# A Novel QoS Provisioning Algorithm for Optimal Multicast Routing in WMNs

^{*}

## Abstract

**:**

^{2}τ

^{K−1}).

## 1. Introduction

- We present a model for the MCOMR problem from the approximate perspective. This model allows multiple QoS metrics to be considered in WMNs with guaranteed multicast service performance.
- We formulate the problem of the MCOMR problem and develop a novel multicast heuristic approximation (NMHA) algorithm based on the technique of auxiliary graph construction, scaling, rounding and the MPH algorithm to solve the problem.
- We analyze the theoretical properties of the proposed algorithm. Analytical results show that our algorithm is effective and achieves lower complexity and the approximate optimal solution for WMNs.
- We conduct experiments to evaluate the performance of the proposed algorithm and compare the algorithm against variations of current best known algorithms. Obtained numerical results indicate that the proposed algorithm is efficient and accurate for multicast service in WMNs.

## 2. Preliminaries

#### 2.1. Problem Formulation

**Definition 1.**

**D**be the set of destination nodes, where $D=\left\{{d}_{k}\right\}\subseteq V,s\notin D,k=1,2,\dots q$. Denote $T\left({V}_{T},{E}_{T}\right)={\displaystyle \sum _{{e}_{x}\in T}{e}_{x}}$ as a multicast routing tree from source s to the set of destination nodes

**D**in

**G**with $D\subseteq {V}_{T}\subseteq V$, ${E}_{T}\subseteq E$. Denote ${w}_{i}\left({T}_{j}\right)={\displaystyle \sum _{{e}_{x}\in {T}_{j}}{w}_{i}({e}_{x})}$, $2\le i\le {\rm K}$ as the sum of the ith weight on edges along multicast tree ${T}_{j}$. The goal is to find a multicast routing tree ${T}_{j}$ from s to

**D**, such that ${w}_{i}\left({T}_{j}\right)\le {\mathrm{L}}_{i}$. We say that MCMR is feasible if it has a feasible solution; otherwise, it is infeasible.

**Definition 2.**

**G**and ${\mathsf{\theta}}^{*}=\text{min}\left\{{\mathsf{\theta}}_{j}\right\}$ such that ${w}_{i}\left({T}^{*}\right)\le {\mathsf{\theta}}^{*}{\mathrm{L}}_{i}$. We call ${\mathsf{\theta}}^{*}$ the optimal value to MCMR and ${T}^{*}$ an optimal multicast routing tree or an optimal solution to MCMR.

**Definition 3.**

**D**such that ${w}_{i}\left({T}^{\mathrm{opt}}\right)\le a{\mathsf{\theta}}^{*}{\mathrm{L}}_{i}$, $2\le i\le \mathrm{K}$, then ${T}^{opt}$ is called an $a$-Approximation to MCOMR, and it is called an $a$-Approximation algorithm.

#### 2.2. Deterministic Algorithm with Auxiliary Graph

**G**to a directed graph ${G}^{\mathrm{K},\tau}$. Each vertex $v\in G$ is associated with ${(1+\mathsf{\tau})}^{\mathrm{K}-1}$ vertices, since each weight has been normalized ${w}_{i}\left(e\right)/{\mathrm{L}}_{i}$. Then, $\mathsf{\tau}=\lceil (n-1)/\mathsf{\epsilon}\rceil $ ($\mathsf{\epsilon}$ is the approximation factor) could be regarded as the maximum integer of all constraints in ${G}^{\mathrm{K},\mathsf{\tau}}$. For example, from the source node s to any node u, ${\mathrm{C}}_{i}\in [0,\mathsf{\tau}],\forall 2\le i\le \mathrm{K}$ is used for recording the ith weight of path length, and$P\left(u,{\mathrm{C}}_{2},{\mathrm{C}}_{i},\dots ,{\mathrm{C}}_{\mathrm{K}}\right)$ for all the K weight of path length. To each undirected edge $(u,v)$ in

