## Abstract

We consider a variant of the online paging problem where the online algorithm may buy additional cache slots at a certain cost. The overall cost incurred equals the total cost for the cache plus the number of page faults. This problem and our results are a generalization of both, the classical paging problem and the ski rental problem.

We derive the following three tight results: (1) For the case where the cache cost depends linearly on the cache size, we give a λ-competitive online algorithm where λ ≈ 3.14619 is a solution of λ = 2 + ln λ. This competitive ratio λ is best possible. (2) For the case where the cache cost grows like a polynomial of degree d in the cache size, we give an online algorithm whose competitive ratio behaves like d/lnd + o(d/lnd). No online algorithm can reach a competitive ratio better than d/lnd. (3) We exactly characterize the class of cache cost functions for which there exist online algorithms with finite competitive ratios.

We derive the following three tight results: (1) For the case where the cache cost depends linearly on the cache size, we give a λ-competitive online algorithm where λ ≈ 3.14619 is a solution of λ = 2 + ln λ. This competitive ratio λ is best possible. (2) For the case where the cache cost grows like a polynomial of degree d in the cache size, we give an online algorithm whose competitive ratio behaves like d/lnd + o(d/lnd). No online algorithm can reach a competitive ratio better than d/lnd. (3) We exactly characterize the class of cache cost functions for which there exist online algorithms with finite competitive ratios.

Original language | English |
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Title of host publication | Algorithms — ESA 2001 |

Subtitle of host publication | 9th Annual European Symposium Århus, Denmark, August 28–31, 2001 Proceedings |

Editors | Friedhelm Meyer auf der Heide |

Publisher | Springer |

Pages | 98-108 |

ISBN (Electronic) | 978-3-540-44676-7 |

ISBN (Print) | 978-3-540-42493-2 |

DOIs | |

Publication status | Published - 2001 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 2161 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

## Keywords

- METIS-202692