithms for Bluetooth networks. This paper is organized as follows; Section II gives a brief description of the link establishment procedures in Bluetooth. Section V studies two different algorithms to implement communication rounds in Bluetooth networks which are OrderedExchange and RandomExchange. Section VI introduces a number of time-efficient algorithms to form connected outdegree limited Bluetooth Scatternet Formation deterministically. II. BLUETOOTH BASICS Bluetooth technology is a wireless technology that uses the ISM band from MHz divided into 79 channels (1 MHz each). Bluetooth devices use the Frequency Hopping Spread Spectrum (FHSS) technique for communication. A pair of nodes communicating with each other alternates between a set of pseudo-random frequency channels known to both nodes. During this alternation, the nodes exchange their messages. Bluetooth is a connection-oriented communication standard. That is, any two communication nodes must build a link before communicating. For a node u to establish a link with a neighbor v, node u switches to a state called PAGE while node v switches to a state called PAGE SCAN. Node u sends packets specifically designated to v in different channels. Node v on the other hand alternate between a set of frequency channels in the PAGE SCAN state and in case it received a packet from u, then both nodes exchange some packet in order to terminate the procedure. Such a link represent a piconet of a master (node u in this example) and one slave (node v in this example). The link establishment should be preceded by a device discovery procedure, which we omit its description in this paper. For any pair of Bluetooth devices to communicate with each other, they both need to be in the same scatternet or the same piconet. Given the nonavailability of scatternet before a BSF algorithm is executed, most BSF algorithms uses the following technique to exchange messages in order to build the scatternets. If a node u needs to send a message to a neighbor v, node u builds a temporary piconet with v. Both nodes exchange the desired messages and then destroy the piconet. Forwarding technique is used if a node to send a message to a non-neighbor node (that is, one node forward the message to another until the required node receives the message). III. PROBLEM DEFINITION AND ASSUMPTIONS Formally, the problem can be defined as follows: Given a network of Bluetooth nodes G(V, E), where V is the set of nodes and E is the set of edges, such that a pair of nodes u and v share the edges (u, v) and (v, u) if they are able to communicate with each other, we need to find a scatternet S(V, E ) such that E E. Scatternet S(V, E ) must be directed such that if (u, v) E then (u, v) E and (v, u) / E. We assume that the input network G(V, E) is a unit disk graph. A unit disk graph consists of points on a plane that represents the vertices of a graph. An edge is assigned between two nodes if the distance between them is less than a threshold t. All algorithms in the literature that solve our problem assume this assumption. An interesting property in unit disk graph is that any node can have at most 5 neighbors that are not neighbors to each other. We assume no position knowledge available for nodes, neither the knowledge of distance between nodes. We assume that each node u has a unique identifier id(u). Each node u knows the identities of all its neighbors denoted as N(u). This is realized by the device discovery phase which we do not consider in this paper. A node u is said to be larger than a node v (denoted as u v if id(u) > id(v)). The set of larger neighbors of a node u are denoted as N l (u). The set of smaller neighbors are denoted as N s (u). Each node u keeps track of its masters M(u) and its slaves S(u). IV. LITERATURE SURVEY Some of the algorithms that solve our problem with the same set of assumptions are, BlueMesh [6], BlueTrees [10] and SHAPER [1]. BlueMesh [6] uses a clustering technique to form interconnected outdegree limited scatternets. BlueMesh is considered as one of the best BSF algorithms. SHAPER [1] builds a rooted-tree-like scatternet. BlueTrees also builds rooted-tree-like scatternets, but it assumes a node is elected a leader in-priori. BlueMesh, BlueTrees and SHAPER suffer all from a large execution time. Our aim is to improve this performance metric. The problem was solved under different set of assumptions as well. BluePleidas [2] uses networks with maximum degree of 7. These degree limited networks are generated using a probabilistic neighbors discovery algorithm that was also introduced in [2]. BlueNet [8] also uses probabilistic techniques to solve the problem. Both BluePleidas and BlueNet does not guarantee the connectivity of the scatternets. Li et al. [4] use /12/$ IEEE

