Abstract

In this paper, we study the problem of regenerator site (RS) selection for mixed line rate optical networks (MLR-RSS), with the objective of minimizing the number of RSs for a given set of requests. We first provide the problem definition of MLR-RSS and show that the MLR-RSS problem is NP-complete. An integer linear programming model is formulated. We then present two heuristic algorithms, named the independent algorithm and the sequential algorithm, and two approximation algorithms, named the MLR-combined algorithm and the weighted MLR-combined algorithm. The performance of the algorithms is compared via simulation, and results show that the weighted MLR-combined algorithm has the best performance. Results suggest that our proposed MLR algorithm outperforms existing single line rate algorithms by more than 20%. Also, the RS distribution suggests that certain nodes in the network have a much higher probability of being chosen as RSs than the others.

© 2014 Optical Society of America

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2013 (2)

2012 (3)

2011 (4)

2010 (2)

A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol., vol.  28, no. 4, pp. 466–475, 2010.
[CrossRef]

S. Chen, I. Ljubic, and S. Raghavan, “The regenerator location problem,” Networks, vol.  55, no. 3, pp. 205–220, May 2010.

2008 (1)

2005 (1)

X. Yang and B. Ramamurthy, “Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing,” Photonic Netw. Commun., vol.  10, no. 1, pp. 39–53, 2005.

1979 (1)

V. Chvatal, “A greedy heuristic for the set-covering problem,” Math. Oper. Res., vol.  4, no. 3, pp. 233–235, 1979.
[CrossRef]

Batayneh, M.

Bathula, B.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Bergman, K.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

Bihon, D.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

Cankaya, H.

Chandrasekaran, R.

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

Chang, F.

F. Chang and M. Salmanian, “Routing requirements in photonic networks,” in IEEE/OSA NFOEC, 2003.

Chen, S.

S. Chen, I. Ljubic, and S. Raghavan, “The regenerator location problem,” Networks, vol.  55, no. 3, pp. 205–220, May 2010.

Chen, X.

Chiu, A.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

Christodoulopoulos, K.

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Reach adapting algorithms for mixed line rate WDM transport networks,” J. Lightwave Technol., vol.  29, no. 21, pp. 3350–3363, 2011.
[CrossRef]

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Adapting the transmission reach in mixed line rates WDM transport networks,” in 15th Int. Conf. on Optical Network Design and Modeling (ONDM), Bologna, Italy, Feb. 2011.

Chvatal, V.

V. Chvatal, “A greedy heuristic for the set-covering problem,” Math. Oper. Res., vol.  4, no. 3, pp. 233–235, 1979.
[CrossRef]

Doverspike, R.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Farahmand, F.

Fedrizzi, R.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Feuer, M.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Flammini, M.

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

Gagnaire, M.

M. Youssef, S. Zahr, and M. Gagnaire, “Cross optimization for RWA and regenerator placement in translucent WDM networks,” in 14th Conf. on Optical Network Design and Modeling (ONDM), Kyoto, Japan, 2010.

M. Youssef, S. Zahr, and M. Gagnaire, “Traffic-driven vs. topology-driven strategies for regeneration sites placement,” in IEEE Int. Conf. on Communications (ICC), Cape Town, South Africa, 2010.

Galimberti, G. M.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

Gao, C.

Gao, X.

Gerstel, O.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Hoffmann, M.

Huang, Y.

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

Ip, E.

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

Ji, F.

Ji, P.

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

Jue, J.

S. Varma and J. Jue, “Regenerator site selection in mixed line rate waveband optical networks,” J. Opt. Commun. Netw., vol.  5, no. 3, pp. 198–209, 2013.
[CrossRef]

Z. Zhu, X. Chen, F. Ji, L. Zhang, F. Farahmand, and J. Jue, “Energy-efficient translucent optical transport networks with mixed regenerator placement,” J. Lightwave Technol., vol.  30, no. 19, pp. 3147–3156, Oct. 2012.
[CrossRef]

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

Y. Zhu, X. Gao, W. Wu, and J. Jue, “Efficient impairment-constrained 3R regenerator placement for light-trees in optical networks,” J. Opt. Commun. Netw., vol.  3, no. 4, pp. 359–371, Apr. 2011.
[CrossRef]

W. Xie, Y. Zhu, and J. Jue, “Energy-efficient impairment-constrained 3R regenerator placement in optical networks,” in IEEE Int. Conf. on Communications (ICC), June 2012, pp. 3020–3024.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

Kim, I.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

Kirstadter, A.

