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Minimum power multicast algorithms for wireless networks with a Lagrangian relaxation approach

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Abstract

Energy efficiency is crucial to the implementation of broadcast and multicast services in wireless ad hoc and sensor networks. This paper proposes a path-based power-efficient approach that minimizes energy consumption in the network by optimizing the transmission ranges of all nodes. Unlike existing link-based and node-based solutions, the proposed approach, which is applicable to both static and mobile wireless networks, constructs minimum-power broadcast trees. This study uses Lagrangian Relaxation (LR) to decompose the tree-construction problem, which is NP-complete, into multiple independently solvable sub-problems. The resulting Lagrange dual solution ensures a lower bound on the objective function value. The proposed approximation heuristic solves the primal problem by obtaining the upper bound of the objective value based on the information from the set of Lagrange multipliers. This study develops a decentralized, near-optimal algorithm to solve the problem. Simulation results show that in randomly generated networks, the proposed method outperforms the existing Minimum Shortest Path Tree (MSPT) algorithm, Prim’s Minimum Spanning Tree (PMST) algorithm, and the Broadcast Incremental Power (BIP) algorithm by 30, 10, and 5%, respectively.

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Notes

  1. Note that this problem formulation requires a given variable of all possible paths P w to determine a path \(p(\in P_w)\) for each destination w. However, not all possible paths P w need to be initially listed because the solution phase of the proposed routing algorithm can select the path of interest from among these paths.

  2. The slot duration is long enough to construct a multicast/broadcast tree so that, after deducting, the lead time \(\Updelta\) is sufficient for message transmission.

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Authors and Affiliations

  1. Department of Management Information Systems, National Chiayi University, Chiayi, Taiwan

    Yean-Fu Wen

  2. Institute of Information Management, National Taipei University, Taipei, Taiwan

    Yean-Fu Wen

  3. Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan

    Wanjiun Liao

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  1. Yean-Fu Wen
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  2. Wanjiun Liao
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Correspondence to Wanjiun Liao.

Appendix: The pseudo code of the GIBT algorithm

Appendix: The pseudo code of the GIBT algorithm

Algorithm 1 shows the pseudo code of the Greedy Incremental Broadcast Tree (GIBT) algorithm. This algorithm was implemented in the same manner as the basic version of Prim’s shortest path algorithm, from the destination nodes to any nodes included in the dominated set A. For example, only the source node s initially belongs to the dominated set A. Thus, a selected node must route to node s. Then, other destination nodes only route to a node that belongs to the dominated set A, which is the constructed multicast tree, iteration by iteration. Figure 14 shows an example of the GIBT algorithm, where the destination set W = {8, 9, 10, 11, 12}. In the following step, node 11 only has to select a sub-path 11-6-5 to a node in the existing multicast tree set A = {1, 2, 5, 8, 9, 10}. Finally, node 12 only has to find a sub-path (e.g., 12-11) to the multicast tree set A = {1, 2, 5, 8, 9, 10, 11}.

Algorithm 1 GBIT (G, s, W)
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Fig. 14
figure 14

An example of the GIBT algorithm. The destination node 11 finds the shortest reverse path route {11-6-5} to node 5, which belongs to the current set of A = {1, 2, 5, 8, 9, 10} and joins the existing multicast tree

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Wen, YF., Liao, W. Minimum power multicast algorithms for wireless networks with a Lagrangian relaxation approach. Wireless Netw 17, 1401–1421 (2011). https://doi.org/10.1007/s11276-011-0344-9

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