3 computational geometry techniques (namely, Yao graph) in order to form connected and outdegree limited scatternets. The main issue in their algorithm is that they assume that nodes are aware of their position. V. THE APPROACH: EFFICIENT IMPLEMENTATIONS OF COMMUNICATIONS ROUNDS IN BLUETOOTH NETWORKS Many distributed algorithms in the literature consist of communications rounds. In a communication rounds, each node in the network sends and receives a message to and from all its neighbors. In this section, we study efficient implementations of communication rounds in Bluetooth networks, namely RandomExchange and OrderedExchange. We give a brief idea of these algorithms in the follow. A detailed study can be found in [3]. The main difficulty that challenges RandomExchange and OrderedExchange is that 1) a Bluetooth node cannot broadcast a message to all its neighbors, and 2) if a node u needs to establish a link with a neighbor v then u must be in the PAGE state while v is in the PAGE SCAN state. The problem can be defined as follows: Given a network G(V, E) with V and E are the set of nodes and edges respectively, we need to let each edge (u, v) E be contacted by one message. We assume, except otherwise mentioned, that the edges (u, v) and (v, u) are equivalent. RandomExchange and OrderedExchange can be easily modified if this assumption is relaxed. In RandomExchange, each node u alternates randomly between the PAGE and PAGE SCAN states. The time a node resides in each of the states is drawn uniformly in random. When a node u is in the PAGE state, then it attempts to send a message to a neighbor v that is not yet contacted. If the operation succeed, then v is considered contacted. When u is in PAGE SCAN state, node u waits to receive messages from neighbors that are not yet contacted. It may happen, in a worst case scenario, that two nodes have the same alternation sequence between PAGE and PAGE SCAN state indefinitely, and whence the algorithm never terminates. This case, which we call deadlock, happens with very low probability. Assuming no deadlocks and every operation is successful, RandomExchange terminates and has a O( ) time complexity where is the maximum degree of the graph G. OrderedExchange, on the other hand, does not suffer of deadlocks and terminates deterministically in a finite time. The algorithm assumes that all nodes have unique identifiers. We define the set of larger neighbors of a node as the set of neighbors with larger identifier. We similarly define the set of smaller neighbors. In OrderedExchange, every node waits to receive a message from all its larger neighbors. Upon the receipt of messages from all the larger neighbors, a node sends a message to all its smaller neighbors. A node that have no larger neighbors starts directly sending messages to its smaller neighbors. Note that each edge (u, v) E such that u v (that is, the identifier of u is larger than v s) is contacted. This proves the correctness of the algorithm. If (u, v) was not equivalent to (v, u) then another round of communication is required. In the second round, a node waits to receive a messages from all its smaller neighbors and sends a message to its larger neighbors. We call a OrderedExchange communication round in which a node waits for its larger neighbors (resp. smaller neighbors) as a descending (resp. ascending) round. The pseudocode of OrderedExchange is given in Algorithm 1. Algorithm 1 OrderedExchange at node u (descending) 1: N l (u) {v : v is a larger neighbor of u } 2: N s (u) {v : v is a smaller neighbor of u } 3: while N l do 4: Upon reception of a message from neighbor v N l (u), N l (u) {N l (u) v} 5: end while 6: for each v N s (u) do 7: Send a message to v 8: end for A. Examples Due to its importance for the understanding of this paper, we give in the following an example to explain the execution of OrderedExchange. Consider the graph G = (V, E) such that V = {v 1, v 2, v 3, v 4 } and E = {(v 1, v 2 ), (v 1, v 3 ), (v 2, v 4 ), (v 3, v 4 )}. Assume that the order of the nodes is v 1 v 2 v 3 v 4. Note that the only node that has no larger neighbors is v 1. According to OrderedExchange, the nodes v 2, v 3 and v 4 wait for their larger neighbors to send them messages, while node v 1 starts sending messages to all its smaller neighbors. In our example, node v 1 sends a message to v 2 and v 3. Node v 1 may send a message to v 2 before v 3 or it may follow the other order. Sending a message to the next largest neighbors gives a better execution time of OrderedExchange. We discussed this in more details in [3]. Upon the reception of the message of v 1 by v 2 and v 3, these nodes send a message to v 4 which is the only smaller neighbor to both nodes. This terminates the execution of one round of OrderedExchange. B. Simulation of RandomExchange and OrderedExchange We