## LOW REDUNDANCY IN STATIC DICTIONARIES WITH CONSTANT QUERY TIME (2001)

Venue: | SIAM J. COMPUT. |

Citations: | 49 - 7 self |

### BibTeX

@MISC{Pagh01lowredundancy,

author = {Rasmus Pagh},

title = { LOW REDUNDANCY IN STATIC DICTIONARIES WITH CONSTANT QUERY TIME },

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

A static dictionary is a data structure storing subsets of a finite universe U, answering membership queries. We show that on a unit cost RAM with word size Θ(log |U|), a static dictionary for n-element sets with constant worst case query time can be obtained using B +O(log log |U|)+o(n) (|U|) bits of storage, where B = ⌈log2 ⌉ is the minimum number of bits needed to represent all n-n element subsets of U.

### Citations

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(Show Context)
Citation Context ...xity of O(n) words (i.e., O(n log m) bits) either had non-constant query time or worked only for restricted universe sizes [4, 16, 17]. The breakthrough paper of Fredman, Komlos and Szemeredi (FKS) [7=-=-=-] described a general constant time hashing scheme using O(n) words. A rened solution in the paper uses B + O(n log n + log log m) bits. This is O(B) when n = m 1s4 . Brodnik and Munro [2] constructed... |

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Citation Context ..., where it can be used directly to obtain an O(B) bit scheme, equaling the result of Brodnik and Munro [2]. However, lower bounds on the time for maintaining ranks under insertions and deletions (see =-=[8]-=-) show that our construction involving the dictionary supporting rank will not dynamize well. It would be interesting to determine the exact redundancy necessary to allow constant time queries. In par... |

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(Show Context)
Citation Context ...hus, Denmark. Email: pagh@brics.dk. A preliminary version of this paper appeared at ICALP 1999. 1 2 RASMUS PAGH restrictions on m and n. They later improved the bound to to B+O(B= log log log m) bits =-=[3]-=-. No non-trivial space lower bound is known in a general model of computation. However, various restrictions on the data structure and the query algorithm have been successfully studied. Yao [17] show... |

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Citation Context ...y choosing numbers from the appropriate interval of lemma 5.1. Each number chosen is checked for primality (using a probabilistic check which uses expected time poly-logarithmic in the number checked =-=[1]-=-, that is, time (log n+ log log m) O(1) ). When a prime is found, it is checked whether u 7! u mod p is 1-1 on S (the element distinctness problem on the residues, taking expected O(n) time using univ... |

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Citation Context ...stence of a suitable prime between n 2 ln m and 3 n 2 ln m is guaranteed by the prime number theorem (in fact, at least half of the primes in the interval will work). See [7, Lemma 2] for details. By [9] the number of primes between m and m+m issm = log m) for any > 11=20. Take = 2=3 and let p be such a prime; naturally the map is then 1-1. A prime in thesrst range will be our choice for p wh... |

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Citation Context ...to B unless n t m=2. By the late 70's, known dictionaries with a space complexity of O(n) words (i.e., O(n log m) bits) either had non-constant query time or worked only for restricted universe sizes =-=[4, 16, -=-17]. The breakthrough paper of Fredman, Komlos and Szemeredi (FKS) [7] described a general constant time hashing scheme using O(n) words. A rened solution in the paper uses B + O(n log n + log log m) ... |

21 | Low redundancy in static dictionaries with O(1) lookup time - Pagh - 1999 |

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(Show Context)
Citation Context ...redi (FKS) [7] described a general constant time hashing scheme using O(n) words. A rened solution in the paper uses B + O(n log n + log log m) bits. This is O(B) when n = m 1s4 . Brodnik and Munro [2=-=]-=- constructed thesrst static dictionary using O(B) bits with no 1 BRICS (Basic Research in Computer Science,Centre of the Danish National Research Foundation) , University of Aarhus, Denmark. Email: pa... |

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Citation Context ...and N , the O(n + log log m) term is less than n log n when n > N . If n N we can simply list the elements of S. The previously best result of this kind needed n (log m) c for some constant c > 0 [5=-=]-=- (an interesting feature of this non-constructive scheme is that it is implicit, i.e., the n words contain a permutation of the elements in S). The question whether n words suce in all cases was posed... |

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Citation Context ...udied. Yao [17] showed that if words of the data structure must contain elements of S, the number of words necessary for o(log n) time queries cannot be bounded by a function of n. Fich and Miltersen =-=[-=-6] studied a RAM with standard unit cost arithmetic operations but without division and bit operations, and showed that query time o(log n) requires m=n ) words of memory for any > 0. Miltersen [11]... |

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Citation Context ...allows ecient random access to the original bits. The important ingredient in the solution is the concept of quotienting. Quotienting was recently applied in a space ecient dictionary supporting rank =-=[14]-=-. In general, quotienting can be used to save around n log n bits in hash tables. Thus, the existence of an eciently evaluable quotient function is a desirable property for a hash function. For the qu... |

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(Show Context)
Citation Context ... [6] studied a RAM with standard unit cost arithmetic operations but without division and bit operations, and showed that query time o(log n) requires m=n ) words of memory for any > 0. Miltersen [1=-=-=-1] showed that on a RAM with bit operations but without multiplication, one needs m words, for some > 0, when n = m o(1) . This Paper. In this paper we show that it is possible to achieve space usag... |

3 | Dispersing hash functions - Pagh - 2000 |