**G**, the directed edge in ${G}^{K,\mathsf{\tau}}$ is constructed from $P\left(u,{\mathrm{C}}_{2},{\mathrm{C}}_{i},\dots ,{\mathrm{C}}_{\mathrm{K}}\right)$ to $P\left(v,{\mathrm{D}}_{2},{\mathrm{D}}_{i},\dots ,{\mathrm{D}}_{\mathrm{k}}\right)$, where ${\mathrm{D}}_{i}={\mathrm{C}}_{i}+{w}_{i}\left(u,v\right)$, $\u200a\forall {\mathrm{C}}_{i},{\mathrm{D}}_{i}\in [0,\mathsf{\tau}]$. It is clear that the infeasible paths of the latter K − 1 metrics have been filtered. Therefore, an optimal path in

**G**corresponds to an optimal path from $P\left(s,0,\dots ,0\right)$ to $P(d,\mathsf{\tau},\dots ,\mathsf{\tau})$ in ${G}^{\mathrm{K},\mathsf{\tau}}$.

#### 2.3. Description of MPH Algorithm

**Step 1:**Choose any node d

_{1}from the set

**D**of multicast destination nodes, let $j=1$, and initialize the generation of tree ${T}_{1}=\left\{{d}_{1}\right\}$, ${V}_{{T}_{1}}=\left\{{d}_{1}\right\}$;

**Step 2:**The cost $w\left({d}_{k},{T}_{j-1}\right)$ from ${d}_{k}(k\le q)$ in $\overline{D\cap {T}_{j-1}}$ to ${T}_{j-1}$ can reach the minimum value by comparing formula (1):

**Step 3:**Until $i>q$, find the Steiner generating tree; otherwise, let $j=j+1$, then repeat

**Step 2**.

## 3. The Proposed NMHA Algorithm

#### 3.1. Description of NMHA Algorithm

**Step 1:**Convert the graph $G(V,E,\stackrel{\rightharpoonup}{W},\stackrel{\rightharpoonup}{L})$ to the new graph ${G}^{\mathrm{N}}(V,E,{W}^{\mathrm{N}},\mathsf{\tau})$ based on the technique of the scaling and rounding;

**Step 2:**Convert the graph ${G}^{\mathrm{N}}(V,E,{W}^{\mathrm{N}},\mathsf{\tau})$ to the new graph ${G}_{\mathrm{K}}^{\mathrm{N}}({V}_{\mathrm{K}}^{\mathrm{N}},{E}_{\mathrm{K}}^{\mathrm{N}},{W}_{\mathrm{K}}^{\mathrm{N}},\mathsf{\tau})$ based on the technique of the auxiliary graph construction;

**Step 3:**Choose any node ${d}_{1}$ from the set

**D**of multicast destination nodes, let $j=1$, and initialize the generation of tree ${{T}_{0}}^{\text{opt}}=\left\{s\right\}$, ${V}_{{T}_{0}}=\left\{s\right\}$;

**Step 4:**Calculate the cost ${w}_{1}\left({d}_{k},{T}_{j-1}^{\text{opt}}\right)$ from ${d}_{k}(k\le q)$ in $\overline{D\cap {T}_{j-1}^{\text{opt}}}$ to ${T}_{j-1}^{\text{opt}}$ for all nodes ${d}_{k}(k\le q)$ by Xue’s deterministic algorithm;

**Step 5:**It can reach the minimum value by comparing formula (2):

**Step 4**;

**Step 6:**Until $i>q$, go to

**Step 7**; otherwise, let $j=j+1$, then repeat

**Step 5**;

**Step 7:**If ${w}_{1}^{\mathrm{N}}({T}_{j-1}^{opt})<\mathsf{\tau}$, find the Steiner generating tree; Else No feasible multicast routing tree.