compare the average execution time of RandomExchange against OrderedExchange. We optimize first the properties of both algorithms for them to reach their optimal performance. RandomExchange is mainly affected by the length of the period of time a node reside in either PAGE or PAGE SCAN, whereas OrderedExchange is affected by the order a node sends message to its neighbors. For instance, let s assume a node v 1 has neighbors v 2, v 3 and v 4 where v 1 v 2 v 3 v 4. Node v 1 can contacts its neighbors following different orders such v 2, then v 3, v 4 or it may follow the order v 4, v 3, v 2 or it may follow any other order. However, the execution time varies with the order used. The worst case time complexity of OrderedExchange is O(n) but may reach O(n 2 ) if an unsuitable order of contacting the neighbors was followed /12/$ IEEE

4 TABLE I: Percentage of improvement of OrderedExchange on RandomExchange. number of nodes /degree Note that RandomExchange execution time depends on the average nodal degree whereas OrderedExchange depends on the number of nodes. Our experiments are conducted over networks in which we control their number of nodes and average nodal degree. We use UCBT [7] as simulator, which is an NS-2 based simulator. All the networks we considered are connected unit disk graphs, which are constructed by placing points in random positions in an area of and setting an edge between any pair of points if the euclidean distance between them is less or equal to 10. We give in Table I the ratio 100 (T w T r )/T w, where T w and T r are the average execution times of OrderedExchange and RandomExchange. The results of Table I shows that OrderedExchange outperforms RandomExchange in all the networks we considered. However, note hat the results suggest that if we fixed the nodal degree and extrapolated the number of nodes to a very large value then RandomExchange will outperform OrderedExchange. This makes simulation results match the theoretical analysis. The result is logical since complexity analysis always consider a very large of input size. Bluetooth networks are rarely found in very large number of nodes in practice. Therefore, we say that it is more practical to use OrderedExchange as an implementation of communication rounds in Bluetooth networks. We use this result to design time-efficient Bluetooth Scatternet Formation algorithms in the Section VI. VI. ALGORITHMS FOR THE FORMATION OF CONNECTED OUTDEGREE LIMITED SCATTERNETS We study in this section BlueMIS I [9]. BlueMIS I, according to its specification, consists of one RanomdExchange communication round. We introduce then ComputeMIS I and ComputeMIS II. Both give the same results as BlueMIS I but use OrderedExchange to implement the communication rounds. Finally, we introduce algorithm Eliminate. A. BlueMIS I In BlueMIS 1, each node u constructs a maximal independent set MIS(u) of its neighbors N(u), where a MIS(u) is a set in which no pair of nodes are neighbors and is of maximal size. The construction of such sets is as follows; initially MIS(u) is empty. Node u contacts its neighbors in ascending order. That is, for each v 1 and v 2 in N(u), and v 1 v 2, then v 2 is contacted before v 1. Once a neighbor v of node u is contacted, then v is added to MIS(u) only if v is not neighbor to any other node already added in MIS(u). The node of MIS(u) are considered slaves to u (denoted as S(u)). The set of masters of u are denoted as M(u). An efficient implementation of BlueMIS 1 is given in Algorithm 2. There is a case of symmetry that must be broken which occurs if a node v is a slave to u and master to u, that is v S(u) and u S(v). To solve this problem, the larger node is considered the master of the other. Algorithm 2 BlueMIS I at node u 1: B, U N(u), S(u), M(u), where S(u) and M(u) is the set of slaves and masters of u 2: while U do 3: v min(u) 4: send a message to v, call the message attack 5: S(u) {S(u) v} 6: U {U {v N(v)}} 7: B {B {N(u) N(v)}} 8: end while 9: for each v B do 10: send a message to v, call the message dummy 11: end for 12: upon receiving of an attack message from v, M(u) {M(u) v} 13: if received messages from all neighbors, terminate. 