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April  2017, 13(2): 1065-1084. doi: 10.3934/jimo.2016062

Solving routing and wavelength assignment problem with maximum edge-disjoint paths

 1 Department of Transportation and Logistics Management, National Chiao Tung University, Hsinchu 300, Taiwan 2 Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27606, USA

* Corresponding author

Received  October 2014 Revised  August 2016 Published  October 2016

The routing and wavelength assignment (RWA) problem in wave-length-division multiplexing optical networks is critically important for increasing the efficiency of wavelength-routed all-optical networks. Given the physical network structure and a set of connection requests, the problem aims to establishing routes for the requests and assigning a wavelength to each of them, subject to the wavelength continuous and wavelength clash constraints. The objective is to minimize he number of required wavelengths to satisfy all requests. The RWA problem was proven to be NP-hard and has been investigated for decades. In this paper, an algorithm based on the maximum edge-disjoint paths is proposed to tackle the RWA problem. Its performance is verified by experiments on some realistic network topologies. Compared with the state-of-the-art bin-packing based methods and the particle swarm optimization algorithm, the proposed method can find the best solution among all testing instances in reasonable computing time.

Citation: Chia-Chun Hsu, Hsun-Jung Cho, Shu-Cherng Fang. Solving routing and wavelength assignment problem with maximum edge-disjoint paths. Journal of Industrial and Management Optimization, 2017, 13 (2) : 1065-1084. doi: 10.3934/jimo.2016062
References:
 [1] D. Banerjee and B. Mukherjee, A practical approach for routing and wavelength assignment in large wavelength-routed optical networks, IEEE Journal on Selected Areas in Communications, 14 (1996), 903-908.  doi: 10.1109/49.510913. [2] M. J. Blesa and C. Blum, Findinge edge-disjoint paths in networks: An ant colony optimization algorithm, Journal of Mathematical Modeling and Algorithms, 6 (2007), 367-391.  doi: 10.1007/s10852-007-9060-y. [3] I. Chlamtac, A. Ganz and G. Karmi, Lightpath communications: an approach to high bandwidth optical WAN's, IEEE Transactions on Communications, 40 (1992), 1171-1182.  doi: 10.1109/26.153361. [4] J. S. Choi, N. Golmie, F. Lapeyrere, F. Mouveaux and D. Su, A functional classification of routing and wavelength assignment schemes in DWDM networks: Static Case, In Proceedings of the 7th International Conference on Optical Communications and Networks, (2000), 1109-1115. [5] H. Choo and V. V. Shakhov, Routing and wavelength assignment in optical WDM networks with maximum quantity of edge disjoint paths, Computational Science -ICCS 2004, 3038 (2004), 1138-1145.  doi: 10.1007/978-3-540-24688-6_147. [6] T. Fischer, K. Bauer, P. Merz and K. Bauer, Solving the routing and wavelength assignment problem with a multilevel distributed memetic algorithm, Memetic Computing, 1 (2009), 101-123.  doi: 10.1007/s12293-008-0006-3. [7] X. Guan, S. Guo, W. Gong and C. Qiao, A new method for solving routing and wavelength assignment problems in optical networks, Journal of Lightwave Technology, 25 (2007), 1895-1909.  doi: 10.1109/JLT.2007.901344. [8] A. Hassan, Particle Swarm Optimization for Routing and Wavelength Assignment in Next Generation WDM Networks, Ph. D thesis, Queen Mary University of London, 2010. [9] C.-C. Hsu and H.-J. Cho, A genetic algorithm for the maximum edge-disjoint paths problem, Neurocomputing, 148 (2015), 17-22.  