## Abstract

The Connected Facility Location (CFL) is a network design problem that arises from a combination of the Uncapacitated Facility Location (FL) and the Steiner Tree (ST) problems. The Online Connected Facility Location problem (OCFL) is an online version of the CFL. San Felice et al. (2014) presented a randomized algorithm for the OCFL and proved that it is \(\mathrm {O}(\log ^2 n)\)-competitive, where *n* is the number of clients. That algorithm combines the sample-and-augment framework of Gupta, Kumar, Pál, and Roughgarden with previous algorithms for the Online Facility Location (OFL) and the Online Steiner Tree (OST) problems. In this paper we use a more precise analysis to show that the same algorithm is \(\mathrm {O}(\log n)\)-competitive. Since there is a lower bound of \(\mathrm {\Omega }(\log n)\) for this problem, our result achieves the best possible competitive ratio, asymptotically.

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## Acknowledgments

We would like to thank two anonymous referees whose suggestions and remarks greatly improved the presentation of this paper.

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## Additional information

First author supported by Grant No. 2009/15535-1, São Paulo Research Foundation (FAPESP).

Second author supported in part by NSF Grant CCF-1115256.

Third author supported in part by Bolsa de Produtividade do CNPq Proc. 303947/2008-0 and Edital Universal CNPq 477692/2012-5.

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San Felice, M.C., Williamson, D.P. & Lee, O. A Randomized \(\mathrm {O}(\log n)\)-Competitive Algorithm for the Online Connected Facility Location Problem.
*Algorithmica* **76**, 1139–1157 (2016). https://doi.org/10.1007/s00453-016-0115-1

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DOI: https://doi.org/10.1007/s00453-016-0115-1

### Keywords

- Online algorithms
- Competitive analysis
- Connected facility location
- Steiner tree
- Approximation algorithms
- Randomized algorithms