### Abstract

We consider the problem of computing information theoretic functions such as entropy on a data stream, using sublinear space. Our first result deals with a measure we call the "entropy norm" of an input stream: it is closely related to entropy but is structurally similar to the well-studied notion of frequency moments. We give a polylogarithmic space one-pass algorithm for estimating this norm under certain conditions on the input stream. We also prove a lower bound that rules out such an algorithm if these conditions do not hold. Our second group of results are for estimating the empirical entropy of an input stream. We first present a sublinear space one-pass algorithm for this problem. For a stream of m items and a given real parameter α, our algorithm uses space Õ(m^{2α}) and provides an approximation of 1/α in the worst case and (1 + ε) in "most" cases. We then present a two-pass polylogarithmic space (1+ε)-approximation algorithm. All our algorithms are quite simple.

Original language | English (US) |
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Title of host publication | STACS 2006 |

Subtitle of host publication | 23rd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings |

Pages | 196-205 |

Number of pages | 10 |

DOIs | |

State | Published - Jul 10 2006 |

Event | STACS 2006: 23rd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings - Marseille, France Duration: Feb 23 2006 → Feb 25 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 3884 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | STACS 2006: 23rd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings |
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Country | France |

City | Marseille |

Period | 2/23/06 → 2/25/06 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*STACS 2006: 23rd Annual Symposium on Theoretical Aspects of Computer Science, Proceedings*(pp. 196-205). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 3884 LNCS). https://doi.org/10.1007/11672142_15