#### 3.2. The Procedure of the NMHA Alogorithm

Algorithm 1 NMHA algorithm |

Input: $G(V,E,\stackrel{\rightharpoonup}{W},\stackrel{\rightharpoonup}{L})$, the source node s and the set of destination nodes D, and the approximation ratio $\mathsf{\epsilon}$Output: ${T}^{\text{opt}}$1. For each $e\in E$ in $G(V,E,\stackrel{\rightharpoonup}{W},\stackrel{\rightharpoonup}{L})$ 2. ${w}_{i}^{\mathrm{N}}(e)=\lfloor \frac{{w}_{i}(e)}{{\mathrm{L}}_{i}}.\frac{n-1}{\mathsf{\epsilon}}\rfloor $; 3. set $\mathsf{\tau}={\mathrm{L}}_{1}^{\mathrm{N}}=\mathrm{...}={\mathrm{L}}_{\mathrm{K}}^{\mathrm{N}}=\lceil \frac{n-1}{\mathsf{\epsilon}}\rceil $; 4. end for 5. $\to $ ${G}_{\mathrm{K}}^{\mathrm{N}}({V}_{\mathrm{K}}^{\mathrm{N}},{E}_{\mathrm{K}}^{\mathrm{N}},{W}_{\mathrm{K}}^{\mathrm{N}},\mathsf{\tau})$; 6. Choose any node ${d}_{1}$ from the set D of multicast destination nodes, initialize generating tree ${{T}_{0}}^{\text{opt}}=\left\{s\right\}$, ${V}_{{T}_{0}}=\left\{s\right\}$;7. for every node $v\in V$ and ${\mathrm{C}}_{i}=0$ to $\mathsf{\tau}$, $2\le i\le \mathrm{K}$, do 8. $P(v,{\mathrm{C}}_{2},{\mathrm{C}}_{i},\mathrm{...},{\mathrm{C}}_{\mathrm{K}})\leftarrow \infty $; $P(s,{\mathrm{C}}_{2},{\mathrm{C}}_{i},\mathrm{...},{\mathrm{C}}_{\mathrm{K}})\leftarrow 0$; 9. end for 10. for $j=1:q$ do 11. for every adjacent node $v(v\notin {T}_{j-1}^{\text{opt}})$ of node ${d}_{j}$, where ${d}_{j}\in {T}_{j-1}^{\text{opt}},v\notin {T}_{j-1}^{\text{opt}}$ and ${\mathrm{C}}_{i}\le \mathsf{\tau}$ do 12. if $P(v,{\mathrm{D}}_{2},{\mathrm{D}}_{i},\mathrm{...},{\mathrm{D}}_{\mathrm{K}})>P({d}_{j},{\mathrm{C}}_{2},{\mathrm{C}}_{i},\mathrm{...},{\mathrm{C}}_{\mathrm{K}})+{w}_{1}(u,v)$ where ${\mathrm{D}}_{i}={\mathrm{C}}_{i}+{w}_{i}({d}_{j},v),\u200a\u200a\u200a2\le i\le \mathrm{K}$ 13. then $P(v,{\mathrm{D}}_{2},{\mathrm{D}}_{i},\mathrm{...},{\mathrm{D}}_{\mathrm{K}})\leftarrow P({d}_{j},{\mathrm{C}}_{2},{\mathrm{C}}_{i},\mathrm{...},{\mathrm{C}}_{\mathrm{K}})+{w}_{1}(u,v)$ 14. $v\leftarrow {d}_{j}$; 15. ${T}_{j}^{\text{opt}}\leftarrow $ node $v$ and edge $\left({d}_{j},v\right)$; 16. end if 17. end for 18. ${w}_{1}({d}_{j},{T}_{j-1}^{\text{opt}})=\mathrm{min}\{{w}_{1}({d}_{k},{T}_{j-1}^{\text{opt}})|{d}_{k}\in \overline{D\cap {T}_{j-1}^{\text{opt}}}\}$; ${T}_{j}^{\text{opt}}=\text{PATH}\left({d}_{j},{T}_{j-1}^{\text{opt}}\right)\cup {T}_{j-1}^{\text{opt}}$ 19. end for 20. if ${w}_{1}^{\mathrm{N}}({T}_{j-1}^{opt})>\tau $ 21. then return. No feasible multicast routing tree, exit; 22. else, ${T}_{j}^{\text{opt}}=\text{PATH}({d}_{j},{T}_{j-1}^{\text{opt}})\cup {T}_{j-1}^{\text{opt}}$; $j=j+1$; 23. end if 24. end for 25. OUTPUT ${T}^{\text{opt}}$; |

#### 3.3. Analysis of the NMHA Algorithm

**Theorem 1.**

**D,**which minimizes ${\mathrm{max}}_{1\le i\le \mathrm{K}}{w}_{i}^{\mathrm{N}}({T}_{j})$ among all multicast routing trees in ${G}_{\mathrm{K}}^{\mathrm{N}}({V}_{\mathrm{K}}^{\mathrm{N}},{E}_{\mathrm{K}}^{\mathrm{N}},{W}_{\mathrm{K}}^{\mathrm{N}},\mathsf{\tau})$. The worst case time complexity of the algorithm is O(qmn

^{2}τ

^{K−1}).