Krummrich, P. M.

S. Pachnicke, T. Paschenda, and P. M. Krummrich, “Physical impairment based regenerator placement and routing in translucent optical networks,” in Optical Fiber Communication Conf. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2008, paper OWA2.

Li, G.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Ljubic, I.

S. Chen, I. Ljubic, and S. Raghavan, “The regenerator location problem,” Networks, vol.  55, no. 3, pp. 205–220, May 2010.

Magill, P.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Manousakis, K.

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Reach adapting algorithms for mixed line rate WDM transport networks,” J. Lightwave Technol., vol.  29, no. 21, pp. 3350–3363, 2011.
[CrossRef]

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Adapting the transmission reach in mixed line rates WDM transport networks,” in 15th Int. Conf. on Optical Network Design and Modeling (ONDM), Bologna, Italy, Feb. 2011.

Marchetti-Spaccamela, A.

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

Martinelli, G.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Mertzios, G.

G. Mertzios, I. Sau, M. Shalom, and S. Zaks, “Placing regenerators in optical networks to satisfy multiple sets of requests,” IEEE/ACM Trans. Netw., vol.  20, no. 6, pp. 1870–1879, 2012.
[CrossRef]

Monaco, G.

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

Moscardelli, L.

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

Mukherjee, B.

Nag, A.

A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol., vol.  28, no. 4, pp. 466–475, 2010.
[CrossRef]

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

A. Nag and M. Tornatore, “Transparent optical network design with mixed line rates,” in 2nd Int. Symp. on Advanced Networks and Telecommuniction Systems (ANTS), Mumbai, India, Dec. 2008.

Pachnicke, S.

S. Pachnicke, T. Paschenda, and P. M. Krummrich, “Physical impairment based regenerator placement and routing in translucent optical networks,” in Optical Fiber Communication Conf. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2008, paper OWA2.

Palacharla, P.

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

Paschenda, T.

S. Pachnicke, T. Paschenda, and P. M. Krummrich, “Physical impairment based regenerator placement and routing in translucent optical networks,” in Optical Fiber Communication Conf. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2008, paper OWA2.

Patel, A.

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

Raghavan, S.

S. Chen, I. Ljubic, and S. Raghavan, “The regenerator location problem,” Networks, vol.  55, no. 3, pp. 205–220, May 2010.

Ramamurthy, B.

X. Yang and B. Ramamurthy, “Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing,” Photonic Netw. Commun., vol.  10, no. 1, pp. 39–53, 2005.

Salmanian, M.

F. Chang and M. Salmanian, “Routing requirements in photonic networks,” in IEEE/OSA NFOEC, 2003.

Salvadori, E.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Saradhi, C. V.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

Sardesai, H.

Sau, I.

G. Mertzios, I. Sau, M. Shalom, and S. Zaks, “Placing regenerators in optical networks to satisfy multiple sets of requests,” IEEE/ACM Trans. Netw., vol.  20, no. 6, pp. 1870–1879, 2012.
[CrossRef]

Schupke, D.

Sekiya, M.

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

Shalom, M.

G. Mertzios, I. Sau, M. Shalom, and S. Zaks, “Placing regenerators in optical networks to satisfy multiple sets of requests,” IEEE/ACM Trans. Netw., vol.  20, no. 6, pp. 1870–1879, 2012.
[CrossRef]

She, Q.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

Shen, G.

Shen, Y.

Sinha, R.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Tanzi, A.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Tornatore, M.

A. Nag, M. Tornatore, and B. Mukherjee, “Optical network design with mixed line rates and multiple modulation formats,” J. Lightwave Technol., vol.  28, no. 4, pp. 466–475, 2010.
[CrossRef]

A. Nag and M. Tornatore, “Transparent optical network design with mixed line rates,” in 2nd Int. Symp. on Advanced Networks and Telecommuniction Systems (ANTS), Mumbai, India, Dec. 2008.

Varma, S.

Varvarigos, E.

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Reach adapting algorithms for mixed line rate WDM transport networks,” J. Lightwave Technol., vol.  29, no. 21, pp. 3350–3363, 2011.
[CrossRef]

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Adapting the transmission reach in mixed line rates WDM transport networks,” in 15th Int. Conf. on Optical Network Design and Modeling (ONDM), Bologna, Italy, Feb. 2011.

Wang, X.

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

Woodward, S.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

Wu, W.

Wu, Y.