14: at termination, locally break symmetry in M(u) and S(u) Note that each node in BlueMIS I sends and receives a message to and from all its neighbors. This shows that BlueMIS I consists of a single communication round. The description of BlueMIS I in [9] indicates that the communication round is implemented using RandomExchange. It should be noted from Algorithm 2 that in BlueMIS I the edge (u, v) is not equivalent to (v, u). Algorithm BlueMIS I forms connected outdegree limited scatternets if executed over connected unit disk graphs. The pseudocode of BlueMIS I and its proof of correctness can be found in [9]. B. ComputeMIS I: A First Version Using OrderedExchange ComputeMIS I uses the results of Section V to improve the time-efficiency of BlueMIS I in networks with practical size. The algorithm consists of two communication rounds implemented using OrderedExchange. In the first round, each node u exchanges its neighbors list N(u) with all its neighbors. Each node u locally constructs the set MIS(u). In the second round, each node u exchanges the set MIS(u) with all its neighbors. Then, each node u locally breaks the symmetries found in MIS(u) and MIS(v) for all v N(u). C. ComputeMIS II: A Second Version Using OrderedExchange ComputeMIS II is another algorithm that implements BlueMIS I using OrderedExchange. ComputeMIS II is more sophisticated than ComputeMIS I. The algorithm consists of a descending round followed by an ascending round. In the first round, each node u constructs a maximal independent set of its smaller neighbors, we call it MIS S (u). Essentially, each node u, upon the receipt of messages from all its larger /12/$ IEEE

5 neighbors, starts sending messages to its smaller neighbors by visiting them in order from largest to smaller ones. A neighbor v of u is added to MIS S (u) if an only if v is not a neighbor to any node in MIS S (u). In the second round, a node u appends the nodes of the set MIS S (u) to the empty set M IS(u). The definition of the ascending OrderedExchange round implies that node u waits to received messages from its smaller neighbors. Upon the receipt of messages from its smaller neighbors, a node u visits its larger neighbors in order from the smallest nodes to the larger ones. A neighbor v is added to MIS(u) if and only if v is not a neighbor to any node in MIS(u). Each node u knows all the neighbors v that added u to the set MIS(v). Therefore, each node can break locally the symmetries found in the maximal independent sets. The pseudocode of the algorithm is shown in Algorithm 3. Algorithm 3 ComputeMIS II at node u 1: N l (u) {v : v is a larger neighbor of u } 2: N s (u) {v : v is a smaller neighbor of u } 3: E N(u), S(u), M(u), p 0, where S(u) and M(u) is the set of slaves and masters of u. 4: while p 1 do 5: L 6: if p = 0 then 7: A N l (u), B N s (u) 8: else 9: A N s (u), B N l (u) 10: end if 11: while A do 12: wait for receipt of messages of all nodes v A 13: if message received from v is attack then 14: M(u) {M(u) v} 15: end if 16: A {A v} 17: end while 18: while B do 19: v min(e) 20: S(u) {S(u) v} 21: send a message to v, call the message attack 22: B {B {v N(v)}} 23: E {E {v N(v)}} 24: L {L {N(u) N(v)}} 25: end while 26: for each v i L do 27: send a message to v i, call the message dummy 28: end for 29: p p : end while 31: locally break symmetry found in S and M D. Algorithm Eliminate In this section, we introduce algorithm Eliminate. The algorithm is a modification on the core of BlueMIS I. In Eliminate, we let each node u constructs a maximal independent set of all its smaller neighbors. This is contrary to BlueMIS I where each node constructs a maximal independent set of all its neighbors. We observe an interesting property in ComputeMIS II that is related to the implementation of Eliminate. Let us consider the first round of ComputeMIS II, where each node u constructs a maximal independent set of all its smaller neighbors, denoted as MIS S (u). We assume that each node u consider the neighbors of MIS S (u) as its slaves. This forms a scatternet that is connected and outdegree limited given that the input network G is a connected unit disk graph. Also, note that there is no symmetries to be broken. That is, if a node u selected node v in MIS S (u) (and therefore, as a slave), then it is not possible that u belongs to MIS S (v), since u cannot be a smaller and larger neighbors of v at the same time. Therefore, there is no pair of nodes u and v such that u is a master and slave to v. Therefore, Eliminate consists mainly of the first round of ComputeMIS II. The advantages of Eliminate is that: 1) It forms connected scatternets given that the input network G is connected. This is because the scatternets formed by Eliminate are supergraphs of those formed by BlueMIS I which are known to be connected. 