doi: 10.1016/j.neucom.2012.10.046. [10] C.-C. Hsu, H.-J. Cho and S.-C. Fang, Routing and wavelength assignment in optical networks from maximum edge-disjoint paths, Genetic and Evolutionary Computing: Advances in Intelligent Systems and Computing, 238 (2014), 95-103.  doi: 10.1007/978-3-319-01796-9_10. [11] Y. S. Kavian, A. Rashedi, A. Mahani and Z. Ghassemlooy, Routing and wavelength assignment in optical networks using artificial bee colony algorithm, European Journal of Operational Research, 124 (2013), 1243-1249.  doi: 10.1016/j.ijleo.2012.03.022. [12] J. Kleinberg, Approximation Algorithms for Disjoint Paths Problems, Ph. D thesis, Massachusetts Institute of Technology, 1996. [13] P. Leesutthipornchai, C. Charnsripinyo and N. Wattanapongsakorn, Solving multi-objective routing and wavelength assignment in WDM network using hybrid evolutionary computation approach, Computer Communications, 33 (2010), 2246-2259.  doi: 10.1016/j.comcom.2010.07.029. [14] G. Li and R. Shmha, The partition coloring problem and its application to wavelength routing and assignment, In Proceedings of the First Workshop on Optical Networks, 2000. [15] P. Manohar, D. Manjunath and R. K. Shevgaonkar, Routing and wavelength assignment in optical networks from edge disjoint path algorithms, IEEE Communications Letters, 6 (2002), 211-213.  doi: 10.1109/4234.1001667. [16] A. X. Martins, C. Duhamel, P. Mahey, R. R. Saldanha and M. C. de Souza, Variable neighborhood descent with iterated local search for routing and wavelength assignment, Computers and Operations Research, 39 (2012), 2133-2141.  doi: 10.1016/j.cor.2011.10.022. [17] T. F. Noronha, M. G. C. Resende and C. C. Ribeiro, A biased random-key genetic algorithm for routing and wavelength assignment, Journal of Global Optimization, 50 (2011), 503-518.  doi: 10.1007/s10898-010-9608-7. [18] T. F. Noronha and C. C. Ribeiro, Routing and wavelength assignment by partition colouring, European Journal of Operational Research, 171 (2006), 797-810.  doi: 10.1016/j.ejor.2004.09.007. [19] A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks, IEEE/ACM Transactions on Networking, 11 (2003), 259-272.  doi: 10.1109/TNET.2003.810321. [20] R. Poli, J. Kennedy and T. Blackwell, Particle swarm optimization -An overview, Swarm intelligence, 1 (2007), 33-57. [21] N. Skorin-Kapov, Routing and wavelength assignment in optical networks using bin packing based algorithms, European Journal of Operational Research, 177 (2007), 1167-1179.  doi: 10.1016/j.ejor.2006.01.003. [22] Y. Wang, T. H. Cheng and M. H. Lim, A Tabu search algorithm for static routing and wavelength assignment problem, IEEE Communications Letters, 9 (2005), 841-843.  doi: 10.1109/LCOMM.2005.1506721. [23] W. J. Yoon, D. H. Kim, M. Y. Chung, T. J. Lee and H. Choo, Routing with maximum EDPs and wavelength assignment with path conflict graphs, In Proceedings of the 2006 international conference on Computational Science and Its Applications, 3981 (2006), 856-865.  doi: 10.1007/11751588_89. [24] H. Zang, J. P. Jue and B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, 1 (2000), 47-60.

show all references

References:
