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A Lower Bound on the AverageCase Complexity of Shellsort
, 1999
"... We give a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for any incremental sequence is \Omega\Gamma pn 1+1=p ) for every p. The proof is an example of the use of Kolmogorov complexity (the incompr ..."
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We give a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for any incremental sequence is \Omega\Gamma pn 1+1=p ) for every p. The proof is an example of the use of Kolmogorov complexity (the incompressibility method) in the analysis of algorithms. 1 Introduction The question of a nontrivial general lower bound (or upper bound) on the average complexity of Shellsort (due to D.L. Shell [14]) has been open for about four decades [5, 13]. We present such a lower bound for ppass Shellsort for every p. Shellsort sorts a list of n elements in p passes using a sequence of increments h 1 ; : : : ; h p . In the kth pass the main list is divided in h k separate sublists of length dn=h k e, where the ith sublist consists of the elements at positions j, where j mod h k = i \Gamma 1, of the main list (i = 1; : : : ; h k ). Every sublist is sorted using a straightforward insertion sort. The effi...
The averagecase complexity of Shellsort
 LECTURE NOTES IN COMPUTER SCIENCE 1644
, 1999
"... We prove a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for 1 1+ any incremental sequence is Ω(pn p) for all p ≤ log n. Using similar arguments, we analyze the averagecase complexity of several oth ..."
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Cited by 6 (2 self)
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We prove a general lower bound on the averagecase complexity of Shellsort: the average number of datamovements (and comparisons) made by a ppass Shellsort for 1 1+ any incremental sequence is Ω(pn p) for all p ≤ log n. Using similar arguments, we analyze the averagecase complexity of several other sorting algorithms.
Enhanced Shell Sorting Algorithm
 Computer Journal of Enformatika
, 2007
"... Abstract—Many algorithms are available for sorting the unordered elements. Most important of them are Bubble sort, Heap sort, Insertion sort and Shell sort. These algorithms have their own pros and cons. Shell Sort which is an enhanced version of insertion sort, reduces the number of swaps of the el ..."
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Abstract—Many algorithms are available for sorting the unordered elements. Most important of them are Bubble sort, Heap sort, Insertion sort and Shell sort. These algorithms have their own pros and cons. Shell Sort which is an enhanced version of insertion sort, reduces the number of swaps of the elements being sorted to minimize the complexity and time as compared to insertion sort. Shell sort improves the efficiency of insertion sort by quickly shifting values to their destination. Average sort time is O(n 1.25), while worstcase time is O(n 1.5). It performs certain iterations. In each iteration it swaps some elements of the array in such a way that in last iteration when the value of h is one, the number of swaps will be reduced. Donald L. Shell invented a formula to calculate the value of ‘h’. this work focuses to identify some improvement in the conventional Shell sort algorithm. “Enhanced Shell Sort algorithm ” is an improvement in the algorithm to calculate the value of ‘h’. It has been observed that by applying this algorithm, number of swaps can be reduced up to 60 percent as compared to the existing algorithm. In some other cases this enhancement was found faster than the existing algorithms available. Keywords—Algorithm, Computation, Shell, Sorting. I.
Analysis of Sorting Algorithms by Kolmogorov Complexity  A Survey
, 2003
"... Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, ..."
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Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewiczsort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.
method
"... Background: Although a variety of methods and expensive kits are available, molecular cloning can be a timeconsuming and frustrating process. Results: Here we report a highly simplified, reliable, and efficient PCRbased cloning technique to insert any DNA fragment into a plasmid vector or into a ge ..."
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Background: Although a variety of methods and expensive kits are available, molecular cloning can be a timeconsuming and frustrating process. Results: Here we report a highly simplified, reliable, and efficient PCRbased cloning technique to insert any DNA fragment into a plasmid vector or into a gene (cDNA) in a vector at any desired position. With this method, the vector and insert are PCR amplified separately, with only 18 cycles, using a high fidelity DNA polymerase. The amplified insert has the ends with ~16base overlapping with the ends of the amplified vector. After DpnI digestion of the mixture of the amplified vector and insert to eliminate the DNA templates used in PCR reactions, the mixture is directly transformed into competent E. coli cells to obtain the desired clones. This technique has many advantages over other cloning methods. First, it does not need gel purification of the PCR product or linearized vector. Second, there is no need of any cloning kit or specialized enzyme for cloning. Furthermore, with reduced number of PCR cycles, it also decreases the chance of random mutations. In addition, this method is highly effective and reproducible. Finally, since this cloning method is also sequence independent, we demonstrated that it can be used for chimera construction, insertion, and multiple mutations spanning a stretch of DNA up to 120 bp.
BOLYAI SOCIETY Entropy, Search, Complexity, pp. 209–232. MATHEMATICAL STUDIES, 16 Analysis of Sorting Algorithms by Kolmogorov Complexity
"... Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Do ..."
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Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard averagecase analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewiczsort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks. 1.
Asymptotic Analysis of (3, 2, 1)Shell Sort
, 2001
"... Shell Sort is an algorithm for sorting alist of numbers in stages. The algorithm, proposed by Shell [16], was given in Knuth’s 1973 book [6] and has generated asubstantial literature in the years since (many references are given in Sedgewick [14] and Mahmoud [10]). Shell Sort is ageneralization of i ..."
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Shell Sort is an algorithm for sorting alist of numbers in stages. The algorithm, proposed by Shell [16], was given in Knuth’s 1973 book [6] and has generated asubstantial literature in the years since (many references are given in Sedgewick [14] and Mahmoud [10]). Shell Sort is ageneralization of insertion sort. It has little overhead and is easy to
Best Increments for the Average Case of Shellsort
, 2001
"... This paper presents the results of using sequential analysis to find increment sequences that minimize the average running time of Shellsort, for array sizes up to several thousand elements. The obtained sequences outperform by about 3% the best ones known so far, and there is a plausible evidence t ..."
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This paper presents the results of using sequential analysis to find increment sequences that minimize the average running time of Shellsort, for array sizes up to several thousand elements. The obtained sequences outperform by about 3% the best ones known so far, and there is a plausible evidence that they are the optimal ones.