## Zero-One Frequency Laws

Citations: | 5 - 0 self |

### BibTeX

@MISC{Braverman_zero-onefrequency,

author = {Vladimir Braverman and Rafail Ostrovsky},

title = {Zero-One Frequency Laws},

year = {}

}

### OpenURL

### Abstract

Data streams emerged as a critical model for multiple applications that handle vast amounts of data. One of the most influential and celebrated papers in streaming is the “AMS ” paper on computing frequency moments by Alon, Matias and Szegedy. The main question left open (and explicitly asked) by AMS in 1996 is to give the precise characterization for which functions G on frequency vectors mi (1 ≤ i ≤ n) can ∑ i∈[n] G(mi) be approximated efficiently, where “efficiently ” means by a single pass over data stream and poly-logarithmic memory. No such characterization was known despite a tremendous amount of research on frequency-based functions in streaming literature. In this paper we finally resolve the AMS main question and give a precise characterization (in fact, a zero-one law) for all monotonically increasing functions on frequencies that are zero at the origin. That is, we consider all monotonic functions G: R ↦ → R such that G(0) = 0 and G can be computed in poly-logarithmic time and space and ask, for which G in this class is there an (1±ɛ)-approximation algorithm for computing ∑ i∈[n] G(mi) for any polylogarithmic ɛ? We give an algebraic characterization for all such G so that: • For all functions G in our class that satisfy our algebraic condition, we provide a very general and constructive way to derive an efficient (1±ɛ)-approximation algorithm for computing ∑ i∈[n] G(mi) with polylogarithmic memory and a single pass over data stream; while • For all functions G in our class that do not satisfy our algebraic characterization, we show a lower bound

### Citations

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Citation Context ...many applications with vast amounts of data. The importance of the streaming model is discussed, e.g., by Aggarwal (ed.) [1] and Muthukrishnan [41]. In the seminal AMS paper, Alon, Matias and Szegedy =-=[2]-=- studied the following basic model: Definition 1.1. Let m, n be positive integers. A stream D = D(n, m) is a sequence of size m of integers p1, . . . , pm, where pi ∈ {1, . . . , n}. A frequency vecto... |

616 | Communication Complexity
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Citation Context ... to note that heavy hitters were solved for many specific metrics [18, 22]. 3. THE LOWER BOUND To establish lower bounds, we will use SET DISJOINTNESS and INDEX problems from communication complexity =-=[37]-=-. Recall that SET DISJOINTNESS is the following promise problem: each of t ≥ 2 players is given a set from the universe [N]; all sets have exactly one common element or disjoint. The lower bound on th... |

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Citation Context ...NTRODUCTION Data streams emerged as a critical model for many applications with vast amounts of data. The importance of the streaming model is discussed, e.g., by Aggarwal (ed.) [1] and Muthukrishnan =-=[41]-=-. In the seminal AMS paper, Alon, Matias and Szegedy [2] studied the following basic model: Definition 1.1. Let m, n be positive integers. A stream D = D(n, m) is a sequence of size m of integers p1, ... |

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Citation Context ...k ) for one-pass algorithms. Indyk and Woodruff [32] and Woodruff [43] gave optimal lower bound in terms of error parameter. Many other results on frequency moments include, e.g., Flajolet and Martin =-=[24]-=-, Bar-Yossef, Jayram, Kumar, and Sivakumar [4], Coppersmith and Kumar [19], Cormode, Datar, Indyk and Muthukrishnan [20], Feigenbaum, Kannan, Strauss and Viswanathan [23], Ganguly [25], Ganguly and Co... |

299 | An improved data stream summary: the count-min sketch and its applications
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267 | Finding frequent items in data streams
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Citation Context ... Xu and Zhang [38], Braverman, Chung, Liu, Mitzenmacher and Ostrovsky [8], Braverman and Ostrovsky [13]. The related question of frequent elements has been studied by Charikar, Chen and Farach-Colton =-=[18]-=-, Cormode and Hadjieleftheriou [21], Cormode and Muthukrishnan [22]. The frequency-based functions were studied in extended models such as the read/write model (Beame, Jayram and Rudra [5]), and the r... |

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150 | An information statistics approach to data stream and communication complexity - Bar-Yossef, Jayram, et al. |