 [1] D. Banerjee and B. Mukherjee, A practical approach for routing and wavelength assignment in large wavelength-routed optical networks, IEEE Journal on Selected Areas in Communications, 14 (1996), 903-908.  doi: 10.1109/49.510913. [2] M. J. Blesa and C. Blum, Findinge edge-disjoint paths in networks: An ant colony optimization algorithm, Journal of Mathematical Modeling and Algorithms, 6 (2007), 367-391.  doi: 10.1007/s10852-007-9060-y. [3] I. Chlamtac, A. Ganz and G. Karmi, Lightpath communications: an approach to high bandwidth optical WAN's, IEEE Transactions on Communications, 40 (1992), 1171-1182.  doi: 10.1109/26.153361. [4] J. S. Choi, N. Golmie, F. Lapeyrere, F. Mouveaux and D. Su, A functional classification of routing and wavelength assignment schemes in DWDM networks: Static Case, In Proceedings of the 7th International Conference on Optical Communications and Networks, (2000), 1109-1115. [5] H. Choo and V. V. Shakhov, Routing and wavelength assignment in optical WDM networks with maximum quantity of edge disjoint paths, Computational Science -ICCS 2004, 3038 (2004), 1138-1145.  doi: 10.1007/978-3-540-24688-6_147. [6] T. Fischer, K. Bauer, P. Merz and K. Bauer, Solving the routing and wavelength assignment problem with a multilevel distributed memetic algorithm, Memetic Computing, 1 (2009), 101-123.  doi: 10.1007/s12293-008-0006-3. [7] X. Guan, S. Guo, W. Gong and C. Qiao, A new method for solving routing and wavelength assignment problems in optical networks, Journal of Lightwave Technology, 25 (2007), 1895-1909.  doi: 10.1109/JLT.2007.901344. [8] A. Hassan, Particle Swarm Optimization for Routing and Wavelength Assignment in Next Generation WDM Networks, Ph. D thesis, Queen Mary University of London, 2010. [9] C.-C. Hsu and H.-J. Cho, A genetic algorithm for the maximum edge-disjoint paths problem, Neurocomputing, 148 (2015), 17-22.  doi: 10.1016/j.neucom.2012.10.046. [10] C.-C. Hsu, H.-J. Cho and S.-C. Fang, Routing and wavelength assignment in optical networks from maximum edge-disjoint paths, Genetic and Evolutionary Computing: Advances in Intelligent Systems and Computing, 238 (2014), 95-103.  doi: 10.1007/978-3-319-01796-9_10. [11] Y. S. Kavian, A. Rashedi, A. Mahani and Z. Ghassemlooy, Routing and wavelength assignment in optical networks using artificial bee colony algorithm, European Journal of Operational Research, 124 (2013), 1243-1249.  doi: 10.1016/j.ijleo.2012.03.022. [12] J. Kleinberg, Approximation Algorithms for Disjoint Paths Problems, Ph. D thesis, Massachusetts Institute of Technology, 1996. [13] P. Leesutthipornchai, C. Charnsripinyo and N. Wattanapongsakorn, Solving multi-objective routing and wavelength assignment in WDM network using hybrid evolutionary computation approach, Computer Communications, 33 (2010), 2246-2259.  doi: 10.1016/j.comcom.2010.07.029. [14] G. Li and R. Shmha, The partition coloring problem and its application to wavelength routing and assignment, In Proceedings of the First Workshop on Optical Networks, 2000. [15] P. Manohar, D. Manjunath and R. K. Shevgaonkar, Routing and wavelength assignment in optical networks from edge disjoint path algorithms, IEEE Communications Letters, 6 (2002), 211-213.  doi: 10.1109/4234.1001667. [16] A. X. Martins, C. Duhamel, P. Mahey, R. R. Saldanha and M. C. de Souza, Variable neighborhood descent with iterated local search for routing and wavelength assignment, Computers and Operations Research, 39 (2012), 2133-2141.  doi: 10.1016/j.cor.2011.10.022. [17] T. F. Noronha, M. G. C. Resende and C. C. Ribeiro, A biased random-key genetic algorithm for routing and wavelength assignment, Journal of Global Optimization, 50 (2011), 503-518.  doi: 10.1007/s10898-010-9608-7. [18] T. F. Noronha and C. C. Ribeiro, Routing and wavelength assignment by partition colouring, European Journal of Operational Research, 171 (2006), 797-810.  doi: 10.1016/j.ejor.2004.09.007. [19] A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks, IEEE/ACM Transactions on Networking, 11 (2003), 259-272.  