# Online Regenerator Placement

Online Regenerator Placement 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 Ω log ( | E | / d ) ⋅ log d log ( log ( | E | / d ) ⋅ log d ) ${\Omega } \left (\frac {\log (|{E}|/d) \cdot \log d}{\log (\log (|{E}|/d) \cdot \log d)}\right )$ , 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 l / 2 $\sqrt {l}/2$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

# Online Regenerator Placement

, Volume 61 (3) – Sep 28, 2016
16 pages

1

/lp/springer_journal/online-regenerator-placement-OI59qNceF0
Publisher
Springer Journals
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-016-9711-3
Publisher site
See Article on Publisher Site

### 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 Ω log ( | E | / d ) ⋅ log d log ( log ( | E | / d ) ⋅ log d ) ${\Omega } \left (\frac {\log (|{E}|/d) \cdot \log d}{\log (\log (|{E}|/d) \cdot \log d)}\right )$ , 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 l / 2 $\sqrt {l}/2$ 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.

### Journal

Theory of Computing SystemsSpringer Journals

Published: Sep 28, 2016

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