OPODIS 2011: Principles of Distributed Systems pp 4-17

# Online Regenerator Placement

• George B. Mertzios
• Mordechai Shalom
• Prudence W. H. Wong
• Shmuel Zaks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7109)

## 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 $$\Omega \left({{\log (|{E}|/d) \cdot \log d} \over{\log ({\log (|{E}|/d) \cdot \log d})}}\right)$$, 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.

## Keywords

online algorithms optical networks

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## Authors and Affiliations

• George B. Mertzios
• 1
• Mordechai Shalom
• 2
• Prudence W. H. Wong
• 3
• Shmuel Zaks
• 4
1. 1.School of Engineering and Computing SciencesDurham UniversityUK
2. 2.TelHai CollegeUpper GalileeIsrael
3. 3.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
4. 4.Department of Computer ScienceTechnionHaifaIsrael