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Bluetooth scatternet formation from a time-efficiency perspective

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Abstract

The Bluetooth Scatternet Formation (BSF) problem consists of interconnecting piconets in order to form a multi-hop topology. While a large number of BSF algorithms have been proposed, only few address time as a key parameter, and when doing so, virtually none of the solutions were tested under realistic settings. In particular, the baseband and link layers of Bluetooth are highly specific and known to have crucial impacts on performance. In this paper, we revisit performance studies for a number of time-efficient BSF algorithms, focusing on BlueStars, BlueMesh, and BlueMIS. We also introduce a novel time-efficient BSF algorithm called BSF-UED (for BSF based on Unnecessary-Edges Deletion), which forms connected scatternets deterministically and limits the outdegree of nodes to 7 heuristically. The performance of the algorithm is evaluated through detailed simulation experiments that take into account the low-level specificities of Bluetooth. We show that BSF-UED compares favorably against BlueMesh while requiring only 1/3 of its execution time. Only BlueStars is faster than BSF-UED, but at the cost of a very large number of slaves per master (much more than 7), which makes it impractical in many scenarios.

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Notes

  1. Bluetooth Smart Ready: http://www.bluetooth.com/pages/Bluetooth-Smart-Devices.aspx. Fetched on Dec. 20, 2012

  2. In the following, a piconet is modeled as a star graph with a master and slaves. We model a master-slave relationship as a directed edge from the master to slave, hence the number of slaves of a master is its outdegree.

  3. Bluehoc: http://bluehoc.sourceforge.net/. Fetched on Dec. 20, 2012

  4. Given two functions f, g, we say that g is in O(f) if there is a constant c > 0 such that \(g(n) < c\cdot f(n)\) for a sufficiently large n.

  5. An independent set of a network (or a graph) is a set of nodes that none of which is neighbor to another. A maximal independent set is an independent set that is not a subset of any other independent set.

  6. In this rule, the fact that v captures u (and not the opposite) is important; it follows the anticipated decrease of \(\varphi(v)\) in Phase 1 when the edge (u, v) was colored red. In a sense, v is “more prepared” than u to handle new slaves. The impact of this operation is seen while starting the elimination algorithm from smaller to larger piconets.

  7. This case may occur if u delegated to a neighbor w the responsibility of v, and hence (u, v) is red. Also, there is a node a \(w^{\prime}\) that is larger than v and delegated to v the responsibility of slaving common neighbors between v and \(w^{\prime}\).

  8. We thank the anonymous reviewer for directing our attention to this important issue in the field of Bluetooth Scatternet Formation algorithms

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Authors and Affiliations

  1. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada

    Ahmed Jedda, Guy-Vincent Jourdan & Hussein T. Mouftah

  2. LaBRI, University of Bordeaux, Bordeaux, France

    Arnaud Casteigts

Authors
  1. Ahmed Jedda
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  2. Arnaud Casteigts
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  3. Guy-Vincent Jourdan
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  4. Hussein T. Mouftah
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Corresponding author

Correspondence to Ahmed Jedda.

Appendices

Appendix 1: Time complexity analysis

In this section, we study the time complexity of BlueStars, BlueMesh, BlueMIS and BSF-UED. We start with BlueStars.

Theorem 6

The time complexity of BlueStars is O(n) where n is the number of nodes in the network.

Proof

The most expensive procedure in BlueStars is the procedure of forming disjoint piconets (i.e. the first phase). A node v i with identifier i starts executing this phase if all its larger neighbors (i.e. with larger identifiers) sent a message to v i . We say v i waits for v j if v j is a larger neighbor of v i . According to this definition, v i may wait for v k where v k is a larger neighbor of v j . We may construct thus a chain \(v_1, \dot, v_n\) of length n such that v i waits for v j if i > j. Therefore, a node v i waits for all the larger nodes in the network, and respectively the smallest node v 1 waits for all the other nodes. Therefore, the time complexity of this phase is O(n). In the second phase of BlueStars, there is a message exchange between (1) the slaves and masters (in order for the masters to identify the neighbor piconets), (2) the masters and gateways (to inform the gateways which piconets they must interconnect), and (3) the gateways and neighbor gateways (to interconnect the neighbor piconets). This requires O(1) time complexity, and therefore the time complexity of BlueStars is O(n). \(\square\)

BlueMesh runs in iterations (as explained in Sect. 4.4). Each iteration is similar with respect to time complexity to BlueStars, since both constructs a maximal independent set in a similar manner. Therefore, each iteration requires O(n). In the following, we analyzes the worst case number of iterations of BlueMesh.