**Proof of Theorem 1.**

**D**that minimizes ${\mathrm{max}}_{1\le i\le \mathrm{K}}{w}_{i}^{\mathrm{N}}({T}_{j})$ among all multicast routing trees in ${G}_{\mathrm{K}}^{\mathrm{N}}$.

^{2}τ

^{K}

^{−}

^{1}).

**Theorem 2.**

**G**, it is a feasible multicast routing tree in ${G}_{\mathrm{K}}^{\mathrm{N}}$.

**Proof of Theorem 2.**

**G**, it has

**G**is a feasible multicast routing tree in ${G}_{\mathrm{K}}^{\mathrm{N}}$.

**Theorem 3.**

**Proof of Theorem 3.**

**G**, it has

**G**, it implies that

^{2}τ

^{K−1}).

**Theorem 4.**

**Proof of Theorem 4.**

**P**=

**NP**. The approximation algorithm NMHA proposed in this paper for the MCOMR problem is the same as that in [35]. It has been proven in Theorem 3 that the NMHA algorithm can finally find a $2(1+\mathsf{\epsilon})\left(1-1/q\right)$-approximation multicast routing tree to MCOMR. Thus, the upper bound of approximation ration should be discussed to evaluate the performance of NMHA. To the optimal multicast routing tree ${T}^{*}$ in

**G**, it has

## 4. The Simulation Experiments

**G**of a static undirected network, in which the nodes in blue are the set of destination nodes

**D**and others are non-multicast destination nodes. The numbers marked between two nodes are linked to cost or distance. The node is originally added to the generating tree T

_{0}= {A}. In order to find the multicast generating tree with the NMHA algorithm (Lines 10–11), the first node U

_{1}is selected, and then it is added to the tree A-U

_{1}in Figure 1b (Lines 12–20) with the total cost of the generating tree 3. In the following phase of the algorithm, destination nodes U

_{2}and U

_{3}are added to the tree, respectively, and the multicast destination nodes are sorted as U

_{1}-U

_{2}-U

_{3}. The final multicast tree obtained, whose total cost is 7, is shown in Figure 1c. According to the NMHA algorithm proposed in this paper, the generating graph is found.

**D**(No. 30, 51 and 55) in the network RNET, which is shown in Figure 3. Comparison with the costs from the source node s to every destination node by Xue’s deterministic algorithm, the minimal-cost node (No. 30) is selected first. Figure 4 shows the minimal-cost path from source node (No. 1) to destination nodes (No. 30), which is selected first by NMHA in the network. The blue paths in these figures indicate the minimal or the optimal paths from source to destination. Figure 5 illustrates that the following node (No. 55) is added to the tree by NMHA. As expected, the optimal multicast routing tree with three constraints added by the algorithm NMHA can be found after the last node (No. 51) is added to the multicast tree. At last, the optimal multicast routing tree described in Figure 6 shows the corresponding multicast tree from the source node (No. 1) to the set of destination nodes

