### Abstract

In this paper, we present near-optimal space bounds for L_{p}- samplers. Given a stream of updates (additions and subtraction) to the coordinates of an underlying vector x ε R^{n}, a perfect L _{p} sampler outputs the i-th coordinate with probability |x _{i}|^{p}/||x||_{p}^{p}. In SODA 2010, Monemizadeh and Woodru showed polylog space upper bounds for approximate Lp-samplers and demonstrated various applications of them. Very recently, Andoni, Krauthgamer and Onak improved the upper bounds and gave a O(ε^{-p} log^{3} n) space ε relative error and constant failure rate L_{p}-sampler for p ε [1; 2]. In this work, we give another such algorithm requiring only O(ε^{-p} log ^{3} n) space for p ε (1; 2). For p ε (0; 1), our space bound is O(ε^{-p} log^{3} n), while for the p = 1 case we have an O(log(1=ε) ε^{-1} log^{2} n) space algorithm. We also give a O(log^{2} n) bits zero relative error L0-sampler, improving the O(log^{3} n) bits algorithm due to Frahling, Indyk and Sohler. As an application of our samplers, we give better upper bounds for the problem of finding duplicates in data streams. In case the length of the stream is longer than the alphabet size, L_{1} sampling gives us an O(log^{2} n) space algorithm, thus improving the previous O(log^{3} n) bound due to Gopalan and Radhakrishnan. In the second part of our work, we prove an (log ^{2} n) lower bound for sampling from 0, ±1 vectors (in this special case, the parameter p is not relevant for Lp sampling). This matches the space of our sampling algorithms for constant ε > 0. We also prove tight space lower bounds for the finding duplicates and heavy hitters problems. We obtain these lower bounds using reductions from the communication complexity problem augmented indexing.

Original language | English |
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Title of host publication | PODS'11 - Proceedings of the 30th Symposium on Principles of Database Systems |

Pages | 49-58 |

Number of pages | 10 |

DOIs | |

Publication status | Published - Jul 15 2011 |

Event | 30th Symposium on Principles of Database Systems, PODS'11 - Athens, Greece Duration: May 13 2011 → May 15 2011 |

### Publication series

Name | Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems |
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### Other

Other | 30th Symposium on Principles of Database Systems, PODS'11 |
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Country | Greece |

City | Athens |

Period | 5/13/11 → 5/15/11 |

### Keywords

- Finding duplicates
- Sampling
- Streaming algorithms

### ASJC Scopus subject areas

- Software
- Information Systems
- Hardware and Architecture

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## Cite this

*PODS'11 - Proceedings of the 30th Symposium on Principles of Database Systems*(pp. 49-58). (Proceedings of the ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems). https://doi.org/10.1145/1989284.1989289