2) It forms outdegree limited scatternets given that the input G is a unit disk graph, since each node u considers only the neighbors of MIS S (u) as slaves. Given that MIS S (u) is a subset of MIS(u), and MIS(u) 5 in unit disk graphs, then MIS S (u) 5. Therefore, each master is outdegree limited to 5 at most. 3) It use less messages than those used by BlueMIS I, since each node u does not consider its larger neighbors when constructing the set MIS S (u), and 4) It is faster than ComputeMIS II since it needs only one OrderedExchange communication round. The pseudocode of Eliminate is similar to that of ComputeMIS II. The only change needed is in line 4 in Algorithm 3. The loop will iterate while p 0 instead of p 1. Theorem VI.1. Eliminate forms a connected scatternet if the input network G(V, E) was connected. Proof: We show that Eliminate deletes edges of the original network G that does not affect its connectivity by showing that the resulting graph from Eliminate contains a minimum spanning tree. We give a lexicographical order of the edges. We say (x 1, y 1 ) (x 2, y 2 ) if y 2 y 1 or if y 1 = y 2 but x 2 x 1, where comparison is performed according to the identifiers of nodes. For each edge (u, v) in G, we assume that u v. Given any triangle in G of three edges (x, y), (y, z) and (x, z) of three vertices x, y, z where x y z, Eliminate deletes only the edge (x, z). Note that (x, y) (y, z) (x, z). We follow Kruskal algorithm for minimum spanning tree, we order the edges ascendingly, and select greedily the edges that form a tree. Edge (x, z) will never be considered in the MST of G as it is always the maximum edge in the 3-circle {(x, y), (y, z), (x, z)} /12/$ IEEE

6 Algorithm 4 Eliminate at node u 1: N l (u) {v : v is a larger neighbor of u } 2: N s (u) {v : v is a smaller neighbor of u } 3: B, U N(u), S(u), M(u), where S(u) and M(u) is the set of slaves and masters of u 4: while N l (u) do 5: wait for reception of a message from a larger neighbor v 6: upon reception of message N l (u) {N l (u) v} 7: if received message is attack from v, M(u) {M(u) v} 8: end while 9: while N s (u) do 10: v max(n s (u)) 11: send a message to v, call the message attack 12: S(u) {S(u) v} 13: U {U {v N s (v)}} 14: B {B {N s (u) N s (v)}} 15: end while 16: for each v B do 17: send a message to v, call the message dummy 18: end for Theorem VI.2. Eliminate forms an outdegree limited scatternet with maximum outdegree of 5 if the input network G(V, E) is a unit disk graph. Proof: Observe that each node u creates a maximal independent set of its smaller neighbors N s (u) N(u). In a unit disk graph, the size of the maximal independent set of the neighbors of a node does not exceed 5. This proof is whence complete. 1) Example: We give in this section an example for the execution of Eliminate. The initial network is modeled as the graph shown in Figure 1-a. Each circle with identifier i, 1 i 7, represents node v i. We assume that v i > v j if i > j. The nodes which are not the largest in their neighborhood are waiting for the reception of messages from their larger neighbors. In this example, nodes v 7 and v 4 starts constructing the maximal independent set of their smaller neighbors. Node v 7 sends an attack message to its smallest neighbor which is v 1 and includes it in its MIS s (u) (or directly to S(u) as in line 12 in Algorithm 4). This makes it not allowed for node v 2 to be included in MIS s (v 7 ) = S(v 7 ) because v 2 is a neighbor to a node in MIS s (v 7 ). Node v 7 then sends an attack message to v 3 and includes it in MIS s (v 3 ). Node v 7 sends a dummy message to v 2 (line 17 in Algorithm 4). Node v 4 sends an attack message to v 2 and includes it in MIS s (v 4 ). Nodes v 7 and v 4 terminate the execution of the algorithm. Nodes v 2 and v 3 are allowed to start constructing their maximal independent sets MIS s (v 2 ) and MIS(v 3 ) respectively, since they received messages from all their larger neighbors. Node v 2 sends an attack message to v 1 and includes it to its MIS s (u), while node v 3 does nothing since it has no smaller neighbors. Node v 1 also does nothing since it has no smaller neighbors. The scatternet formed is illustrated in Figure 1-b. Fig. 2: Execution time of different algorithms Using ComputeMIS I or ComputeMIS II forms the scatternet illustrated in Figure 1-c. Note that some neighbors selected each other as masters and slaves at the same time (e.g. nodes v 2 and v 4, nodes v 2 and v 1, and nodes v 3 and v 7 ). We eliminate this situation by breaking symmetries. That is, we let the larger node becomes the master, whereas the smaller node become slave only. After breaking symmetry the scatternet formed is exactly the same as that generated by Eliminate and shown in Figure 1-b. It is not necessary that scatternets formed by Eliminate are the same as those generated by ComputeMIS I and ComputeMIS II. However, simulation results show that they are similar. VII. SIMULATION EXPERIMENTS We use the same simulation environment used in Section V to compare the average execution time of BlueMesh, BlueMIS I, ComputeMIS I, ComputeMIS II and Eliminate (see Figure 2). Each value in Figure 2 is averaged over 1000 experiments. We note the superiority of ComputeMIS I and ComputeMIS II over BlueMIS I. This shows the impact of using OrderedExchange in Bluetooth networks. ComputeMIS I and ComputeMIS II are not significantly different than each other. However, the execution times