**D**(No. 30, 51 and 55).

## 5. Conclusions

^{2}τ

^{K−1}).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zhu, Z.; Li, S.; Chen, X. Design QoS-Aware Muli-Path Provisioning Strategies for Efficient CLOUD-Assisted SVC Video Streaming to Heterogeneous Clients. IEEE Trans. Multimedia
**2013**, 15, 758–768. [Google Scholar] - Huang, J.; Huang, X.; Ma, Y. Routing with multiple quality of services constraints: An approximation perspective. J. Netw. Comput. Appl.
**2012**, 35, 465–475. [Google Scholar] [CrossRef] - Fang, X.; Yang, D.; Xue, G. MAP: Multiconstrained Anypath Routing in Wireless Mesh Networks. IEEE Trans. Mob. Comput.
**2012**, 12, 1893–1906. [Google Scholar] [CrossRef] - Lu, T.; Zhu, J. Genetic Algorithm for Energy-Efficient QoS Multicast Routing. IEEE Commun. Lett.
**2013**, 17, 31–34. [Google Scholar] [CrossRef] - Su, Y.-S.; Su, S.-L.; Li, J.-S. Joint Topology-Transparent Scheduling and QoS Routing in Ad Hoc Networks. IEEE Trans. Veh. Technol.
**2014**, 63, 372–389. [Google Scholar] [CrossRef] - Huijun, D.; Hua, Q.; Jihong, Z. QoS routing algorithm with multi-dimensions for overlay networks. China Commun.
**2013**, 10, 167–176. [Google Scholar] [CrossRef] - Xiao, Y.; Thulasiraman, K.; Fang, X.; Yang, D.; Xue, G. Computing a Most Probable Delay Constrained Path: NP-Hardness and Approximation Schemes. IEEE Trans. Comput.
**2012**, 61, 738–744. [Google Scholar] [CrossRef] - Chen, S.; Song, M.; Sahni, S. Two techniques for fast computation of constrained shortest paths. IEEE/ACM Trans. Netw.
**2008**, 16, 105–115. [Google Scholar] [CrossRef] - Huang, J.; Liu, Y. MOEAQ: A QoS-aware multicast routing algorithm for MANET. Expert Syst. Appl.
**2010**, 37, 1391–1399. [Google Scholar] [CrossRef] - Liu, L.; Song, Y.; Zhang, H.; Ma, H.; Vasilakos, A.V. Physarum Optimization: A Biology-inspired Algorithm for the Steiner Tree Problem in Networks. IEEE Trans. Comput.
**2015**, 64, 818–831. [Google Scholar] - Youssef, M.; Ibrahim, M.; Abdelatif, M.; Lin, C.; Vasilakos, A.V. Routing Metrics of Cognitive Radio Networks: A Survey. IEEE Commun. Surv. Tutor.
**2014**, 16, 92–109. [Google Scholar] [CrossRef] - Zhang, X.M.; Zhang, Y.; Yan, F.; Vasilakos, A.V. Interference-based topology control algorithm for delay-constrained mobile Ad hoc networks. IEEE Trans. Mob. Comput.
**2015**, 14, 742–754. [Google Scholar] [CrossRef] - Pei, G.; Parthasarathy, S.; Srinivasan, A.; Vullikanti, A.K.S. Approximation Algorithms for Throughput Maximization in Wireless Networks with Delay Constraints. IEEE/ACM Trans. Netw.
**2013**, 21, 1988–2000. [Google Scholar] [CrossRef] - Takahashi, H.; Matsuyama, A. An approximate solution for the Steiner problem in graphs. Math Jpn.
**1980**, 24, 573–577. [Google Scholar] - Yuan, X. Heuristic algorithms for multiconstrained quality-of-service routing. IEEE/ACM Trans. Netw.
**2002**, 10, 244–256. [Google Scholar] [CrossRef] - Xue, G.; Sen, A.; Zhang, W.; Tang, J.; Thulasiraman, K. Finding a path subject to many additive QoS constraints. IEEE/ACM Trans. Netw.
**2007**, 15, 201–211. [Google Scholar] [CrossRef] - Xue, G.; Zhang, W.; Tang, J.; Thulasiraman, K. Polynomial time approximation algorithms for multi-constrained QoS routing. IEEE/ACM Trans. Netw.
**2008**, 16, 656–669. [Google Scholar] - Feng, G.; Korkmaz, T. A Fast Hybrid ε-Approximation Algorithm for Computing Constrained Shortest Paths. IEEE Commun. Lett.