doi: 10.1109/TNET.2003.810321. [20] R. Poli, J. Kennedy and T. Blackwell, Particle swarm optimization -An overview, Swarm intelligence, 1 (2007), 33-57. [21] N. Skorin-Kapov, Routing and wavelength assignment in optical networks using bin packing based algorithms, European Journal of Operational Research, 177 (2007), 1167-1179.  doi: 10.1016/j.ejor.2006.01.003. [22] Y. Wang, T. H. Cheng and M. H. Lim, A Tabu search algorithm for static routing and wavelength assignment problem, IEEE Communications Letters, 9 (2005), 841-843.  doi: 10.1109/LCOMM.2005.1506721. [23] W. J. Yoon, D. H. Kim, M. Y. Chung, T. J. Lee and H. Choo, Routing with maximum EDPs and wavelength assignment with path conflict graphs, In Proceedings of the 2006 international conference on Computational Science and Its Applications, 3981 (2006), 856-865.  doi: 10.1007/11751588_89. [24] H. Zang, J. P. Jue and B. Mukherjee, A review of routing and wavelength assignment approaches for wavelength-routed optical WDM networks, Optical Networks Magazine, 1 (2000), 47-60.
Pseudocode of FD(FFD) Algorithm
Pseudocode of FD(FFD) Algorithm
Pseudocode of PSO Algorithm
Pseudocode of a basic MEDP-RWA algorithm
Pseudocode of GMR algorithm
Number of times that the best value is achieved by different methods
Frequency graphs of testing results on zib54
Frequency graphs of testing results on ta2
Frequency graphs of testing results on Germany50
Relative difference of computational time on Norway
Relative difference of computational time on giul
Relative difference of computational time on graph Germany
Relative difference of computational time on graph ta2
Main quantitative characteristics of testing networks
 Graph $\left | V \right |$ $\left | E \right |$ Min. Avg. Max. Diameter CHNNET 15 27 3 3.6 5 5 NSF 16 25 2 3.1 4 4 New York 16 49 2 6.1 11 3 APPANET 20 32 3 3.2 4 6 EON 20 39 2 3.9 7 5 France 25 45 2 3.6 10 5 Norway 27 51 2 3.8 6 7 cost266 37 57 2 3.1 5 8 janos-us-ca 39 67 2 3.1 5 10 giul 39 86 3 4.4 8 6 piro40 40 89 4 4.5 5 7 USAnet 46 75 2 3.3 5 11 Germany50 50 88 2 3.5 5 9 zib54 54 80 1 3.0 10 8 ta2 65 108 1 3.3 10 8 (Min., Avg. and Max. denote the minimum, average and maximum degree, respectively)
 Graph $\left | V \right |$ $\left | E \right |$ Min. Avg. Max. Diameter CHNNET 15 27 3 3.6 5 5 NSF 16 25 2 3.1 4 4 New York 16 49 2 6.1 11 3 APPANET 20 32 3 3.2 4 6 EON 20 39 2 3.9 7 5 France 25 45 2 3.6 10 5 Norway 27 51 2 3.8 6 7 cost266 37 57 2 3.1 5 8 janos-us-ca 39 67 2 3.1 5 10 giul 39 86 3 4.4 8 6 piro40 40 89 4 4.5 5 7 USAnet 46 75 2 3.3 5 11 Germany50 50 88 2 3.5 5 9 zib54 54 80 1 3.0 10 8 ta2 65 108 1 3.3 10 8 (Min., Avg. and Max. denote the minimum, average and maximum degree, respectively)
Results of GMR and bin-packing based methods (time unit: sec)
 GA_MEDP_RWA (GMR) FF FFD BF BFD instance max avg min avg time # wl $t_{FF}$ # wl $t_{FFD}$ # wl $t_{BF}$ # wl $t_{BFD}$ CHNNET_02 4 4.0 4 2.07 4 0.24 4 0.14 4 0.11 5 0.14 CHNNET_04 5 4.4 4 3.02 5 0.18 4 0.15 5 0.23 4 0.26 CHNNET_06 11 11.0 11 10.66 11 0.70 11 0.62 11 0.81 11 0.96 CHNNET_08 14 14.0 14 13.50 15 1.02 14 1.17 14 1.21 14 1.51 CHNNET_1 16 15.0 15 16.31 15 1.48 15 1.42 16 1.75 15 2.20 NSF_02 5 5.0 5 2.41 6 0.34 5 0.17 6 0.19 6 0.18 NSF_04 8 7.3 7 5.80 9 0.37 9 0.35 8 0.52 8 0.50 NSF_06 9 8.8 8 7.97 10 0.46 10 0.49 9 0.59 9 0.64 NSF_08 14 13.2 13 15.11 17 1.06 15 1.10 14 1.50 14 1.47 NSF_1 17 16.6 16 22.89 19 1.70 17 1.45 17 1.96 17 2.08 NewYork_02 2 2.0 2 1.37 2 0.14 2 0.08 2 0.07 2 0.08 NewYork_04 3 3.0 3 2.65 3 0.23 3 0.20 3 0.23 