## Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions (1996)

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Venue: | Algorithmica |

Citations: | 61 - 5 self |

### BibTeX

@ARTICLE{Alon96derandomization,witnesses,

author = {Noga Alon and Moni Naor},

title = {Derandomization, witnesses for Boolean matrix multiplication and construction of perfect hash functions},

journal = {Algorithmica},

year = {1996},

volume = {16},

pages = {16--434}

}

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### Abstract

Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computing witnesses for the Boolean product of two matrices. That is, if A and B are two n by n matrices, and C = AB is their Boolean product, the algorithm finds for every entry Cij = 1 a witness: an index k so that Aik = Bkj = 1. Its running time exceeds that of computing the product of two n by n matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a given n-subset of {1,..., m}.

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Citation Context ...= �n k=1(Aik ∧ Bkj). The n3 time method that evaluates these expressions gives for every i, j for which Cij = 1 all the k’s for which Aik = Bkj = 1. The subcubic methods on the other hand (see, e.g., =-=[8]-=-) consider A and B as matrices of integers and do not provide any of these k’s. We call a k such that Aik = Bkj = 1 a witness (for the fact that Cij = 1). We want to compute in addition to the matrix ... |

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Citation Context ...oportional to the size of the space. The size of the space is usually some polynomial in the input size. Thus again this approach suffers from considerable increase in time. This approach is taken in =-=[17]-=-, [1], [15] using probability spaces that are k-wise independent, and in [5], [23] using small bias probability spaces and almost k-wise independence (see definition below in subsection 1.3). Our goal... |

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Citation Context ...hich usually takes time proportional to the input size. Thus, the cost of derandomization though polynomial, can considerably increase the complexity of the algorithm. This approach is taken in [29], =-=[25]-=-, (cf., also [6].) A different approach for finding a good point is to show that the random choices made need not be fully independent, i.e. even if some limited form of independence is obeyed, then t... |

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Citation Context ...od of computing h(x). • An efficient construction - given S there should be an efficient way of finding h ∈ H that is perfect for S Perfect hash functions have been investigated extensively (see e.g. =-=[9, 10, 11, 12, 16, 19, 21, 26, 27, 30]-=-). It is known (and not too difficult to show, see [11], [16], [24]) that the minimum possible number of bits required to represent such a mapping is Θ(n + log log m) for all m ≥ 2n. Fredman, Komlós a... |

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Citation Context .... Therefore if 1/ɛ is logarithmic in n and c is at most logarithmic in n the size of the probability space is still polylogarithmic in n. To be more precise, the construction of [23], as optimized in =-=[2]-=-, yields a probability space of size O( ) and the ones in [5] yield probability spaces of size O( c2 log2 n ɛ2 ). 4 i∈S c log n ɛ 3s2 Boolean matrix multiplication with witnesses All the matrices in t... |

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Citation Context ...ator which usually takes time proportional to the input size. Thus, the cost of derandomization though polynomial, can considerably increase the complexity of the algorithm. This approach is taken in =-=[29]-=-, [25], (cf., also [6].) A different approach for finding a good point is to show that the random choices made need not be fully independent, i.e. even if some limited form of independence is obeyed, ... |

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Citation Context ...ent ω < 2.376 and is due to Coppersmith and Winograd [8]. For two functions f(n) and g(n) we let g(n) = Õ(f(n)) denote the statement that g(n) is O(f(n)(log n) O(1) ). Several researchers (see, e.g., =-=[28]-=-, [4]) observed that there is a simple randomized algorithm that computes witnesses in Õ(nω ) time. In Section 2 we describe a deterministic algorithm for computing the witnesses in Õ(nω ) time. It is... |

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Citation Context ...< 2.376 and is due to Coppersmith and Winograd [8]. For two functions f(n) and g(n) we let g(n) = Õ(f(n)) denote the statement that g(n) is O(f(n)(log n) O(1) ). Several researchers (see, e.g., [28], =-=[4]-=-) observed that there is a simple randomized algorithm that computes witnesses in Õ(nω ) time. In Section 2 we describe a deterministic algorithm for computing the witnesses in Õ(nω ) time. It is esse... |

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Citation Context ... since each probability space is small the total work is increased by only a polylogarithmic factor. We note that the two approaches described above had been combined in a different way previously in =-=[18, 7, 22]-=-. The random variables used there were k-wise independent resulting in a probability space of size O(n k ). This probability space was then searched using an estimator function in O(k log n) steps. Th... |

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Citation Context ...probabilities. A different, more complicated algorithm for this problem, whose running time is slightly inferior, i.e. not Õ(nω ) (but is also O(nω+o(1) )), has been found by Galil and Margalit [20], =-=[14]-=-. The main motivation for studying the computation of witnesses for Boolean matrix multiplication is the observation of Galil and Margalit that this problem is crucial for the design of efficient algo... |

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Citation Context ...g h ∈ H that is perfect for S Perfect hash functions have been investigated extensively (see e.g. [9, 10, 11, 12, 16, 19, 21, 26, 27, 30]). It is known (and not too difficult to show, see [11], [16], =-=[24]-=-) that the minimum possible number of bits required to represent such a mapping is Θ(n + log log m) for all m ≥ 2n. Fredman, Komlós and Szemerédi [12] developed a method for constructing perfect hash ... |

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Citation Context ...s with small integer weights which are based on fast matrix multiplication. Efficient algorithms for computing the distances in this way were initiated in [3] and improved (for some special cases) in =-=[13]-=-, [28]. The attempt to extend this method for computing the shortest paths as well leads naturally to the above problem, which already found other (related) applications as well. See [4, 14, 20, 28] f... |