## Almost Random Graphs with Simple Hash Functions (2003)

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Venue: | STOC'03 |

Citations: | 21 - 1 self |

### BibTeX

@MISC{Dietzfelbinger03almostrandom,

author = {Martin Dietzfelbinger and Philipp Woelfel},

title = {Almost Random Graphs with Simple Hash Functions},

year = {2003}

}

### Years of Citing Articles

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### Abstract

### Citations

1957 | Random Graphs
- Bollobás
- 2001
(Show Context)
Citation Context ... denoted by G(S, h1, h2). If the edges (h1(x), h2(x)), x ∈ S, are assumed to be fully random in V ×W , the properties of G(S, h1, h2) can be studied using known tools from the theory of random graphs =-=[1]-=-. On the other hand, in the context of randomized algorithms one usually assumes some kind of limited randomness that can be generated by choosing h1 and h2 from (universal) hash classes comprising fu... |

776 |
Graph Theory
- Diestel
- 1991
(Show Context)
Citation Context ... d r,m, is in several aspects similar to that of perfectly random bipartite graphs with |S| edges. In order to state the main result precisely, we need some graph theoretical definitions. (See, e.g., =-=[3]-=-, for basic notions in graph theory.) Let G = (V, E) be a connected graph. We identify each simple cycle and each simple path of G with the set of its edges. The length of a path or a cycle is the num... |

718 |
Universal classes of hash functions
- Carter, Wegman
- 1979
(Show Context)
Citation Context ...ing a random hash function from a hash class (i. e., a family of hash functions), instead of using fixed hash functions, dates back to 1979, when Carter and Wegman introduced “universal hash classes” =-=[2]-=-. For d ≥ 2, a universe U and a range size m, a family H of functions mapping U to [m] is called a d-wise independent class (of hash functions) if for h chosen randomly from H the values h(x1), . . . ... |

136 | Cuckoo hashing
- Pagh, Rodler
(Show Context)
Citation Context ... to [m] at random, according to some distribution. Often, the structure of the (random) graph determined by the edge (multi)set {(h1(x), h2(x)) | x ∈ S} is essential in the analysis of some algorithm =-=[9, 11, 13, 14, 15, 19]-=-. Slightly different situations result according as one assumes the node set of the graph is [m] or one considers a bipartite graph with node set V ⊎ W , where V and W are disjoint copies of [m]. In t... |

133 | Dynamic perfect hashing: upper and lower bounds
- Dietzfelbinger, Karlin, et al.
- 1994
(Show Context)
Citation Context ...t used hash classes with simpler functions (but for which it is not clear whether the analysis carries through) one already knew that in practice cuckoo hashing is superior to dynamic perfect hashing =-=[8]-=-; our new result shows that cuckoo hashing even in a version that can be theoretically analyzed beats dynamic perfect hashing as far as space requirements go and is at least competitive with respect t... |

88 |
auf der Heide. Efficient PRAM simulation on a distributed memory machine
- Karp, Luby, et al.
- 1992
(Show Context)
Citation Context ... to [m] at random, according to some distribution. Often, the structure of the (random) graph determined by the edge (multi)set {(h1(x), h2(x)) | x ∈ S} is essential in the analysis of some algorithm =-=[9, 11, 13, 14, 15, 19]-=-. Slightly different situations result according as one assumes the node set of the graph is [m] or one considers a bipartite graph with node set V ⊎ W , where V and W are disjoint copies of [m]. In t... |

59 |
A reliable randomized algorithm for the closest-pair problem
- Dietzfelbinger, Hagerup, et al.
- 1997
(Show Context)
Citation Context ...dent hash classes in such a way that for evaluating the function one integer multiplication as well as some simple bit operations and table lookups are sufficient. Consider the 2-universal class from =-=[6]-=- that consists of functions fa : {0, 1, . . . , 2 κ − 1} → {0, 1, . . . , 2 µ − 1} given by fa : x ↦→ (ax) mod 2 κ div 2 κ−µ . We use the nice (and well-known) property of this class that one can eval... |

47 | Space efficient hash tables with worst case constant access time
- Fotakis, Pagh, et al.
- 2003
(Show Context)
Citation Context ...n that can be theoretically analyzed beats dynamic perfect hashing as far as space requirements go and is at least competitive with respect to evaluation time. In a recent development, Fotakis et al. =-=[10]-=- brought down the space requirement in cuckoo hashing to (1+ɛ)n by allowing a fixed number d = d(ɛ) ≥ 2 of hash functions instead of just two. However, in the analysis it was assumed that the hash fun... |

47 |
On universal classes of fast high performance hash functions, their time-space tradeo, and their applications
- Siegel
- 1989
(Show Context)
Citation Context ...ltigraph, i. e., there may be multiple edges. Accordingly, when we talk about sets of edges, we mean multisets.sSiegel’s high-performance hash classes. Siegel (in the technical report version [23] of =-=[22]-=-) gave a construction to the following effect. Fact 1. Let 0 < µ < 1 and k ≥ 1 with µk < 1 be given. Then if ζ < 1 and d ≥ 1 satisfy ζ ≥ 2k 1+log d+µ log n + ·k (for d ζ log n n large enough), then th... |