3 0.23 NewYork_06 5 4.1 4 3.93 4 0.33 4 0.34 4 0.34 4 0.33 NewYork_08 7 6.0 6 6.75 7 0.64 6 0.58 7 0.62 6 0.81 NewYork_1 8 8.0 8 7.03 8 0.73 8 0.80 8 1.02 8 2.30 ARPANET_02 10 9.1 9 10.58 9 0.55 9 0.53 9 0.64 9 0.86 ARPANET_04 13 12.1 12 16.96 12 1.05 12 1.29 12 1.31 12 1.76 ARPANET_06 21 21.0 21 25.98 21 2.70 21 2.68 21 3.39 21 4.58 ARPANET_08 29 29.0 29 32.31 30 5.46 29 6.14 30 7.82 29 10.94 ARPANET_1 33 33.0 33 62.85 34 6.85 33 7.77 34 10.19 33 12.68 EON_02 4 3.1 3 3.19 3 0.22 4 0.17 4 0.34 4 0.36 EON_04 9 8.3 8 14.74 10 1.10 9 1.14 8 1.53 8 1.54 EON_06 11 11.0 11 17.52 13 1.21 11 1.18 11 1.94 11 1.90 EON_08 14 13.7 13 26.63 16 2.20 13 2.99 13 3.61 13 3.72 EON_1 19 18.1 18 38.17 22 3.64 18 3.58 18 5.77 18 5.73 France_02 8 8.0 8 8.87 8 1.30 8 1.27 8 1.57 8 1.72 France _04 13 12.8 12 16.05 14 4.29 13 4.90 13 5.14 13 6.03 France _06 22 22.0 22 39.07 22 8.73 22 10.55 22 11.69 22 15.27 France _08 27 26.3 26 47.09 28 13.25 27 14.00 27 18.53 26 24.20 France _1 34 34.0 34 60.09 34 22.54 34 23.99 34 29.93 34 35.84
 GA_MEDP_RWA (GMR) FF FFD BF BFD instance max avg min avg time # wl $t_{FF}$ # wl $t_{FFD}$ # wl $t_{BF}$ # wl $t_{BFD}$ CHNNET_02 4 4.0 4 2.07 4 0.24 4 0.14 4 0.11 5 0.14 CHNNET_04 5 4.4 4 3.02 5 0.18 4 0.15 5 0.23 4 0.26 CHNNET_06 11 11.0 11 10.66 11 0.70 11 0.62 11 0.81 11 0.96 CHNNET_08 14 14.0 14 13.50 15 1.02 14 1.17 14 1.21 14 1.51 CHNNET_1 16 15.0 15 16.31 15 1.48 15 1.42 16 1.75 15 2.20 NSF_02 5 5.0 5 2.41 6 0.34 5 0.17 6 0.19 6 0.18 NSF_04 8 7.3 7 5.80 9 0.37 9 0.35 8 0.52 8 0.50 NSF_06 9 8.8 8 7.97 10 0.46 10 0.49 9 0.59 9 0.64 NSF_08 14 13.2 13 15.11 17 1.06 15 1.10 14 1.50 14 1.47 NSF_1 17 16.6 16 22.89 19 1.70 17 1.45 17 1.96 17 2.08 NewYork_02 2 2.0 2 1.37 2 0.14 2 0.08 2 0.07 2 0.08 NewYork_04 3 3.0 3 2.65 3 0.23 3 0.20 3 0.23 3 0.23 NewYork_06 5 4.1 4 3.93 4 0.33 4 0.34 4 0.34 4 0.33 NewYork_08 7 6.0 6 6.75 7 0.64 6 0.58 7 0.62 6 0.81 NewYork_1 8 8.0 8 7.03 8 0.73 8 0.80 8 1.02 8 2.30 ARPANET_02 10 9.1 9 10.58 9 0.55 9 0.53 9 0.64 9 0.86 ARPANET_04 13 12.1 12 16.96 12 1.05 12 1.29 12 1.31 12 1.76 ARPANET_06 21 21.0 21 25.98 21 2.70 21 2.68 21 3.39 21 4.58 ARPANET_08 29 29.0 29 32.31 30 5.46 29 6.14 30 7.82 29 10.94 ARPANET_1 33 33.0 33 62.85 34 6.85 33 7.77 34 10.19 33 12.68 EON_02 4 3.1 3 3.19 3 0.22 4 0.17 4 0.34 4 0.36 EON_04 9 8.3 8 14.74 10 1.10 9 1.14 8 1.53 8 1.54 EON_06 11 11.0 11 17.52 13 1.21 11 1.18 11 1.94 11 1.90 EON_08 14 13.7 13 26.63 16 2.20 13 2.99 13 3.61 13 3.72 EON_1 19 18.1 18 38.17 22 3.64 18 3.58 18 5.77 18 5.73 France_02 8 8.0 8 8.87 8 1.30 8 1.27 8 1.57 8 1.72 France _04 13 12.8 12 16.05 14 4.29 13 4.90 13 5.14 13 6.03 France _06 22 22.0 22 39.07 22 8.73 22 10.55 22 11.69 22 15.27 France _08 27 26.3 26 47.09 28 13.25 27 14.00 27 18.53 26 24.20 France _1 34 34.0 34 60.09 34 22.54 34 23.99 34 29.93 34 35.84
Results of GMR and bin-packing based methods (cont'd)
 GA_MEDP_RWA (GMR) FF FFD BF BFD instance max avg min avg time # wl $t_{FF}$ # wl $t_{FFD}$ # wl $t_{BF}$ # wl $t_{BFD}$ Norway_02 9 8.6 8 9.26 10 1.67 8 1.78 9 1.90 8 2.17 Norway _04 15 14.6 14 18.47 15 3.75 15 4.12 15 4.56 15 5.43 Norway _06 22 21.4 21 32.84 22 7.87 22 8.20 22 9.73 22 12.32 Norway _08 30 29.5 29 46.41 31 15.05 30 16.59 31 19.79 30 24.59 Norway _1 37 36.6 36 63.77 37 21.99 36 23.75 38 28.58 36 36.13 cost266_02 19 18.2 18 46.68 19 7.34 18 7.44 19 10.38 19 12.40 cost266_04 35 34.1 33 159.69 35 27.28 33 28.58 36 36.35 35 47.28 cost266_06 54 53.1 53 237.55 54 67.18 53 70.90 55 98.00 53 117.30 cost266_08 68 67.2 67 275.29 67 120.73 68 144.22 69 190.55 68 273.40 janos-us-ca_02 26 26.0 26 54.75 27 16.61 