Theorem 7

The number of iterations run by BlueMesh is in the worst case O(log n) in arbitrary graphs and O(1) in unit disk graphs, where n is the number of nodes in the network.

Proof

We build the worst case scenario as follows. We start with the simple graph of two nodes u and v linked by the edge (u, v). Let’s assume that nodes u and v are the last surviving nodes in iteration k, where k is the index of the last phase of the algorithm. Note that if a node u survived iteration k − 1 and moved to iteration k, then it must have a larger neighbor \(u^{\prime}\). This means that there is at least 4 nodes \(u, u^{\prime}, v\) and \(v^{\prime}\) in iteration k − 1. Therefore, the maximum number of nodes that move to iteration i is |P i-1|/2 where |P i | is the number of nodes in iteration i. Therefore, the maximum number of phases is at O(log n). The worst case scenario of BlueMesh is shown in Fig. 11.

Fig. 11
figure 11

An example scenario with Bluemesh

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Note that if a node has more than 5 neighbors in a unit disk graph then at least two of them are also neighbors. Following the previous argument, if a node survived k iterations then it must have at least k largest neighbors that are not neighbors to each other. This means that k is at most 5 in unit disk graphs. Therefore, the maximum number of BlueMesh iterations if run over unit disk graphs is O(1). \(\square\)

Therefore, the time complexity of BlueMesh in unit disk graphs is O(n). Note that the time complexity of BlueMesh remains O(n) in arbitrary graphs. The result of Theorem 7 matces simulation results in this paper and in [6], whereas a theoritical analysis of this aspect of BlueMesh is first studied here.

BlueMIS I complexity is straightforward. Each node exchanges its neighbors list with all its neighbors in a first round. In a second round, each node u sends to all its neighbors the set S(u) which is the set of nodes that u may be master to. Therefore, the time complexity of BlueMIS I is O(1). Thus, BlueMIS I is a local distributed algorithm. Local distributed algorithms has the advantage that their execution time does not depends on the size of the input network, making them suitable to solve scalability issues. The case is different in BlueMIS II. To achieve the exact same results given in [7], BlueMIS II time complexity may be at least in O(n). This is because, in order to avoid certain worst-case conflicting scenarios, each node must execute the rules of BlueMIS II while every other is waiting (that is, in a sequential manner). This is one of the major issues in BlueMIS II. In our implementation, every node executes its rules locally after collecting sufficient neighborhood information. Therefore, we assumed that the time complexity of BlueMIS II remains O(1).

BSF-UED time complexity is similar to that of BlueStars. The first phase is similar to the first phase of BlueStars and thus has O(n) time complexity. In the second phase of BSF-UED, there is a communication exchange between, (1) slaves and masters, (2) masters and gateways, and 3) gateways and gateways from neighbor piconets. Therefore, the time complexity of BSF-UED is O(n).

Appendix 2: Flow diagrams of BSF-UED

In this appendix section, we present the flow diagrams of our algorithm BSF-UED without the heuristics. We believe that this simplifies the understanding of our algorithm and its pseudocode (see Figs. 12 and 13).

Fig. 12
figure 12

Flow diagram of Phase 1 of BSF-UED

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Fig. 13
figure 13

Flow diagram of Phase 2 of BSF-UED

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Jedda, A., Casteigts, A., Jourdan, GV. et al. Bluetooth scatternet formation from a time-efficiency perspective. Wireless Netw 20, 1133–1156 (2014). https://doi.org/10.1007/s11276-013-0664-z

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