Xie, W.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

W. Xie, Y. Zhu, and J. Jue, “Energy-efficient impairment-constrained 3R regenerator placement in optical networks,” in IEEE Int. Conf. on Communications (ICC), June 2012, pp. 3020–3024.

Yang, X.

X. Yang and B. Ramamurthy, “Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing,” Photonic Netw. Commun., vol.  10, no. 1, pp. 39–53, 2005.

Youssef, M.

M. Youssef, S. Zahr, and M. Gagnaire, “Traffic-driven vs. topology-driven strategies for regeneration sites placement,” in IEEE Int. Conf. on Communications (ICC), Cape Town, South Africa, 2010.

M. Youssef, S. Zahr, and M. Gagnaire, “Cross optimization for RWA and regenerator placement in translucent WDM networks,” in 14th Conf. on Optical Network Design and Modeling (ONDM), Kyoto, Japan, 2010.

Zahr, S.

M. Youssef, S. Zahr, and M. Gagnaire, “Cross optimization for RWA and regenerator placement in translucent WDM networks,” in 14th Conf. on Optical Network Design and Modeling (ONDM), Kyoto, Japan, 2010.

M. Youssef, S. Zahr, and M. Gagnaire, “Traffic-driven vs. topology-driven strategies for regeneration sites placement,” in IEEE Int. Conf. on Communications (ICC), Cape Town, South Africa, 2010.

Zaks, S.

G. Mertzios, I. Sau, M. Shalom, and S. Zaks, “Placing regenerators in optical networks to satisfy multiple sets of requests,” IEEE/ACM Trans. Netw., vol.  20, no. 6, pp. 1870–1879, 2012.
[CrossRef]

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

Zanardi, A.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

Zhang, L.

Zhang, Q.

C. Gao, H. Cankaya, A. Patel, J. Jue, X. Wang, Q. Zhang, P. Palacharla, and M. Sekiya, “Survivable impairment-aware traffic grooming and regenerator placement with connection-level protection,” J. Opt. Commun. Netw., vol.  4, no. 3, pp. 259–270, Mar. 2012.
[CrossRef]

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

Zhang, W.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

Zhu, Y.

Y. Zhu, X. Gao, W. Wu, and J. Jue, “Efficient impairment-constrained 3R regenerator placement for light-trees in optical networks,” J. Opt. Commun. Netw., vol.  3, no. 4, pp. 359–371, Apr. 2011.
[CrossRef]

W. Xie, Y. Zhu, and J. Jue, “Energy-efficient impairment-constrained 3R regenerator placement in optical networks,” in IEEE Int. Conf. on Communications (ICC), June 2012, pp. 3020–3024.

Zhu, Z.

IEEE/ACM Trans. Netw. (2)

M. Flammini, A. Marchetti-Spaccamela, G. Monaco, L. Moscardelli, and S. Zaks, “On the complexity of the regenerator placement problem in optical networks,” IEEE/ACM Trans. Netw., vol.  19, no. 2, pp. 498–511, Apr. 2011.
[CrossRef]

G. Mertzios, I. Sau, M. Shalom, and S. Zaks, “Placing regenerators in optical networks to satisfy multiple sets of requests,” IEEE/ACM Trans. Netw., vol.  20, no. 6, pp. 1870–1879, 2012.
[CrossRef]

J. Lightwave Technol. (5)

J. Opt. Commun. Netw. (4)

Math. Oper. Res. (1)

V. Chvatal, “A greedy heuristic for the set-covering problem,” Math. Oper. Res., vol.  4, no. 3, pp. 233–235, 1979.
[CrossRef]

Networks (1)

S. Chen, I. Ljubic, and S. Raghavan, “The regenerator location problem,” Networks, vol.  55, no. 3, pp. 205–220, May 2010.

Photonic Netw. Commun. (1)

X. Yang and B. Ramamurthy, “Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing,” Photonic Netw. Commun., vol.  10, no. 1, pp. 39–53, 2005.

Other (17)

F. Chang and M. Salmanian, “Routing requirements in photonic networks,” in IEEE/OSA NFOEC, 2003.

M. Feuer, S. Woodward, I. Kim, P. Palacharla, X. Wang, D. Bihon, B. Bathula, W. Zhang, R. Sinha, G. Li, and A. Chiu, “Simulations of a service velocity network employing regenerator site concentration,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NTu2J.5.

A. Nag and M. Tornatore, “Transparent optical network design with mixed line rates,” in 2nd Int. Symp. on Advanced Networks and Telecommuniction Systems (ANTS), Mumbai, India, Dec. 2008.