26 16.53 27 27.40 27 27.58 janos-us-ca_04 40 39.6 39 97.42 42 49.52 39 59.49 43 70.75 40 88.45 janos-us-ca_06 68 67.8 67 210.36 71 110.99 69 124.10 72 157.50 68 200.32 janos-us-ca_08 89 88.2 88 326.39 92 199.09 92 212.16 93 257.39 91 345.67 giul_02 11 10.3 10 26.93 10 7.40 10 8.11 10 8.89 10 11.24 giul_04 21 20.2 19 82.66 19 29.10 19 31.70 19 37.42 19 46.16 giul_06 25 24.6 24 150.73 25 48.91 25 55.96 25 59.10 24 73.12 giul_08 36 34.9 34 226.85 35 113.85 34 125.20 36 134.25 34 181.70 piro40_02 17 17 17 55.51 17 10.67 17 12.58 17 15.20 17 18.50 piro40_04 28 28 28 118.50 28 34.98 28 38.00 28 43.05 28 54.44 piro40_06 46 46 46 208.81 46 61.98 46 68.00 46 84.33 46 109.15 piro40_08 60 60 60 338.61 60 105.97 60 119.35 60 140.20 60 181.52 USAnet_02 25 24.0 23 86.13 25 35.62 26 37.95 26 45.15 25 56.09 USAnet_04 47 45.9 45 355.64 47 130.43 45 147.76 48 176.43 45 228.50 USAnet_06 67 66.1 65 422.63 68 244.00 65 287.58 68 346.66 66 432.89 USAnet_08 88 86.5 85 490.20 88 475.69 85 544.98 89 587.11 86 791.38 Germany50_02 21 20.7 20 109.28 21 35.86 21 36.10 21 46.10 22 58.58 Germany50_04 45 44.2 43 350.24 45 152.90 44 163.00 48 217.60 48 276.50 Germany50_06 61 60.5 60 584.92 63 347.70 61 410.61 63 430.89 65 544.74 Germany50_08 76 74.7 73 921.59 77 552.50 75 725.90 79 843.70 76 1098.40 zib54_02 32 31.1 30 149.44 33 52.28 31 55.01 35 77.70 33 91.67 zib54_04 67 65.7 65 406.90 67 209.80 65 220.50 73 304.64 71 379.55 zib54_06 92 90.0 88 762.56 91 442.95 89 522.85 99 696.78 98 889.98 zib54_08 122 119.2 117 1446.41 117 994.60 117 939.60 130 1229.60 127 1457.22 ta2_02 35 34.6 34 411.59 35 148.93 34 163.00 35 188.75 35 250.99 ta2_04 72 70.1 69 945.79 72 507.30 69 542.50 73 635.57 70 813.04 ta2_06 99 97.4 96 1865.20 98 1311.70 98 1484.00 100 1881.30 97 2047.40 ta2_08 131 129.7 128 2835.57 130 1935.30 129 2577.90 135 2644.43 128 3713.46
 GA_MEDP_RWA (GMR) FF FFD BF BFD instance max avg min avg time # wl $t_{FF}$ # wl $t_{FFD}$ # wl $t_{BF}$ # wl $t_{BFD}$ Norway_02 9 8.6 8 9.26 10 1.67 8 1.78 9 1.90 8 2.17 Norway _04 15 14.6 14 18.47 15 3.75 15 4.12 15 4.56 15 5.43 Norway _06 22 21.4 21 32.84 22 7.87 22 8.20 22 9.73 22 12.32 Norway _08 30 29.5 29 46.41 31 15.05 30 16.59 31 19.79 30 24.59 Norway _1 37 36.6 36 63.77 37 21.99 36 23.75 38 28.58 36 36.13 cost266_02 19 18.2 18 46.68 19 7.34 18 7.44 19 10.38 19 12.40 cost266_04 35 34.1 33 159.69 35 27.28 33 28.58 36 36.35 35 47.28 cost266_06 54 53.1 53 237.55 54 67.18 53 70.90 55 98.00 53 117.30 cost266_08 68 67.2 67 275.29 67 120.73 68 144.22 69 190.55 68 273.40 janos-us-ca_02 26 26.0 26 54.75 27 16.61 26 16.53 27 27.40 27 27.58 janos-us-ca_04 40 39.6 39 97.42 42 49.52 39 59.49 43 70.75 40 88.45 janos-us-ca_06 68 67.8 67 210.36 71 110.99 69 124.10 72 157.50 68 200.32 janos-us-ca_08 89 88.2 88 326.39 92 199.09 92 212.16 93 257.39 91 345.67 giul_02 11 10.3 10 26.93 10 7.40 10 8.11 10 8.89 10 11.24 giul_04 21 20.2 19 82.66 19 29.10 19 31.70 19 37.42 19 46.16 giul_06 25 24.6 24 150.73 25 48.91 25 55.96 25 59.10 24 73.12 giul_08 36 34.9 34 226.85 35 113.85 34 125.20 36 134.25 34 181.70 piro40_02 17 17 17 55.51 17 10.67 17 12.58 17 15.20 17 18.50 piro40_04 28 28 28 118.50 28 34.98 28 38.00 28 43.05 28 54.44 piro40_06 46 46 46 208.81 46 61.98 46 68.00 46 84.33 46 109.15 piro40_08 60 60 60 338.61 60 105.97 60 119.35 60 140.20 60 181.52 USAnet_02 25 24.0 23 86.13 25 35.62 26 37.95 26 45.15 25 56.09 USAnet_04 47 45.9 45 355.64 47 130.43 45 147.76 48 176.43 45 228.50 USAnet_06 67 66.1 65 422.63 68 244.00 65 287.58 68 346.66 66 432.89 USAnet_08 88 86.5 85 490.20 88 475.69 85 544.98 89 587.11 86 791.38 Germany50_02 