## Sketching and Streaming High-Dimensional Vectors (2011)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Nelson11sketchingand,

author = {Jelani Nelson and Erik D. Demaine},

title = {Sketching and Streaming High-Dimensional Vectors},

year = {2011}

}

### OpenURL

### Abstract

A sketch of a dataset is a small-space data structure supporting some prespecified set of queries (and possibly updates) while consuming space substantially sublinear in the space required to actually store all the data. Furthermore, it is often desirable, or required by the application, that the sketch itself be computable by a small-space algorithm given just one pass over the data, a so-called streaming algorithm. Sketching and streaming have found numerous applications in network traffic monitoring, data mining, trend detection, sensor networks, and databases. In this thesis, I describe several new contributions in the area of sketching and streaming algorithms. • The first space-optimal streaming algorithm for the distinct elements problem. Our algorithm also achieves O(1) update and reporting times. • A streaming algorithm for Hamming norm estimation in the turnstile model which achieves the best known space complexity.

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Citation Context ...(0, 2) Table 1.3: Comparison of our contribution to previous works on Fp estimation in data streams. It is known that not all ℓp norms can be efficiently approximated in a data stream. In particular, =-=[11, 27]-=- show that polynomial space in d, m is required for p > 2, whereas space polylogarithmic in these parameters is achievable for 0 < p ≤ 2 [8, 67]. 1 In this thesis, we focus on this feasible regime for... |

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Citation Context ...ssumptions are due to Bar-Yossef et al. [12], who provide algorithms with various tradeoffs (see Table 1.2). We also give a new algorithm for estimating ℓ0, also known as the Hamming norm of a vector =-=[32]-=-, with optimal running times and near-optimal space. We sometimes refer to the Hamming norm of x as ‖x‖0. This problem is simply a generalization of F0 estimation to turnstile streams; in particular, ... |

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Citation Context ...he analysis following Eq. (5.17). Since h is independent of σ, the total probability of having error larger than 2ε is greater than √ δ 2 = δ. � 5.2 Numerical Linear Algebra Applications The works of =-=[30, 113]-=- gave algorithms to solve various approximate numerical linear algebra problems given small memory and a only one or few passes over an input matrix. They considered models where one only sees a row o... |

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Citation Context ... [22, 37], and data warehousing [1]. The problem of space-efficient F0 estimation is well-studied, beginning with the 12work of Flajolet and Martin [49], and continuing with a long line of research, =-=[8, 12, 13, 15, 21, 31, 41, 44, 48, 57, 58, 69, 124]-=-. Our Contribution: We settle both the space- and time-complexities of F0 estimation by giving an algorithm using O(ε−2 + log d) space, with O(1) worst-case update and reporting times. Our space upper... |

57 |
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Citation Context ...ty properties of the Internet graph [104]. Other applications include selecting a minimum-cost query plan [114], database design [47], OLAP [102, 115], data integration [22, 37], and data warehousing =-=[1]-=-. The problem of space-efficient F0 estimation is well-studied, beginning with the 12work of Flajolet and Martin [49], and continuing with a long line of research, [8, 12, 13, 15, 21, 31, 41, 44, 48,... |

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Citation Context ...re discussion of empirical entropy estimation in data streams in the context of network anomaly detection. The first algorithms for approximating entropy in the streaming model are due to Guha et al. =-=[60]-=-; they achieved O((ε −2 +log d) log 2 d) space in the insertion-only model, assuming that the stream is randomly ordered. Chakrabarti, Do Ba and Muthukrishnan [26] then gave an algorithm for worst-cas... |