K. Christodoulopoulos, K. Manousakis, and E. Varvarigos, “Adapting the transmission reach in mixed line rates WDM transport networks,” in 15th Int. Conf. on Optical Network Design and Modeling (ONDM), Bologna, Italy, Feb. 2011.

W. Xie, Y. Zhu, and J. Jue, “Energy-efficient impairment-constrained 3R regenerator placement in optical networks,” in IEEE Int. Conf. on Communications (ICC), June 2012, pp. 3020–3024.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection for mixed line rate optical networks with flexible routing,” in 16th Int. Conf. on Optical Network Design and Modeling (ONDM), Apr. 2012.

W. Xie, J. Jue, X. Wang, Q. Zhang, Q. She, P. Palacharla, and M. Sekiya, “Regenerator site selection and regenerator placement for mixed line rate optical networks,” in Int. Conf. on Computing, Networking and Communications (ICNC), Jan. 2013, pp. 395–399.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, R. Doverspike, P. Magill, and K. Bergman, “Cost optimization using regenerator site concentration and routing in ROADM networks,” in 9th Int. Conf. on Design of Reliable Communication Networks (DRCN), Mar. 2013, pp. 154–162.

B. Bathula, R. Sinha, A. Chiu, M. Feuer, G. Li, S. Woodward, W. Zhang, K. Bergman, I. Kim, and P. Palacharla, “On concentrating regenerator sites in ROADM networks,” in Nat. Fiber Optic Engineers Conf. (NFOEC), Los Angeles, CA, 2012, paper NW3F.6.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Traffic independent heuristics for regenerator site selection for providing any-to-any optical connectivity,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG4.

C. V. Saradhi, A. Zanardi, R. Fedrizzi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “A framework for regenerator site selection based on multiple paths,” in Optical Fiber Communication Conf. (OFC), San Diego, CA, 2010, paper OTuG7.

C. V. Saradhi, R. Fedrizzi, A. Zanardi, E. Salvadori, G. M. Galimberti, A. Tanzi, G. Martinelli, and O. Gerstel, “Regenerator sites selection based on multiple paths considering impairments and protection requirements,” in 16th European Conf. on Networks and Optical Communications (NOC), 2011, pp. 84–87.

M. Youssef, S. Zahr, and M. Gagnaire, “Traffic-driven vs. topology-driven strategies for regeneration sites placement,” in IEEE Int. Conf. on Communications (ICC), Cape Town, South Africa, 2010.

M. Youssef, S. Zahr, and M. Gagnaire, “Cross optimization for RWA and regenerator placement in translucent WDM networks,” in 14th Conf. on Optical Network Design and Modeling (ONDM), Kyoto, Japan, 2010.

DARPA CORONET Program on Dynamic Multi-Terabit Core Optical Networks, “Sample Optical Network Topology Files” [Online]. Available: http://www.monarchna.com/topology.html .

A. Patel, P. Ji, A. Nag, Y. Huang, E. Ip, R. Chandrasekaran, and J. Jue, “Optimal placement of combined 2R/3R regenerators in WDM networks,” in 17th Opto-Electronics and Communications Conf. (OECC), July 2012, pp. 2348–2352.

S. Pachnicke, T. Paschenda, and P. M. Krummrich, “Physical impairment based regenerator placement and routing in translucent optical networks,” in Optical Fiber Communication Conf. and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), 2008, paper OWA2.

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Figures (13)

Fig. 1.
Fig. 1.

Example of network and regenerator placement.

Fig. 2.
Fig. 2.

Example of reachability path.

Fig. 3.
Fig. 3.

Example of a suboptimal solution obtained by the MLR-combined algorithm.

Fig. 4.
Fig. 4.

24-node network topology.

Fig. 5.
Fig. 5.

75-node CORONET [31].

Fig. 6.
Fig. 6.

Average number of RSs and regenerators versus total number of requests for the 24-node network.

Fig. 7.
Fig. 7.

Average number of RSs and regenerators versus total number of requests for the 75-node network.

Fig. 8.
Fig. 8.

Average number of RSs and regenerators versus different ratios of 100 Gbps requests in total number of requests for the 24-node network.

Fig. 9.
Fig. 9.

Average number of RSs and regenerators versus different ratios of 100 Gbps requests in total number of requests for the 75-node network.

Fig. 10.
Fig. 10.

Average number of RSs and regenerators versus different reachabilities of 100 Gbps signal for the 75-node network.