**2013**, 17, 1471–1474. [Google Scholar] [CrossRef] - Szymanski, T.H. Max-Flow Min-Cost Routing in a Future-Internet with Improved QoS Guarantees. IEEE Trans. Commun.
**2013**, 61, 1485–1497. [Google Scholar] [CrossRef] - Yang, W.; Zhang, Y. A Fast Algorithm for the Optimal Constrained Path Routing in Wireless Mesh Networks. J. Commun.
**2016**, 11, 126–131. [Google Scholar] - Li, P.; Guo, S.; Yu, S.; Vasilakos, A.V. Reliable Multicast with Pipelined Network Coding Using Opportunistic Feeding and Routing. IEEE Trans. Parallel Distrib. Syst.
**2014**, 25, 3264–3273. [Google Scholar] [CrossRef] - Meng, T.; Wu, F.; Yang, Z.; Chen, G.; Vasilakos, A. Spatial Reusability-Aware Routing in Multi-Hop Wireless Networks. IEEE Trans. Comput.
**2015**, 65. [Google Scholar] [CrossRef] - Liu, J.; Wan, J.; Wang, Q.; Zeng, B.; Fang, S. A Time-recordable Cross-Layer Communication Protocol for the Positioning of Vehicular Cyber-Physical Systems. Future Gener. Comput. Syst.
**2016**, 56, 438–448. [Google Scholar] [CrossRef] - Liu, J.; Wan, J.; Wang, Q.; Li, D.; Qiao, Y.; Cai, H. A Novel Energy-saving One-Sided Synchronous Two-Way Ranging Algorithm for Vehicular Positioning. ACM/Springer Mob. Netw. Appl.
**2015**, 20, 661–672. [Google Scholar] [CrossRef] - Wan, J.; Liu, J.; Shao, A.; Vasilakos, A.V.; Imran, M.; Zhou, K. Mobile Crowd Sensing for Traffic Prediction in Internet of Vehicles. Sensors
**2016**, 16, 88. [Google Scholar] [CrossRef] [PubMed] - Alasaad, A.; Nicanfar, H.; Gopalakrishnan, S.; Leung, V.C.M. A ring-based multicast routing topology with QoS support in wireless mesh networks. Wirel. Netw.
**2013**, 19, 1627–1651. [Google Scholar] [CrossRef] - Hwang, I.S.; Nikoukar, A.; Chen, K.C.; Liem, A.T.; Lu, C.H. QoS enhancement of live IPTV using an extended real-time streaming protocol in Ethernet passive optical networks. IEEE/OSA J. Opt. Commun. Netw.
**2014**, 6, 695–704. [Google Scholar] [CrossRef] - Xiang, Z.; Tao, M.; Wang, X. Coordinated Multicast Beamforming in Multicell Networks. IEEE Trans. Wirel. Commun.
**2013**, 12, 12–21. [Google Scholar] [CrossRef] - Cao, Y.; Blostein, S.D.; Chan, W.Y. Optimization of unequal error protection rateless codes for multimedia multicasting. J. Commun. Netw.
**2015**, 17, 221–230. [Google Scholar] [CrossRef] - Afolabi, R.O.; Dadlani, A.; Kim, K. Multicast Scheduling and Resource Allocation Algorithms for OFDMA-Based Systems: A Survey. IEEE Commun. Surv. Tutor.
**2013**, 15, 240–254. [Google Scholar] [CrossRef] - Al-Fuqaha, A.; Khreishah, A.; Guizani, M.; Rayes, A.; Mohammadi, M. Toward better horizontal integration among IoT services. IEEE Commun. Mag.
**2015**, 53, 72–79. [Google Scholar] [CrossRef] - Mahboobi, B.; Mehrizi, S.; Ardebilipour, M. Multicast Relay Beamforming in CDMA Networks: Nonregenerative Approach. IEEE Commun. Lett.
**2015**, 19, 1418–1421. [Google Scholar] [CrossRef] - Bornhorst, N.; Pesavento, M.; Gershman, A.B. Distributed Beamforming for Multi-Group Multicasting Relay Networks. IEEE Trans. Signal Proc.
**2012**, 60, 221–232. [Google Scholar] [CrossRef] - Chuah, S.P.; Chen, Z.; Tan, Y.P. Energy-Efficient Resource Allocation and Scheduling for Multicast of Scalable Video Over Wireless Networks. IEEE Trans. Multimed.
**2012**, 14, 1324–1336. [Google Scholar] [CrossRef] - Quang, P.T.A.; Piamrat, K.; Singh, K.D.; Viho, C. Video Streaming over Ad-hoc Networks: A QoE-based Optimal Routing Solution. IEEE Trans. Veh. Technol.
**2016**. [Google Scholar] [CrossRef]

