A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear nu...
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A new algorithm for generating order preserving minimal perfect hash functions is presented. The algorithm is probabilistic, involving generation of random graphs. It uses expected linear time and requires a linear number of words to represent the hash function, and thus is optimal up to constant factors. It runs very fast in practice.
This paper considers a family of randomized on-line algorithms, Algorithm R(m), where 1 less-than-or-equal-to m less-than-or-equal-to n - 1 and n is the number of input points, for the on-line Steiner tree and on-line...
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This paper considers a family of randomized on-line algorithms, Algorithm R(m), where 1 less-than-or-equal-to m less-than-or-equal-to n - 1 and n is the number of input points, for the on-line Steiner tree and on-line spanning tree problems on Euclidean space. Our main result is that if m is a fixed constant, the competitive ratios of Algorithm R(m) for the on-line Steiner tree and spanning tree problems are THETA(n). We also show that the competitive ratio of Algorithm R(n - 1), which is deterministic greedy algorithm, for the on-line spanning tree problem is the same as that for the on-line Steiner tree problem, which is O(log n).
In this research note we investigate the number of moves and the displacement of particular elements during the execution of the well-known quicksort algorithm. This type of analysis is useful if the costs of data mov...
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In this research note we investigate the number of moves and the displacement of particular elements during the execution of the well-known quicksort algorithm. This type of analysis is useful if the costs of data moves were dependent on the Source and target locations, and possibly the moved element itself. From the mathematical point of view, the analysis of these quantities turns out to be related to the analysis of quickselect, a selection algorithm which is a variant of quicksort that finds the i-th smallest element of n given elements, without sorting them. Our results constitute thus a novel application of M. Kuba's machinery [M. Kuba, On quickselect, partial sorting and multiple quickselect, Inform. Process. Lett. 99(5) (2006) 181-186] for the solution of general quickselect recurrences. (C) 2009 Elsevier B.V. All rights reserved.
Gutin et al. [G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1-2) (2002) 107-116] proved that, in the ATSP problem, ...
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Gutin et al. [G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1-2) (2002) 107-116] proved that, in the ATSP problem, a tour of weight not exceeding the weight of an average tour is of dominance ratio at least 1/(n-1) for all n not equal 6. (Tours with this property can be easily obtained.) In [N. Alon, G. Gutin, M. Krivelevich, algorithms with large domination ratio, Journal on algorithms 50 (2004) 118-131;G. Gutin, A. Vainshtein, A. Yeo, Domination analysis of combinatorial optimization problems, Discrete Applied Mathematics 129 (2-3) (2003) 513-520;G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1-2) (2002) 107-116], algorithms with large dominance ratio were provided for MAX CUT, MAX r-SAT, ATSP, and other problems. All these algorithms share a common property - they provide solutions of quality guaranteed to be not worse than the average solution value. In this paper we show that, in general, this property by itself does not necessarily ensure a good performance in terms of dominance. Specifically, we show that for the MAXSAT problem, algorithms with this property might perform poorly in terms of dominance. (C) 2007 Elsevier B.V. All rights reserved.
Lee and Batcher have designed networks that efficiently merge k separately provided sorted sequences of known lengths totalling n. We show that the design is still possible, and in fact easier to describe, if we do no...
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Lee and Batcher have designed networks that efficiently merge k separately provided sorted sequences of known lengths totalling n. We show that the design is still possible, and in fact easier to describe, if we do not make use of the lengths, or even the directions of monotorticity, of the individual sequences-the sequences can be provided in a single undelimited concatenation of length n. The depth of the simplest resulting network to sort sequenccs that are "k-tonic" and of length n is (1 + [log(2) k])[log(2) n] =O((log k)(log n)), generalizing Batcher's 1968 results for the extreme values of k (k = 2 corresponding to merging, and k = [n/2] corresponding to general sorting). The exposition is self-contained and call serve even as an introduction to sorting networks and Batcher's results. (c) 2005 Elsevier Inc. All rights reserved.
