## Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families (1999)

Venue: | In Proc. International Workshop on Randomization and Approximation Techniques in Computer Science |

Citations: | 7 - 0 self |

### BibTeX

@INPROCEEDINGS{Saks99lowdiscrepancy,

author = {Michael Saks and Aravind Srinivasan and Shiyu Zhou and David Zuckerman},

title = {Low Discrepancy Sets Yield Approximate Min-Wise Independent Permutation Families},

booktitle = {In Proc. International Workshop on Randomization and Approximation Techniques in Computer Science},

year = {1999},

pages = {29--32},

publisher = {Springer}

}

### OpenURL

### Abstract

Motivated by a problem of filtering near-duplicate Web documents, Broder, Charikar, Frieze & Mitzenmacher defined the following notion of ffl-approximate min-wise independent permutation families. A multiset F of permutations of f0; 1; : : : ; n \Gamma 1g is such a family if for all K ` f0; 1; : : : ; n \Gamma 1g and any x 2 K, a permutation chosen uniformly at random from F satisfies j Pr[minf(K)g = (x)] \Gamma 1 jKj j ffl jKj : We show connections of such families with low discrepancy sets for geometric rectangles, and give explicit constructions of such families F of size n O( p log n) for ffl = 1=n \Theta(1) , improving upon the previously best-known bound of Indyk. We also present polynomialsize constructions when the min-wise condition is required only for jKj 2 O(log 2=3 n) , with ffl 2 \GammaO(log 2=3 n) . Keywords: Combinatorial problems; min-wise independent permutations; information retrieval; document filtering; pseudorandom permutations; explicit constructions.

### Citations

381 | A simple parallel algorithm for the maximal independent set problem
- Luby
- 1986
(Show Context)
Citation Context ...mutation families is often more difficult than constructing pseudorandom function families. For example, there are polynomial size constructions of k-wise independent function families for constant k =-=[8, 9, 1, 12]-=-. On the other hand, although there are polynomial-size 3-wise independent permutation families (see, e.g. [14]), there are only exponential size constructions known for higher k. In fact, the only su... |

216 |
A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem
- Alon, Babai, et al.
- 1986
(Show Context)
Citation Context ...mutation families is often more difficult than constructing pseudorandom function families. For example, there are polynomial size constructions of k-wise independent function families for constant k =-=[8, 9, 1, 12]-=-. On the other hand, although there are polynomial-size 3-wise independent permutation families (see, e.g. [14]), there are only exponential size constructions known for higher k. In fact, the only su... |

206 | Min-wise independent permutations
- Broder, Charikar, et al.
- 1998
(Show Context)
Citation Context ...nt type of pseudorandom permutation family, called a min-wise independent permutation family. Motivated by a problem of filtering near-duplicate Web documents, Broder, Charikar, Frieze & Mitzenmacher =-=[3]-=- defined them as follows: Definition 1.1 ([3]) Let [n] denote f0; 1; : : : ; n \Gamma 1g, and S n denote the set of permutations of [n]. A multiset F contained in S n is called min-wise independent if... |

102 | Finite permutation groups and finite simple groups
- Cameron
- 1981
(Show Context)
Citation Context ...bgroups of the symmetric group that are 6-wise independent are the alternating group and the symmetric group itself; for 4-wise and 5-wise independence there are only finitely many besides these (see =-=[4]-=-). There are constructions of almost k-wise independent permutation families with error ffl = O(k 2 =n) [13], again not as good as is known for function families. We address a different type of pseudo... |

101 | On the construction of pseudorandom permutations: LubyRackoff revisited
- Naor, Reingold
- 1999
(Show Context)
Citation Context ...p itself; for 4-wise and 5-wise independence there are only finitely many besides these (see [4]). There are constructions of almost k-wise independent permutation families with error ffl = O(k 2 =n) =-=[13]-=-, again not as good as is known for function families. We address a different type of pseudorandom permutation family, called a min-wise independent permutation family. Motivated by a problem of filte... |

83 | A fast parallel algorithm for the maximal independent set problem
- Karp, Wigderson
- 1985
(Show Context)
Citation Context ...mutation families is often more difficult than constructing pseudorandom function families. For example, there are polynomial size constructions of k-wise independent function families for constant k =-=[8, 9, 1, 12]-=-. On the other hand, although there are polynomial-size 3-wise independent permutation families (see, e.g. [14]), there are only exponential size constructions known for higher k. In fact, the only su... |