21 20.7 20 109.28 21 35.86 21 36.10 21 46.10 22 58.58 Germany50_04 45 44.2 43 350.24 45 152.90 44 163.00 48 217.60 48 276.50 Germany50_06 61 60.5 60 584.92 63 347.70 61 410.61 63 430.89 65 544.74 Germany50_08 76 74.7 73 921.59 77 552.50 75 725.90 79 843.70 76 1098.40 zib54_02 32 31.1 30 149.44 33 52.28 31 55.01 35 77.70 33 91.67 zib54_04 67 65.7 65 406.90 67 209.80 65 220.50 73 304.64 71 379.55 zib54_06 92 90.0 88 762.56 91 442.95 89 522.85 99 696.78 98 889.98 zib54_08 122 119.2 117 1446.41 117 994.60 117 939.60 130 1229.60 127 1457.22 ta2_02 35 34.6 34 411.59 35 148.93 34 163.00 35 188.75 35 250.99 ta2_04 72 70.1 69 945.79 72 507.30 69 542.50 73 635.57 70 813.04 ta2_06 99 97.4 96 1865.20 98 1311.70 98 1484.00 100 1881.30 97 2047.40 ta2_08 131 129.7 128 2835.57 130 1935.30 129 2577.90 135 2644.43 128 3713.46
Results of GMR and PSO (time unit: sec)
 GA_MEDP_RWA PSO instance max avg min avg time max avg min avg time CHNNET_02 4 4.0 4 2.07 7 5.9 5 12.97 CHNNET_04 5 4.4 4 3.02 9 7.5 7 18.60 CHNNET_06 11 11.0 11 10.66 18 15.5 13 30.39 CHNNET_08 14 14.0 14 13.50 24 20.4 18 48.95 CHNNET_1 16 15.0 15 16.31 25 21.5 18 78.96 NSF_02 5 5.0 5 2.41 7 6.0 5 6.75 NSF_04 8 7.3 7 5.80 10 9.0 8 13.75 NSF_06 9 8.8 8 7.97 11 10.0 9 16.19 NSF_08 14 13.2 13 15.11 17 15.5 14 29.02 NSF_1 17 16.6 16 22.89 20 18.9 18 36.03 NewYork_02 2 2.0 2 1.37 3 2.9 2 7.37 NewYork _04 3 3.0 3 2.65 6 5.2 4 18.78 NewYork _06 5 4.1 4 3.93 8 6.8 6 25.51 NewYork _08 7 6.0 6 6.75 10 9.1 8 38.35 NewYork _1 8 8.0 8 7.03 12 10.7 10 50.17 ARPANET_02 10 9.1 9 10.58 13 11.3 10 21.30 ARPANET_04 13 12.1 12 16.96 19 16.3 14 62.15 ARPANET_06 21 21.0 21 25.98 28 24.9 22 93.86 ARPANET_08 29 29.0 29 32.31 40 36.5 33 133.70 ARPANET_1 33 33.0 33 62.85 46 41.2 36 172.60 EON_02 4 3.1 3 3.19 5 4.3 4 10.40 EON_04 9 8.3 8 14.74 12 10.6 10 35.34 EON_06 11 11.0 11 17.52 15 13.0 12 46.61 EON_08 14 13.7 13 26.63 20 18.2 17 48.92 EON_1 19 18.1 18 38.17 25 22.9 22 73.32 France_02 8 8.0 8 8.87 14 11.5 10 50.60 France _04 13 12.8 12 16.05 22 19.3 17 86.48 France _06 22 22.0 22 39.07 34 30.4 27 129.16 France _08 27 26.3 26 47.09 45 40.3 38 290.72 France _1 34 34.0 34 60.09 54 48.9 45 236.64
 GA_MEDP_RWA PSO instance max avg min avg time max avg min avg time CHNNET_02 4 4.0 4 2.07 7 5.9 5 12.97 CHNNET_04 5 4.4 4 3.02 9 7.5 7 18.60 CHNNET_06 11 11.0 11 10.66 18 15.5 13 30.39 CHNNET_08 14 14.0 14 13.50 24 20.4 18 48.95 CHNNET_1 16 15.0 15 16.31 25 21.5 18 78.96 NSF_02 5 5.0 5 2.41 7 6.0 5 6.75 NSF_04 8 7.3 7 5.80 10 9.0 8 13.75 NSF_06 9 8.8 8 7.97 11 10.0 9 16.19 NSF_08 14 13.2 13 15.11 17 15.5 14 29.02 NSF_1 17 16.6 16 22.89 20 18.9 18 36.03 NewYork_02 2 2.0 2 1.37 3 2.9 2 7.37 NewYork _04 3 3.0 3 2.65 6 5.2 4 18.78 NewYork _06 5 4.1 4 3.93 8 6.8 6 25.51 NewYork _08 7 6.0 6 6.75 10 9.1 8 38.35 NewYork _1 8 8.0 8 7.03 12 10.7 10 50.17 ARPANET_02 10 9.1 9 10.58 13 11.3 10 21.30 ARPANET_04 13 12.1 12 16.96 19 16.3 14 62.15 ARPANET_06 21 21.0 21 25.98 28 24.9 22 93.86 ARPANET_08 29 29.0 29 32.31 40 36.5 33 133.70 ARPANET_1 33 33.0 33 62.85 46 41.2 36 172.60 EON_02 4 3.1 3 3.19 5 4.3 4 10.40 EON_04 9 8.3 8 14.74 12 10.6 10 35.34 EON_06 11 11.0 11 17.52 15 13.0 12 46.61 EON_08 14 13.7 13 26.63 20 18.2 17 48.92 EON_1 19 18.1 18 38.17 25 22.9 22 73.32 France_02 8 8.0 8 8.87 14 11.5 10 50.60 France _04 13 12.8 12 16.05 22 19.3 17 86.48 France _06 22 22.0 22 39.07 34 30.4 27 129.16 France _08 27 26.3 26 47.09 45 40.3 38 290.72 France _1 34 34.0 34 60.09 54 48.9 45 236.64
Results of GMR and PSO (con't)