of ComputeMIS I and ComputeMIS II are about 0.53 (on average) of the execution time of BlueMIS I in networks of size of less than 70. We note that Eliminate outpefroms all other algorithms. Its execution time is about 60% the execution time of ComputeMIS I and ComputeMIS II. The reason behind this superiority is that Eliminate runs in one communication round of OrderedExchange, whereas both ComputeMIS I and ComputeMIS II runs in two OrderedExchange communication rounds. Eliminate execution time is about 0.15 the execution time of BlueMesh. However, it forms scatternets with a higher number of masters and M/S bridges. Figure 3 shows the improvements in the execution time on BlueMIS under the three different versions of BlueMIS I (that is, with BlueMIS I, ComputeMIS I, ComputeMIS II, Eliminate), denoted by FBMIS-BMIS-I, FBMIS-CI, FBMIS- CII, FBMIS-EL. Figure 3 shows that BlueMIS with Eliminate has an execution time of about 62% of BlueMIS with BlueMIS I. Note however that BlueMIS consists of two phases where the latter is called BlueMIS II. We note however that the description of BlueMIS II in [9] lacks some details that may have a big impact on the performance of the algorithm /12/$ IEEE

7 Fig. 1: a) Initial network, b) Scatternet formed by Eliminate, c) Scatternet formed by ComputeMIS I and ComputeMIS II before symmetery breaking connected outdegree limited scatternets. These algorithms are inspired from BlueMIS I. ComputeMIS I and ComputeMIS II forms the same scatternets formed by BlueMIS I, given the same input, but with a faster execution time. Eliminate forms similar scatternets but with an even better execution time. The scatternets formed by Eliminate were compared against those formed by BlueMIS I in term of different performance metrics used to measure the performance of Bluetooth scatternets. Fig. 3: The effect of our algorithms on BlueMIS Fig. 4: A comparison of the scatternet properties. Size is the size of the network in number of nodes. Grey cells for BlueMIS I, blue cells for Eliminate We compare in Figure 4 the properties of the scatternets formed by Eliminate and those formed by BlueMIS I. Note that ComputeMIS I and ComputeMIS II form the same scatternets as BlueMIS I. We compare the scatternets in term of the number of masters, number of M/S bridges, number of S/S bridges, average role per node, average piconet size and maximum piconet size. We see that there is not much significant differences between the two of scatternets. In fact, we note that Eliminate introduces a minor improvement in term of average role per node and number of S/S bridges. This is mainly because Eliminate let each node finds the maximal independent set of its smaller neighbors, while nodes in BlueMIS I find the maximal independent set of all their neighbors. We say that the extra work performed by BlueMIS I is not necessary, and better results can be obtained if we avoid it. REFERENCES [1] F. Cuomo, T. Melodia, and I. F. Akyildiz, Distributed self-healing and variable optimization algoritms for qos provisioning in scatternets, IEEE Selected Areas in Communication, vol. 22, no. 7, pp , [2] D. Dubhashi, O. Häggström, G. Mambrini, A. Panconesi, and C. Petrioli, Blue pleiades, a new solution for device discovery and scatternet formation in multi-hop bluetooth networks, Wireless Networks, vol. 13, pp , January [3] A. Jedda, G.-V. Jourdan, and N. Zaguia, Toward better understanding of the behavior of bluetooth networks distributed algorithms, [4] X. Y. Li, I. Stojmenovic, and Y. Wang, Partial delaunay triangulation and degree limited localized bluetooth scatternet formation, IEEE Transactions on Parallel and Distributed Systems, vol. 15, no. 4, pp , [5] N. A. Lynch, Distributed Algorithms. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., [6] C. Petrioli, S. Basagni, and I. Chlamtac, Bluemesh: degree-constrained multi-hop scatternet formation for bluetooth networks, Mobile Networks and Applications, vol. 9, pp , February [7] Q. Wang, Ucbt - bluetooth extension for ns2 at the university of cincinnati, Tech. Rep. [8] Z. Wang, R. K. Thomas, and J. Haas, Performance comparison of bluetooth scatternet formation protocols for multi-hop networks, Wireless Networks, vol. 15, no. 2, pp , [9] N. Zaguia, Y. Daadaa, and I. Stojmenovic, Simplified bluetooth scatternet formation using maximal independent sets, Integr. Comput.-Aided Eng., vol. 15, no. 3, pp , [10] G. Zaruba, S. Basagni, and I. Chlamtac, Bluetrees - scatternet formation to enable bluetooth-based personal area networks, in Proceedings of the IEEE International Conference on Communications, ICC 2001, vol. 1, , pp VIII. CONCLUSION We introduced in this paper ComputeMIS I, ComputeMIS II and Eliminate, which are all BSF algorithms that forms /12/$ IEEE

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