**Figure 1.**An example of the novel multicast heuristic approximation (NMHA) algorithm. (

**a**), (

**b**) and (

**c**) are the step 1, step 2 and step 3 of NMHA, respectively.

s | the source node |

${d}_{k}$ | the kth destination nodes |

D | the set of destination nodes |

m, n | the edge and the node number of graph G, respectively |

q | the number of terminal nodes |

${w}_{i}({e}_{x})$ | ith link weight of edge e_{x} |

K | the number of Quality-of-Service (QoS) requirements |

$\stackrel{\rightharpoonup}{W}$ | the set of weights of the edges |

$\stackrel{\rightharpoonup}{L}$ | the set of constrains |

${\mathrm{L}}_{i}$ | ith QoS requirement |

$T$ | the multicast tree with all nodes in the set of destination nodes D |

${T}_{j}$ | the Steiner tree with j destination nodes in the set D |

${V}_{T}$, ${E}_{T}$ | all the nodes and the edges of the multicast trees T, respectively |

${T}^{*}$, ${T}^{opt}$ | the optimal and the approximate optimal multicast routing tree, respectively |

$w\left({d}_{k},{T}_{j-1}\right)$ | the total weights from ${d}_{k}$ to ${T}_{j-1}$ |

$\text{PATH}\left({d}_{k},{T}_{j-1}\right)$ | the short path connected node ${d}_{k}$ to the ${T}_{j-1}$ |

${\mathrm{C}}_{i}$, ${\mathrm{D}}_{i}$ | the ith weight of path length from source s to any node u in ${G}^{\mathrm{K},\mathsf{\tau}}$ |

$P\left(u,{\mathrm{C}}_{i},\dots ,{\mathrm{C}}_{\mathrm{K}}\right)$ | all the K weight of path length from source s to any node u in ${G}^{\mathrm{K},\mathsf{\tau}}$ |

**Table 2.**A comparison of the performance of novel multicast heuristic approximation (NMHA) algorithm and Xue’s algorithm [16] in the random network.

NO. | q = 2 | q = 5 | q = 8 | q = 11 | q = 14 | q = 17 | q = 20 | q = 23 |

NMHA | 42 | 107 | 163 | 233 | 297 | 366 | 419 | 492 |

Xue’s | 58 | 751 | 1850 | 3611 | 4443 | 8772 | 12,054 | 15,779 |

NO. | q = 26 | q = 30 | q = 34 | q = 38 | q = 42 | q = 46 | q = 50 | q = 54 |

NMHA | 569 | 627 | 725 | 799 | 889 | 963 | 1052 | 1138 |

Xue’s | 21,136 | 28,412 | 33,689 | 39,956 | 51,896 | 63,512 | 75,412 | 87,512 |

**Table 3.**Average Weigh Occupying (AWO) (10

^{−3}) for NMHA algorithm and Fast Minimum Path Cost Heuristic (FMPH) algorithm with 50 nodes.

NO. | q = 2 | q = 5 | q = 8 | q = 11 | q = 14 | q = 17 | q = 20 | q = 23 |

NMHA | 24 | 33 | 34 | 56 | 58 | 59 | 70 | 59 |

FMPH | 24 | 35 | 38 | 64 | 66 | 69 | 80 | 75 |

NO. | q = 26 | q = 29 | q = 32 | q = 35 | q = 38 | q = 42 | q = 46 | q = 47 |

NMHA | 72 | 80 | 81 | 90 | 95 | 101 | 109 | 110 |

FMPH | 85 | 98 | 105 | 103 | 117 | 124 | 134 | 136 |

NO. | q = 2 | q = 35 | q = 70 | q = 105 | q = 140 | q = 175 | q = 210 | q = 245 |

NMHA | 1009 | 1490 | 1822 | 2201 | 2330 | 2474 | 2819 | 2837 |

FMPH | 1067 | 1490 | 2230 | 2485 | 2764 | 3310 | 3545 | 3355 |

NO. | q = 270 | q = 305 | q = 340 | q = 375 | q = 410 | q = 445 | q = 496 | q = 497 |

NMHA | 3320 | 3349 | 3610 | 3535 | 3894 | 4041 | 4488 | 4492 |

FMPH | 4128 | 4129 | 4824 | 4877 | 5139 | 5212 | 5985 | 5989 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yang, W.; Chen, Y. A Novel QoS Provisioning Algorithm for Optimal Multicast Routing in WMNs. *Future Internet* **2016**, *8*, 38.
https://doi.org/10.3390/fi8030038

**AMA Style**

Yang W, Chen Y. A Novel QoS Provisioning Algorithm for Optimal Multicast Routing in WMNs. *Future Internet*. 2016; 8(3):38.
https://doi.org/10.3390/fi8030038

**Chicago/Turabian Style**

Yang, Weijun, and Yuanfeng Chen. 2016. "A Novel QoS Provisioning Algorithm for Optimal Multicast Routing in WMNs" *Future Internet* 8, no. 3: 38.
https://doi.org/10.3390/fi8030038