Abstract
Connections between nodes in optical networks are realized by lightpaths. Due to the decay of the signal, a regenerator has to be placed on every lightpath after at most d hops, for some given positive integer d. A regenerator can serve only one lightpath. The placement of regenerators has become an active area of research during recent years, and various optimization problems have been studied. The first such problem is the Regeneration Location Problem (Rlp), where the goal is to place the regenerators so as to minimize the total number of nodes containing them. We consider two extreme cases of online Rlp regarding the value of d and the number k of regenerators that can be used in any single node. (1) d is arbitrary and k unbounded. In this case a feasible solution always exists. We show an O(log|X|⋅ logd)-competitive randomized algorithm for any network topology, where X is the set of paths of length d. The algorithm can be made deterministic in some cases. We show a deterministic lower bound of \({\Omega } \left (\frac {\log (|{E}|/d) \cdot \log d}{\log (\log (|{E}|/d) \cdot \log d)}\right )\), where E is the edge set. (2) d = 2 and k = 1. In this case there is not necessarily a solution for a given input. We distinguish between feasible inputs (for which there is a solution) and infeasible ones. In the latter case, the objective is to satisfy the maximum number of lightpaths. For a path topology we show a lower bound of \(\sqrt {l}/2\) for the competitive ratio (where l is the number of internal nodes of the longest lightpath) on infeasible inputs, and a tight bound of 3 for the competitive ratio on feasible inputs.
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Notes
The node at distance 1 must have no regenerator, else there are two consecutive internal nodes with regenerators and the algorithm would have rejected the path.
Here we simplify the discussion slightly by assuming that the path P i does not contain a chain of two internal edges that both do not belong to any other paths because the algorithm can simply assign regenerators to alternate internal nodes without conflicting any other paths and this would not affect the number of paths that can be satisfied by the algorithm.
References
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, S.: The online set cover problem. SIAM J. Computing 39(2), 361–370 (2009)
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)
Chen, S., Ljubic, I., Raghavan, S.: The regenerator location problem. Networks 55(3), 205–220 (2010)
Fedrizzi, R., Galimberti, G.M., Gerstel, O., Martinelli, G., Salvadori, E., Saradhi, C.V., Tanzi, A., Zanardi, A.: Traffic Independent Heuristics for Regenerator Site Selection for Providing Any-to-Any Optical Connectivity. In: Proceedings of IEEE/OSA Conference on Optical Fiber Communications (OFC) (2010)
Flammini, M., Marchetti-Spaccamela, A., Monaco, G., Moscardelli, L., Zaks, S.: On the complexity of the regenerator placement problem in optical networks. IEEE/ACM Trans. Networking 19(2), 498–511 (2011)
Flammini, M., Monaco, G., Moscardelli, L., Shalom, M., Zaks, S.: Optimizing regenerator cost in traffic grooming. Theor. Comput. Sci. 412(52), 7109–7121 (2011)
Kim, S. W., Seo, S.W.: Regenerator placement algorithms for connection establishment in all-optical networks. IEEE Proceedings Communications 148(1), 25–30 (2001)
Mertzios, G.B., Sau, I., Shalom, M., Zaks, S.: Placing regenerators in optical networks to satisfy multiple sets of requests. IEEE Trans. Networking 20(6), 1870–1879 (2012)
Pachnicke, S., Paschenda, T., Krummrich, P.M.: Physical impairment based regenerator placement and routing in translucent optical networks. In: Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, page OWA2. Optical Society of America (2008)
Sriram, K., Griffith, D., Su, R., Golmie, N.: Static vs. dynamic regenerator assignment in optical switches: models and cost trade-offs. Workshop on High Performance Switching and Routing (HPSR), 151–155 (2004)
Yang, X., Ramamurthy, B.: Dynamic routing in translucent WDM optical networks. In: Proceedings of the IEEE International Conference on Communications (ICC), pp 955–971 (2002)
Yang, X., Ramamurthy, B.: Sparse regeneration in translucent wavelength-routed optical networks: Architecture, network design and wavelength routing. Photon Netw. Commun. 10(1), 39–53 (2005)
Acknowledgments
This work was supported in part by the Israel Science Foundation grant No. 1249/08 and British Council Grant UKTELHAI09.
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A preliminary version of this paper appeared in the 15th International Conference on Principles of Distributed Systems (OPODIS), Toulouse, France, December 2011, pp. 4–17.
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Mertzios, G.B., Shalom, M., Wong, P.W.H. et al. Online Regenerator Placement. Theory Comput Syst 61, 739–754 (2017). https://doi.org/10.1007/s00224-016-9711-3
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DOI: https://doi.org/10.1007/s00224-016-9711-3