In 1992 F. K. Hwang and J. F. Weng published an O(n(2)) time algorithm for computing the shortest network under a given full Steiner topology interconnecting n fixed points in the Euclidean plane, The Hwang-Weng algor...
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In 1992 F. K. Hwang and J. F. Weng published an O(n(2)) time algorithm for computing the shortest network under a given full Steiner topology interconnecting n fixed points in the Euclidean plane, The Hwang-Weng algorithm can be used to improve substantially existing algorithms for the Steiner minimum tree problem because it reduces the number of different Steiner topologies to be considered dramatically. In this paper we present an improved Hwang-Weng algorithm. While the worst-case time complexity of our algorithm is still O (n(2)), its average time complexity over all the full Steiner topologies interconnecting n fixed points is O (n log n).
We present an algorithm for constructing a perfect word hash function for n integers that takes O (n(4) logn) time. This time is independent of size of the integers or the number of bits in the integers. We call it a ...
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We present an algorithm for constructing a perfect word hash function for n integers that takes O (n(4) logn) time. This time is independent of size of the integers or the number of bits in the integers. We call it a word hash function because we require that the hash function can hash multiple integers packed in a word in constant time. Previous algorithms for constructing a perfect word hash function have time dependent on the number of the bits in integers. Although an O (n(loglogn)(2)) time algorithm is known for constructing a perfect hash function, the hash function constructed is not a word hash function and it cannot hash multiple integers packed in one word in constant time. The property of word hashing is indispensable in the current best deterministic and randomized algorithms for integer sorting. Our result is achieved via an algorithm of O (n(2) log(2) n) time that computes the shift distances for integers of St (n(2) log ri) bits. These shift distances can then be used to pack the extracted bits of each integer to O (n) bits. Perfect word hash function constructed with our method using these shift distances allows a batch of mn integers with m integers packed in a word to be hashed in O (n) time. (C) 2017 Elsevier B.V. All rights reserved.
The Set Covering problem is an NP-complete problem. A recent result of Lund and Yanakakis establishes that Set Covering cannot be approximated with ratio c log(2) n for any c<1/4 unless all NP problems are solvable...
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The Set Covering problem is an NP-complete problem. A recent result of Lund and Yanakakis establishes that Set Covering cannot be approximated with ratio c log(2) n for any c<1/4 unless all NP problems are solvable in DTIME(n(polylog n)). Th, known greedy algorithm for Set Covering delivers a ratio of H(d)=Sigma(i=1)(d)1/i, where d is the size of the largest set, a value bounded by 1 + log d. Here we describe an improved version of the greedy algorithm and show that its worst case ratio bound is, H(d)-1/6.
We further simplify Paterson's version of the Ajtai-KomlA(3)s-Szemer,di sorting network, and its analysis, mainly by tuning the invariant to be maintained.
We further simplify Paterson's version of the Ajtai-KomlA(3)s-Szemer,di sorting network, and its analysis, mainly by tuning the invariant to be maintained.
The vertices of a directed acyclic graph (DAG) are correctly prioritized if every vertex v in the graph is assigned a priority, denoted by priority(v), such that if there is an edge in the DAG from vertex v to vertex ...
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The vertices of a directed acyclic graph (DAG) are correctly prioritized if every vertex v in the graph is assigned a priority, denoted by priority(v), such that if there is an edge in the DAG from vertex v to vertex w then priority(v) < priority(w). The dynamic priority-ordering problem is to maintain a correct prioritization of the graph as the DAG is modified. We show that the Alpern et al. algorithm for this problem does not have a constant competitive ratio, where the cost of the algorithm is measured in terms of the number of primitive priority-manipulation operations. The proof shows that there exists no algorithm for the problem that has a constant competitive ratio, as long as the allowed primitive priority-manipulation operations satisfy a simple property. The proof also shows that there exists no algorithm for the problem of maintaining a topological-sort ordering that has a constant competitive ratio.
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