Fig. 11.
Fig. 11.

Average number of RSs and regenerators versus different reachabilities of 400 Gbps signal for the 75-node network.

Fig. 12.
Fig. 12.

RS distribution under uniform traffic.

Fig. 13.
Fig. 13.

RS distribution under nonuniform traffic.

Tables (3)

Tables Icon

Algorithm 1 Independent Algorithm

Tables Icon

Algorithm 2 Sequential HLRF Algorithm

Tables Icon

Algorithm 3 MLR-Combined Algorithm

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

min i = 1 | V | z i .
j = 1 | V | x k , s , j s , d j = 1 | V | x k , j , s s , d = req s , d , k , 1 s | V | , 1 d | V | , 1 k | L | ,
j = 1 | V | x k , j , d s , d j = 1 | V | x k , d , j s , d = req s , d , k , 1 s | V | , 1 d | V | , 1 k | L | ,
i = 1 | V | x k , i , j s , d = i = 1 | V | x k , j , i s , d , 1 s | V | , 1 d | V | , 1 k | L | , 1 j | V | , j s , d .
x k , i , j s , d r k , i , j s , d , 1 s | V | , 1 d | V | , 1 k | L | , 1 i | V | , 1 j | V | .
y k , i s , d x k , i , j s , d , 1 s | V | , 1 d | V | , 1 k | L | , 1 i | V | , i s , 1 j | V | .
z i s = 1 | V | d = 1 | V | k = 1 | L y k , i s , d V L , 1 i | V | .
A = { a 1 , 1 , a 2 , 1 , a 3 , 1 , a 4 , 1 , a 5 , 1 , a 1 , 2 , a 6 , 2 , a 3 , 2 , a 7 , 2 , a 5 , 2 } , b 1 = { a 1 , 1 , a 2 , 1 , a 1 , 2 , a 6 , 2 } , b 2 = { a 1 , 1 , a 2 , 1 , a 3 , 1 , a 4 , 1 , a 5 , 1 } , b 3 = { a 1 , 1 , a 2 , 1 , a 3 , 1 , a 4 , 1 , a 5 , 1 , a 1 , 2 , a 6 , 2 , a 3 , 2 , a 7 , 2 , a 5 , 2 } , b 4 = { a 1 , 1 , a 2 , 1 , a 3 , 1 , a 4 , 1 , a 5 , 1 } , b 5 = { a 4 , 1 , a 5 , 1 , a 7 , 2 , a 5 , 2 } , b 6 = { a 1 , 2 , a 6 , 2 , a 3 , 2 , a 7 , 2 , a 5 , 2 } , b 7 = { a 1 , 2 , a 6 , 2 , a 3 , 2 , a 7 , 2 , a 5 , 2 } .
A = { a 1 , 1 , 1 , a 2 , 1 , 1 , a 3 , 1 , 1 , a 4 , 1 , 1 , a 5 , 1 , 1 , a 6 , 1 , 1 , a 7 , 1 , 1 , a 1 , 2 , 2 , a 8 , 2 , 2 , a 3 , 2 , 2 , a 9 , 2 , 2 , a 5 , 2 , 2 , a 6 , 2 , 2 , a 7 , 2 , 2 } , b 1 = { a 1 , 1 , 1 , a 2 , 1 , 1 , a 1 , 2 , 2 } , b 2 = { a 1 , 1 , 1 , a 2 , 1 , 1 , a 3 , 1 , 1 } , b 3 = { a 1 , 1 , 1 , a 2 , 1 , 1 , a 3 , 1 , 1 , a 4 , 1 , 1 , a 3 , 2 , 2 } , b 4 = { a 1 , 1 , 1 , a 2 , 1 , 1 , a 3 , 1 , 1 , a 4 , 1 , 1 , a 5 , 1 , 1 , a 6 , 1 , 1 , a 7 , 1 , 1 } , b 5 = { a 4 , 1 , 1 , a 5 , 1 , 1 , a 6 , 1 , 1 , a 7 , 1 , 1 , a 5 , 2 , 2 } , b 6 = { a 5 , 1 , 1 , a 6 , 1 , 1 , a 7 , 1 , 1 , a 6 , 2 , 2 , a 7 , 2 , 2 } , b 7 = { a 6 , 1 , 1 , a 7 , 1 , 1 , a 7 , 2 , 2 } , b 8 = { a 1 , 2 , 2 , a 8 , 2 , 2 } , b 9 = { a 9 , 2 , 2 } .