61 | A small Approximately Min-Wise Independent Family of Hash Functions
- Indyk
- 1999
(Show Context)
Citation Context ...roder et. al. showed the existence of an (n; d; ffl)-mwif of cardinality O(d 2 log(2n=d)=ffl 2 ) [3]. Indyk presented an explicit construction of an (n; n; ffl)-mwif of cardinality n O(log(1=ffl)) in =-=[7]-=-. In this paper, we show a connection between the construction of approximate min-wise independent families and the construction of low discrepancy sets for geometric rectangles, and use this connecti... |

48 |
On a set of almost deterministic k-independent random variables
- Joffe
- 1974
(Show Context)
Citation Context |

31 | Approximation of general independent distributions
- EVEN, GOLDEICH, et al.
- 1992
(Show Context)
Citation Context ...g 2=3 n) . Also, when d = n, our bound is better than that of [7] if ffls2 \Gammac 0 p log n , where c 0 ? 0 is a certain absolute constant. We remark that Lu's construction builds on earlier work of =-=[2, 5, 6, 10]-=-. Given log L random bits to index a random elementsof the permutation family guaranteed by Corollary 1.1, and given any i 2 [n], we can deterministically construct (i) in time polylogarithmic in L. 2... |

23 | Efficient construction of a small hitting set for combinatorial rectangles in high dimension
- Linial, Luby, et al.
- 1997
(Show Context)
Citation Context ...g 2=3 n) . Also, when d = n, our bound is better than that of [7] if ffls2 \Gammac 0 p log n , where c 0 ? 0 is a certain absolute constant. We remark that Lu's construction builds on earlier work of =-=[2, 5, 6, 10]-=-. Given log L random bits to index a random elementsof the permutation family guaranteed by Corollary 1.1, and given any i 2 [n], we can deterministically construct (i) in time polylogarithmic in L. 2... |

20 |
Notes on Geometry
- Rees
- 1983
(Show Context)
Citation Context ...omial size constructions of k-wise independent function families for constant k [8, 9, 1, 12]. On the other hand, although there are polynomial-size 3-wise independent permutation families (see, e.g. =-=[14]-=-), there are only exponential size constructions known for higher k. In fact, the only subgroups of the symmetric group that are 6-wise independent are the alternating group and the symmetric group it... |

19 | Discrepancy sets and pseudorandom generators for combinatorial rectangles
- Armoni, Saks, et al.
- 1996
(Show Context)
Citation Context ...g 2=3 n) . Also, when d = n, our bound is better than that of [7] if ffls2 \Gammac 0 p log n , where c 0 ? 0 is a certain absolute constant. We remark that Lu's construction builds on earlier work of =-=[2, 5, 6, 10]-=-. Given log L random bits to index a random elementsof the permutation family guaranteed by Corollary 1.1, and given any i 2 [n], we can deterministically construct (i) in time polylogarithmic in L. 2... |

12 | Improved Pseudorandom Generators for combinatorial rectangles
- Lu
(Show Context)
Citation Context ...ollowing: Theorem 1.1 Let m be arbitrary. Suppose D ` [0; m) n is any ffi-discrepant set for GR(m; d; n). Then for any 1 msff ! 1, \Gamma(D) is an (n; d; ffl)-mwif, where ffl = (ff + ffi ff )d 2 . Lu =-=[11]-=- gave an explicit construction of ffi-discrepant sets for GR(m; d; n) of cardinality (mn) O(1) \Delta (1=ffi) O( p log(maxf2;d= log(1=ffi)g)) : Therefore, setting m = 2d 2 =ffl, ff = 1=m and ffi = 1=m... |

8 |
Pseudorandomness for network algorithms
- Impagliazza, Nisan, et al.
- 1994
(Show Context)
Citation Context |

4 | On a set of almost deterministic k-independent random variables - Joe - 1974 |

2 |
An optimal construction of exactly min-wise independent permutations
- Takei, Itoh, et al.
- 1998
(Show Context)
Citation Context ...any min-wise independent family must have exponential size: more precisely, its cardinality is at least lcm(1; 2; : : : ; n)se n\Gammao(n) . (This lower bound of lcm(1; 2; : : : ; n) is in fact tight =-=[15]-=-.) This motivates one to study families that are only approximately min-wise independent; moreover, in practice, we may also have an upper bound d on the cardinality of the sets K of Definition 1.1, s... |