## Abstract

The paper deals with on-line routing in wavelength division multiplexing (WDM) optical networks. A sequence of requests arrives over time, each is a pair of nodes to be connected by a path. The problem is to assign a wavelength and a path to each pair, so that no two paths sharing a link are assigned the same wavelength. The goal is to minimize the number of wavelengths used to establish all connections. Raghavan and Upfal (Proc. 26th Annual Symp. on Theory of Computing, 1994, pp. 133-143) considered the off-line version of the problem, which was further studied in Aumann and Rabani (Proc. 6th ACM-SIAM Symp. on Discrete Algorithms, 1995, pp. 567-576), Kaklamanis and Persiano (Proc. 4th Annual European Symp. on Algorithms, Lecture Notes in Computer Science, vol. 1136, Springer, Berlin, 1996, pp. 460-470), Mihail et al. (Proc. 36th IEEE Annual Symp. on Foundations of Computer Science, 1995, pp. 548-557), Rabani, (Proc. 37th Annual Symp. on Foundations of Computer Science, 1996, pp. 400-409). For a line topology, the problem is the well-studied interval graph coloring problem. On-line algorithms for this problem have been analyzed in Kierstead and Trotter (Congr. Numer. 33 (1981) 143-153). We consider trees, trees of rings, and meshes topologies, previously studied in the off-line case. We give on-line algorithms with competitive ratio O(log n) for all these topologies. We give a matching Ω(log n) lower bound for meshes. We also prove that any algorithm for trees cannot have competitive ratio better than Ω(log n/log log n). We also consider the problem where every edge is associated with parallel links. While in WDM technology, a fiber link requires different wavelengths for every transmission, space division multiplexing technology allows parallel links for a single wavelength, at an additional cost. Thus, it may be beneficial in terms of network economics to combine between the two technologies (this is indeed done in practice). For arbitrary networks with Ω(log n) parallel links we give an on-line algorithm with competitive ratio O(log n).

Original language | American English |
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Pages (from-to) | 19-39 |

Number of pages | 21 |

Journal | Theoretical Computer Science |

Volume | 221 |

Issue number | 1-2 |

DOIs | |

State | Published - 28 Jun 1999 |

Externally published | Yes |

### Bibliographical note

Funding Information:The paper deals with on-line routing in wavelength division multiplexing (WDM) optical networks. A sequence of requests arrives over time, each is a pair of nodes to be connected by a path. The problem is to assign a wavelength and a path to each pair, so that no two paths sharing a link are assigned the same wavelength. The goal is to minimize the number of wavelengths used to establish all connections. Raghavan and Upfal (Proc. 26th Annual Symp. on Theory of Computing, 1994, pp. 133-143) considered the off-line version of the problem, which was further studied in Aumann and Rabani (Proc. 6th ACM-SIAM Symp. on Discrete Algorithms, 1995, pp. 567-576), Kaklamanis and Persiano (Proc. 4th Annual European Symp. on Algorithms, Lecture Notes in Computer Science, vol. 1136, Springer, Berlin, 1996, pp. 460-470), Mihail et al. (Proc. 36th IEEE Annual Symp. on Foundations of Computer Science, 1995, pp. 548-557), Rabani, (Proc. 37th Annual Symp. on Foundations of Computer Science, 1996, pp. 400-409). For a line topology, the problem is the well-studied interval graph coloring problem. On-line algorithms for this problem have been analyzed in Kierstead and Trotter (Congr. Numer. 33 (1981) 143-153). We consider trees, trees of rings, and meshes topologies, previously studied in the off-line case. We give on-line algorithms with competitive ratio O(logn) for all these topologies. We give a matching f~(logn) lower bound for meshes. We also prove that any algorithm for trees cannot have competitive ratio better than f~(log n/log log n). We also consider the problem where every edge is associated with parallel links. While in WDM technology, a fiber link requires different wavelengths for every transmission, space division multiplexing technology allows parallel links for a single wavelength, at an additional cost. Thus, it may be beneficial in terms of network economics to combine between the two technologies (this is indeed done in "~:Ã preliminary version of this paper has appeared in the Proceedings of the 24th International Colloquium on Automata, Languages, and Programming (ICALP 97), pp. 516-526, Lecture Notes in Computer Science 1256, Springer, Berlin, 1997. * Corresponding author. Tel.: 39-6-49918341; fax: 39-5-85300849. E-mail addresses." yairb@icsi.berkeley.edu (Y. Bartal), leon@dis.uniromal.it (S. Leonardi) t Research supported in part by the Rothschild Postdoctoral fellowship and by the National Science Foundation operating grants CCR-9304722 and NCR-9416101. 2 This work was partially done while the author was a post-doc at the International Computer Science Institute (ICSI), Berkeley. This work is partly supported by EU ESPRIT Long Term Research Project ALCOM-IT under contract no. 20244, and by Italian Ministry of Scientific Research Project 40% "Algoritmi, Modelli di Calcolo e Strutture Informative".

## Keywords

- Competitive analysis
- Online algorithms
- Optical networks
- Routing