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Citation Context ... [32] and Woodruff [43] gave optimal lower bound in terms of error parameter. Many other results on frequency moments include, e.g., Flajolet and Martin [24], Bar-Yossef, Jayram, Kumar, and Sivakumar =-=[4]-=-, Coppersmith and Kumar [19], Cormode, Datar, Indyk and Muthukrishnan [20], Feigenbaum, Kannan, Strauss and Viswanathan [23], Ganguly [25], Ganguly and Cormode [26], Li [39], and Kane, Nelson and Wood... |

87 |
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72 |
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Citation Context ...yk [30] presented a celebrated method of stable distributions for approximating Lp norms p ∈ (0, 2] in a general model where deletions are allowed and updates can be larger then 1. Indyk and Woodruff =-=[33]-=- gave the first optimal algorithm for Fk, k > 2, proving O ∗ (n 1−2/k ) upper bound. This result was later improved by polylog factors by Bhuvanagiri, Ganguly, Kesh and Saha [7]. BarYossef, Jayram, Ku... |

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71 | Comparing Data Streams Using Hamming Norms (How to Zero In
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- 2003
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Citation Context ...ter. Many other results on frequency moments include, e.g., Flajolet and Martin [24], Bar-Yossef, Jayram, Kumar, and Sivakumar [4], Coppersmith and Kumar [19], Cormode, Datar, Indyk and Muthukrishnan =-=[20]-=-, Feigenbaum, Kannan, Strauss and Viswanathan [23], Ganguly [25], Ganguly and Cormode [26], Li [39], and Kane, Nelson and Woodruff [35, 36], Braverman and Ostrovsky [9, 11]. Currently, many important ... |

59 | Optimal space lower bounds for all frequency moments
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- 2004
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Citation Context ...ve the first nearly matching lower bound of Ω(n 1−(2+ɛ)/k ). Later Chakrabarti, Khot and Sun [17] improved the lower bound to Ω(n 1−2/k ) for one-pass algorithms. Indyk and Woodruff [32] and Woodruff =-=[43]-=- gave optimal lower bound in terms of error parameter. Many other results on frequency moments include, e.g., Flajolet and Martin [24], Bar-Yossef, Jayram, Kumar, and Sivakumar [4], Coppersmith and Ku... |

54 | Streaming and sublinear approximation of entropy and information distances
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Citation Context ...tion theory to prove the first nearly matching lower bound of Ω(n 1−(2+ɛ)/k ). Later Chakrabarti, Khot and Sun [17] improved the lower bound to Ω(n 1−2/k ) for one-pass algorithms. Indyk and Woodruff =-=[32]-=- and Woodruff [43] gave optimal lower bound in terms of error parameter. Many other results on frequency moments include, e.g., Flajolet and Martin [24], Bar-Yossef, Jayram, Kumar, and Sivakumar [4], ... |

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Citation Context ...g, Liu, Mitzenmacher and Ostrovsky [8], Braverman and Ostrovsky [13]. The related question of frequent elements has been studied by Charikar, Chen and Farach-Colton [18], Cormode and Hadjieleftheriou =-=[21]-=-, Cormode and Muthukrishnan [22]. The frequency-based functions were studied in extended models such as the read/write model (Beame, Jayram and Rudra [5]), and the randomized model (Chakrabarti, Cormo... |

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Citation Context ...e optimal lower bound in terms of error parameter. Many other results on frequency moments include, e.g., Flajolet and Martin [24], Bar-Yossef, Jayram, Kumar, and Sivakumar [4], Coppersmith and Kumar =-=[19]-=-, Cormode, Datar, Indyk and Muthukrishnan [20], Feigenbaum, Kannan, Strauss and Viswanathan [23], Ganguly [25], Ganguly and Cormode [26], Li [39], and Kane, Nelson and Woodruff [35, 36], Braverman and... |

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Citation Context ...ift Invariant Theorem is not applicable. The questions of Alon, Matias and Szegedy [2] and Guha To the best of our knowledge, the only work in this direction is the result of Guha, Indyk and McGregor =-=[27]-=-. They proved the Shift Invariant Theorem, a general result that gives a necessary condition for approximating a wide class of two-variable functions. The Shift Invariant Theorem says, very informally... |

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3 |
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