### Abstract

Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, ..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multi-user tracing ((k, α)-FUT families). We show that r(n, k) = Θ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n,k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2log n + O(1), which is optimal up to an additive constant.

Original language | English (US) |
---|---|

Title of host publication | Automata, Languages and Programming - 35th International Colloquium, ICALP 2008, Proceedings |

Pages | 258-270 |

Number of pages | 13 |

Edition | PART 1 |

DOIs | |

State | Published - 2008 |

Event | 35th International Colloquium on Automata, Languages and Programming, ICALP 2008 - Reykjavik, Iceland Duration: Jul 7 2008 → Jul 11 2008 |

### Publication series

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

Number | PART 1 |

Volume | 5125 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 35th International Colloquium on Automata, Languages and Programming, ICALP 2008 |
---|---|

Country | Iceland |

City | Reykjavik |

Period | 7/7/08 → 7/11/08 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

## Fingerprint Dive into the research topics of 'Optimal monotone encodings'. Together they form a unique fingerprint.

## Cite this

*Automata, Languages and Programming - 35th International Colloquium, ICALP 2008, Proceedings*(PART 1 ed., pp. 258-270). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5125 LNCS, No. PART 1). https://doi.org/10.1007/978-3-540-70575-8_22