### Abstract

We consider the problem of extending the analysis of balls and bins processes where a ball is placed in the least loaded of d randomly chosen bins to cover deletions. In particular, we are interested in the case where the system maintains a fixed load, and deletions are determined by an adversary before the process begins. We show that with high probability the load in any bin is O(log log n). In fact, this result follows from recent work by Cole et al. concerning a more difficult problem of routing in a butterfly network. The main contribution of this paper is to give a different proof of this bound, which follows the lines of the analysis of Azar, Broder, Karlin, and Upfal for the corresponding static load balancing problem. We also give a specialized (and hence simpler) version of the argument from the paper by Cole et al. for the balls and bins scenario. Finally, we provide an alternative analysis also based on the approach of Azar, Broder, Karlin, and Upfal for the special case where items are deleted according to their age. Although this analysis does not yield better bounds than our argument for the general case, it is interesting because it utilizes a two dimensional family of random variables in order to account for the age of the items. This technique may be of more general use.

Original language | English (US) |
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Title of host publication | Randomization and Approximation Techniques in Computer Science - 2nd International Workshop, RANDOM 1998, Proceedings |

Publisher | Springer Verlag |

Pages | 145-158 |

Number of pages | 14 |

Volume | 1518 |

ISBN (Print) | 354065142X, 9783540651420 |

DOIs | |

State | Published - 1998 |

Event | 2nd International Workshop on Randomization and Approximation Techniques in Computer Science, Random 1998 - Barcelona, Spain Duration: Oct 8 1998 → Oct 10 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 1518 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 2nd International Workshop on Randomization and Approximation Techniques in Computer Science, Random 1998 |
---|---|

Country | Spain |

City | Barcelona |

Period | 10/8/98 → 10/10/98 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Randomization and Approximation Techniques in Computer Science - 2nd International Workshop, RANDOM 1998, Proceedings*(Vol. 1518, pp. 145-158). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1518). Springer Verlag. https://doi.org/10.1007/3-540-49543-6_12

**On balls and bins with deletions.** / Cole, Richard; Frieze, Alan; Maggs, Bruce M.; Mitzenmacher, Michael; Richa, Andréa W.; Sitaraman, Ramesh; Upfal, Eli.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Randomization and Approximation Techniques in Computer Science - 2nd International Workshop, RANDOM 1998, Proceedings.*vol. 1518, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1518, Springer Verlag, pp. 145-158, 2nd International Workshop on Randomization and Approximation Techniques in Computer Science, Random 1998, Barcelona, Spain, 10/8/98. https://doi.org/10.1007/3-540-49543-6_12

}

TY - GEN

T1 - On balls and bins with deletions

AU - Cole, Richard

AU - Frieze, Alan

AU - Maggs, Bruce M.

AU - Mitzenmacher, Michael

AU - Richa, Andréa W.

AU - Sitaraman, Ramesh

AU - Upfal, Eli

PY - 1998

Y1 - 1998

N2 - We consider the problem of extending the analysis of balls and bins processes where a ball is placed in the least loaded of d randomly chosen bins to cover deletions. In particular, we are interested in the case where the system maintains a fixed load, and deletions are determined by an adversary before the process begins. We show that with high probability the load in any bin is O(log log n). In fact, this result follows from recent work by Cole et al. concerning a more difficult problem of routing in a butterfly network. The main contribution of this paper is to give a different proof of this bound, which follows the lines of the analysis of Azar, Broder, Karlin, and Upfal for the corresponding static load balancing problem. We also give a specialized (and hence simpler) version of the argument from the paper by Cole et al. for the balls and bins scenario. Finally, we provide an alternative analysis also based on the approach of Azar, Broder, Karlin, and Upfal for the special case where items are deleted according to their age. Although this analysis does not yield better bounds than our argument for the general case, it is interesting because it utilizes a two dimensional family of random variables in order to account for the age of the items. This technique may be of more general use.

AB - We consider the problem of extending the analysis of balls and bins processes where a ball is placed in the least loaded of d randomly chosen bins to cover deletions. In particular, we are interested in the case where the system maintains a fixed load, and deletions are determined by an adversary before the process begins. We show that with high probability the load in any bin is O(log log n). In fact, this result follows from recent work by Cole et al. concerning a more difficult problem of routing in a butterfly network. The main contribution of this paper is to give a different proof of this bound, which follows the lines of the analysis of Azar, Broder, Karlin, and Upfal for the corresponding static load balancing problem. We also give a specialized (and hence simpler) version of the argument from the paper by Cole et al. for the balls and bins scenario. Finally, we provide an alternative analysis also based on the approach of Azar, Broder, Karlin, and Upfal for the special case where items are deleted according to their age. Although this analysis does not yield better bounds than our argument for the general case, it is interesting because it utilizes a two dimensional family of random variables in order to account for the age of the items. This technique may be of more general use.

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U2 - 10.1007/3-540-49543-6_12

DO - 10.1007/3-540-49543-6_12

M3 - Conference contribution

AN - SCOPUS:84958656539

SN - 354065142X

SN - 9783540651420

VL - 1518

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 145

EP - 158

BT - Randomization and Approximation Techniques in Computer Science - 2nd International Workshop, RANDOM 1998, Proceedings

PB - Springer Verlag

ER -