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Analysis of Shellsort and related algorithms
 ESA ’96: Fourth Annual European Symposium on Algorithms
, 1996
"... This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellso ..."
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This is an abstract of a survey talk on the theoretical and empirical studies that have been done over the past four decades on the Shellsort algorithm and its variants. The discussion includes: upper bounds, including linkages to numbertheoretic properties of the algorithm; lower bounds on Shellsort and Shellsortbased networks; averagecase results; proposed probabilistic sorting networks based on the algorithm; and a list of open problems. 1 Shellsort The basic Shellsort algorithm is among the earliest sorting methods to be discovered (by D. L. Shell in 1959 [36]) and is among the easiest to implement, as exhibited by the following C code for sorting an array a[l],..., a[r]: shellsort(itemType a[], int l, int r) { int i, j, h; itemType v;
Lower Bounds for Shellsort
 In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science
, 1997
"... We show lower bounds on the worstcase complexity of Shellsort. In particular, we give a fairly simple proof of an \Omega\Gamma n lg 2 n=(lg lg n) 2 ) lower bound for the size of Shellsort sorting networks, for arbitrary increment sequences. We also show an identical lower bound for the running ..."
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Cited by 13 (4 self)
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We show lower bounds on the worstcase complexity of Shellsort. In particular, we give a fairly simple proof of an \Omega\Gamma n lg 2 n=(lg lg n) 2 ) lower bound for the size of Shellsort sorting networks, for arbitrary increment sequences. We also show an identical lower bound for the running time of Shellsort algorithms, again for arbitrary increment sequences. Our lower bounds establish an almost tight tradeoff between the running time of a Shellsort algorithm and the length of the underlying increment sequence. Proposed running head: Lower Bounds for Shellsort. Contact author: Prof. Greg Plaxton, Department of Computer Science, University of Texas at Austin, Austin, Texas 787121188. 1 Introduction Shellsort is a classical sorting algorithm introduced by Shell in 1959 [15]. The algorithm is based on a sequence H = h 0 ; : : : ; hm\Gamma1 of positive integers called an increment sequence. An input file A = A[0]; : : : ; A[n \Gamma 1] of elements is sorted by performing an ...
On the Performance of WEAKHEAPSORT
, 2000
"... . Dutton #1993# presents a further HEAPSORT variant called WEAKHEAPSORT, which also contains a new data structure for priority queues. The sorting algorithm and the underlying data structure are analyzed showing that WEAKHEAPSORT is the best HEAPSORT variant and that it has a lot of nice propert ..."
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Cited by 6 (2 self)
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. Dutton #1993# presents a further HEAPSORT variant called WEAKHEAPSORT, which also contains a new data structure for priority queues. The sorting algorithm and the underlying data structure are analyzed showing that WEAKHEAPSORT is the best HEAPSORT variant and that it has a lot of nice properties. It is shown that the worst case number of comparisons is ndlog ne# 2 dlog ne + n #dlog ne#nlog n +0:1nand weak heaps can be generated with n # 1 comparisons. A doubleended priority queue based on weakheaps can be generated in n + dn=2e#2 comparisons. Moreover, examples for the worst and the best case of WEAKHEAPSORT are presented, the number of WeakHeaps on f1;:::;ng is determined, and experiments on the average case are reported. 1
The worst case in Shellsort and related algorithms
 Journal of Algorithms
, 1993
"... Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prov ..."
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Abstract. We show that sorting a sufficiently long list of length N using Shellsort with m increments (not necessarily decreasing) requires at least N 1+c/ √ m comparisons in the worst case, for some constant c> 0. For m ≤ (log N / log log N) 2 we obtain an upper bound of the same form. We also prove that Ω(N(log N / log log N) 2) comparisons are needed regardless of the number of increments. Our approach is general enough to apply to other sorting algorithms, including Shakersort, for which an even stronger result is proved. 1.
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.