 GA_MEDP_RWA PSO instance max avg min avg time max avg min avg time Norway_02 9 8.6 8 9.26 15 13.1 11 61.56 Norway _04 15 14.6 14 18.47 25 22.5 20 101.40 Norway _06 22 21.4 21 32.84 37 32.6 29 202.11 Norway _08 30 29.5 29 46.41 48 43.6 40 254.75 Norway _1 37 36.6 36 63.77 59 54.1 49 323.64 cost266_02 19 18.2 18 46.68 26 23.8 21 105.22 cost266_04 35 34.1 33 159.69 54 50.9 47 239.95 cost266_06 54 53.1 53 237.55 79 73.2 70 436.38 cost266_08 68 67.2 67 275.29 99 93.4 88 703.53 janos-us-ca_02 26 26.0 26 54.75 40 36.1 31 288.18 janos-us-ca_04 40 39.6 39 97.42 66 60.5 56 529.74 janos-us-ca_06 68 67.8 67 210.36 111 102.7 94 959.93 janos-us-ca_08 89 88.2 88 326.39 144 134.1 122 1239.46 giul_02 11 10.3 10 26.93 16 14.5 13 294.44 giul_04 21 20.2 19 82.66 32 28.8 26 451.61 giul_06 25 24.6 24 150.73 42 38.3 34 738.65 giul_08 36 34.9 34 226.85 59 53.5 49 990.63 piro40_02 17 17 17 55.51 22 20.0 19 159.88 piro40_04 28 28 28 118.50 36 33.7 32 300.88 piro40_06 46 46 46 208.81 66 58.7 52 474.73 piro40_08 60 60 60 338.61 86 79.4 73 684.40 USAnet_02 25 24.0 23 86.13 40 36.4 32 446.56 USAnet_04 47 45.9 45 355.64 76 69.5 65 983.93 USAnet_06 67 66.1 65 422.63 106 98.1 92 1418.73 USAnet_08 88 86.5 85 490.20 139 126.5 120 1715.20 Germany50_02 21 20.7 20 109.28 37 33.5 31 379.56 Germany50_04 45 44.2 43 350.24 74 69.4 66 683.95 Germany50_06 61 60.5 60 584.92 102 95.7 91 969.87 Germany50_08 76 74.7 73 921.59 126 120.5 115 1271.90 zib54_02 32 31.1 30 149.44 75 64.1 58 619.63 zib54_04 67 65.7 65 406.90 159 141.6 133 1462.19 zib54_06 92 90.0 88 762.56 208 195.2 187 2955.29 zib54_08 122 119.2 117 1446.41 289 266.3 250 4147.09 ta2_02 35 34.6 34 411.59 75 70.3 66 1453.68 ta2_04 72 70.1 69 945.79 152 141.8 133 2852.21 ta2_06 99 97.4 96 1865.20 208 198.7 181 3867.39 ta2_08 131 129.7 128 2835.57 284 271.0 258 5953.52
 GA_MEDP_RWA PSO instance max avg min avg time max avg min avg time Norway_02 9 8.6 8 9.26 15 13.1 11 61.56 Norway _04 15 14.6 14 18.47 25 22.5 20 101.40 Norway _06 22 21.4 21 32.84 37 32.6 29 202.11 Norway _08 30 29.5 29 46.41 48 43.6 40 254.75 Norway _1 37 36.6 36 63.77 59 54.1 49 323.64 cost266_02 19 18.2 18 46.68 26 23.8 21 105.22 cost266_04 35 34.1 33 159.69 54 50.9 47 239.95 cost266_06 54 53.1 53 237.55 79 73.2 70 436.38 cost266_08 68 67.2 67 275.29 99 93.4 88 703.53 janos-us-ca_02 26 26.0 26 54.75 40 36.1 31 288.18 janos-us-ca_04 40 39.6 39 97.42 66 60.5 56 529.74 janos-us-ca_06 68 67.8 67 210.36 111 102.7 94 959.93 janos-us-ca_08 89 88.2 88 326.39 144 134.1 122 1239.46 giul_02 11 10.3 10 26.93 16 14.5 13 294.44 giul_04 21 20.2 19 82.66 32 28.8 26 451.61 giul_06 25 24.6 24 150.73 42 38.3 34 738.65 giul_08 36 34.9 34 226.85 59 53.5 49 990.63 piro40_02 17 17 17 55.51 22 20.0 19 159.88 piro40_04 28 28 28 118.50 36 33.7 32 300.88 piro40_06 46 46 46 208.81 66 58.7 52 474.73 piro40_08 60 60 60 338.61 86 79.4 73 684.40 USAnet_02 25 24.0 23 86.13 40 36.4 32 446.56 USAnet_04 47 45.9 45 355.64 76 69.5 65 983.93 USAnet_06 67 66.1 65 422.63 106 98.1 92 1418.73 USAnet_08 88 86.5 85 490.20 139 126.5 120 1715.20 Germany50_02 21 20.7 20 109.28 37 33.5 31 379.56 Germany50_04 45 44.2 43 350.24 74 69.4 66 683.95 Germany50_06 61 60.5 60 584.92 102 95.7 91 969.87 Germany50_08 76 74.7 73 921.59 126 120.5 115 1271.90 zib54_02 32 31.1 30 149.44 75 64.1 58 619.63 zib54_04 67 65.7 65 406.90 159 141.6 133 1462.19 zib54_06 92 90.0 88 762.56 208 195.2 187 2955.29 zib54_08 122 119.2 117 1446.41 289 266.3 250 4147.09 ta2_02 35 34.6 34 411.59 75 70.3 66 1453.68 ta2_04 72 70.1 69 945.79 152 141.8 133 2852.21 ta2_06 99 97.4 96 1865.20 208 198.7 181 3867.39 ta2_08 131 129.7 128 2835.57 284 271.0 258 5953.52
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