41 |
Practical minimal perfect hash functions for large databases
- Fox, Heath, et al.
- 1992
(Show Context)
Citation Context ... to [m] at random, according to some distribution. Often, the structure of the (random) graph determined by the edge (multi)set {(h1(x), h2(x)) | x ∈ S} is essential in the analysis of some algorithm =-=[9, 11, 13, 14, 15, 19]-=-. Slightly different situations result according as one assumes the node set of the graph is [m] or one considers a bipartite graph with node set V ⊎ W , where V and W are disjoint copies of [m]. In t... |

39 |
Universal hashing and k-wise independent random variables via integer arithmetic without primes
- Dietzfelbinger
- 1996
(Show Context)
Citation Context ...tudied in the area of universal hashing. Most known constructions of d-wise independent classes for d > 2 involve polynomials of degree up to d − 1 over some algebraic structures like fields or rings =-=[2, 4, 5, 24]-=-. The most notable exception is Siegel’s class from [23] with functions of a relatively complex structure and a relatively large evaluation time (in particular for small d). Complementing the random g... |

27 |
Polynomial hash functions are reliable
- Dietzfelbinger, Gil, et al.
- 1992
(Show Context)
Citation Context ...r constant d ≥ 2 (and m a prime power) such classes may easily be constructed. For d = 2, many constructions are known, for d ≥ 3 most standard constructions involve polynomials of degree < d (e. g., =-=[5]-=-). Alternatively, the following construction may be used. 1 Throughout this paper, the word graph is used in the sense of multigraph, i. e., there may be multiple edges. Accordingly, when we talk abou... |

26 | On universal classes of extremely random constant-time hash functions
- Siegel
(Show Context)
Citation Context ...se of multigraph, i. e., there may be multiple edges. Accordingly, when we talk about sets of edges, we mean multisets.sSiegel’s high-performance hash classes. Siegel (in the technical report version =-=[23]-=- of [22]) gave a construction to the following effect. Fact 1. Let 0 < µ < 1 and k ≥ 1 with µk < 1 be given. Then if ζ < 1 and d ≥ 1 satisfy ζ ≥ 2k 1+log d+µ log n + ·k (for d ζ log n n large enough),... |

26 |
New classes and applications of hash functions
- Wegman, Carter
- 1979
(Show Context)
Citation Context ...tudied in the area of universal hashing. Most known constructions of d-wise independent classes for d > 2 involve polynomials of degree up to d − 1 over some algebraic structures like fields or rings =-=[2, 4, 5, 24]-=-. The most notable exception is Siegel’s class from [23] with functions of a relatively complex structure and a relatively large evaluation time (in particular for small d). Complementing the random g... |

23 |
A family of perfect hashing methods
- Majewski, Wormald, et al.
- 1996
(Show Context)
Citation Context |

18 | The Analysis of Closed Hashing under Limited Randomness (Extended Abstract - Schmidt, Siegel - 1990 |

17 | Uniform hashing in constant time and linear space - Pagh, Pagh |

13 | On the Cell Probe Complexity of Membership and Perfect Hashing - Pagh - 2001 |

12 |
auf der Heide, Dynamic hashing in real time, in
- Dietzfelbinger, Meyer
- 1992
(Show Context)
Citation Context ...(n ζ ) is respected) makes the evaluation time quite large, although still constant. Note that this construction does not use polynomials. For other hash classes with strong randomness properties see =-=[7, 11]-=-. Now we resume the discussion of our basic construction. Up to now, in applications of the graphs G(S, h1, h2) mentioned above one had to assume for the analysis that h1, h2 are chosen from Siegel’s ... |

11 | More Analysis of Double Hashing - Lueker, Molodowitch - 1988 |

11 | Uniform Hashing in Constant Time and Optimal Space
- Pagh, Pagh
- 2008
(Show Context)
Citation Context |

8 |
and displace: Efficient evaluation of minimal perfect hash functions
- Hash
- 1999
(Show Context)
Citation Context ...ns hf,g,z is due to Dietzfelbinger and Meyer auf der Heide [7], who have thoroughly analyzed a hash class very similar to R d r,m. Modifications of such hash classes appeared later in different works =-=[11, 17]-=-. The analysis of the behaviour of function pairs from ˆ R d r,m on a given key set S is fundamentally different from previous methods applied to pairs of functions from R d r,m in that we consider th... |

6 | Closed hashing is computable and optimally randomizable with universal hash functions
- Siegel, Schmidt
- 1995
(Show Context)
Citation Context ...uniformity assumption in the analysis of classical hashing schemes like linear probing or double hashing can be realized by a randomized algorithm, at the cost of O(n) extra space. Schmidt and Siegel =-=[20, 21]-=- proposed an analysis of double hashing and linear probing based on the weaker O(log n)-wise independence, but the analysis in particular of double hashing becomes considerably more complicated than w... |

1 |
Exploiting storage redundancy to speed up shared memory simulations. Theoret
- Heide, Scheideler, et al.
- 1996
(Show Context)
Citation Context |