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A Lower Bound on the Average-Case Complexity of Shellsort [article]

Tao Jiang, Paul Vitanyi (CWI & U. Amsterdam)
2015 arXiv   pre-print
We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a p-pass Shellsort for any incremental sequence is Ω (pn^1 + 1  ...  The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the average-case complexity of several other sorting algorithms is analyzed.  ...  Acknowledgements We thank Don Knuth, Ian Munro, Vaughan Pratt, and Osamu Watanabe for discussions and references on Shellsort.  ... 
arXiv:cs/9906008v2 fatcat:5evz5fy4rfg27ilbqdxasxxea4

A lower bound on the average-case complexity of shellsort

Tao Jiang, Ming Li, Paul Vitányi
2000 Journal of the ACM  
We demonstrate an ⍀( pn 1ϩ1/p) lower bound on the average-case running time (uniform distribution) of p-pass Shellsort. This is the first nontrivial general lower bound for average-case Shellsort.  ...  We thank Don Knuth, Ian Munro, Vaughan Pratt, and Osamu Watanabe for discussions and references on Shellsort.  ...  Some special cases of the lower bound on the average-case complexity are: (1) For p ϭ 1, our lower bound is asymptotically tight: it is the average number of inversions for Insertion Sort. (2) For p ϭ  ... 
doi:10.1145/355483.355488 fatcat:6h2yttuaqndo5lrx54utxdveri

On The Average-Case Complexity of Shellsort [article]

Paul M.B. Vitanyi (National Research Center for Mathematics and Computer Science in the Netherlands
2017 arXiv   pre-print
We prove a lower bound expressed in the increment sequence on the average-case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked.  ...  A special case of this lower bound yields the general Jiang-Li-Vit\'anyi lower bound. We obtain new results e.g. determining the average-case complexity precisely in the Yao-Janson-Knuth 3-pass case.  ...  Acknowledgment I thank Ronald de Wolf for comments on a preliminary draft of this paper and the referees for constructive suggestions.  ... 
arXiv:1501.06461v4 fatcat:boaew3jajngahj6h43afiiqicq

Average-Case Complexity of Shellsort (Preliminary Version) [chapter]

Tao Jiang, Ming Li, Paul Vitányi
1999 Lecture Notes in Computer Science  
We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and com-paris~ns) made by a p-pass Shellsort for any incremental sequence is fl(pn HIP)  ...  The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the average-case complexity of several other sorting algorithms is analyzed.  ...  larger shows a 8(nlog 2 n/(loglogn) 2 ) lower bound on the worst case complexity of Shellsort.  ... 
doi:10.1007/3-540-48523-6_42 fatcat:seetfrrgunhx7m7clhdqstqkda

On the average-case complexity of Shellsort

Paul Vitányi
2017 Random structures & algorithms (Print)  
We prove a lower bound expressed in the increment sequence on the average-case complexity of the number of inversions of Shellsort. This lower bound is sharp in every case where it could be checked.  ...  A special case of this lower bound yields the general Jiang-Li-Vitányi lower bound. We obtain new results e.g. determining the average-case complexity precisely in the Yao-Janson-Knuth 3-pass case. *  ...  Acknowledgment I thank Ronald de Wolf for comments on a preliminary draft of this paper and the referees for constructive suggestions.  ... 
doi:10.1002/rsa.20737 fatcat:6pudhgpuijcflpxhkx542mb5nu

Average-Case Complexity of Shellsort [article]

Tao Jiang, Paul Vitanyi (CWI, U of Amsterdam)
1999 arXiv   pre-print
We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a p-pass Shellsort for any incremental sequence is Ω (pn^1 + 1  ...  Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.  ...  Sedgewick for telling us many things about Shellsort and stack/queue sorts.  ... 
arXiv:cs/9901010v1 fatcat:3hsatxce2jhbvcwmkedlbkd5um

Analysis of Shellsort and related algorithms [chapter]

Robert Sedgewick
1996 Lecture Notes in Computer Science  
The discussion includes: upper bounds, including linkages to number-theoretic properties of the algorithm; lower bounds on Shellsort and Shellsort-based networks; average-case results; proposed probabilistic  ...  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.  ...  References This paper summarizes the work of many authors, and certainly cannot do justice to the detailed information given in their original works, listed here.  ... 
doi:10.1007/3-540-61680-2_42 fatcat:chbi6rk7abhlnehpyanx3cuxoq

The Incompressibility Method [chapter]

Tao Jiang, Ming Li, Paul Vitányi
2000 Lecture Notes in Computer Science  
The question of a nontrivial general lower bound (or upper bound) on the average-case complexity of Shellsort has been open for about forty years. We obtain the rst such lower bound.  ...  Our second example concerns a \dynamic" problem in average-case running time of algorithms.  ...  Acknowledgement: We thank John Tromp for help with the proof of theorem 1. We thank Don Knuth, Ian Munro, and Vaughan Pratt for discussions and references on Shellsort.  ... 
doi:10.1007/3-540-44411-4_3 fatcat:jtkvaqwbcbb3bhbcjlt7rxd7l4

Analysis of Sorting Algorithms by Kolmogorov Complexity (A Survey) [article]

Paul M.B. Vitanyi
2009 arXiv   pre-print
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.  ...  Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method).  ...  Shellsort The question of a nontrivial general lower bound (or upper bound) on the average complexity of Shellsort (due to D.L.  ... 
arXiv:0905.4452v1 fatcat:amug6iugczdb3b65kxjrxuemwu

ANALYSIS OF SHELLSORT ALGORITHMS

Avik Mitra, BCA Department The Heritage Academy Kolkata, India
2019 International Journal of Advanced Research in Computer Science  
In this paper, we have analyzed the performance of variants of Shellsort based on their time complexity.  ...  The time complexity of Shellsort is dependent upon the method of interleaving (called increment sequence) giving variants of Shellsort.  ...  The discussed variants did not achieve the lower bound Θ(n logn) [20] of a comparison sort algorithm.  ... 
doi:10.26483/ijarcs.v10i3.6433 fatcat:b3wj246iz5deleelxtmbv7lyse

Page 5964 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
When the number of increments is unrestricted, Shellsort has a complexity gap be- tween the lower bound of Q(N(log N)?/loglog N) and the upper bound of O(N (log N)*).  ...  The author conjectures that the complex- ity is actually @( N(log N)?). He points out that virtually nothing is known about the average-case behaviour of Shellsort and Shaker- sort.  ... 

Lower Bounds for Shellsort

C.Greg Plaxton, Torsten Suel
1997 Journal of Algorithms  
We show lower bounds on the worst-case complexity of Shellsort.  ...  Our lower bounds establish an almost tight trade-o between the running time of a Shellsort algorithm and the length of the underlying increment sequence.  ...  Overview of the Paper In this paper we show lower bounds on the worst-case complexity of Shellsort.  ... 
doi:10.1006/jagm.1996.0825 fatcat:uuhstzejondplopfecok3topke

Page 1260 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews  
Summary: “We show lower bounds on the worst-case complex- ity of Shellsort. In particular, we give a fairly simple proof of an Q(n(Ig?  ...  Our lower bounds establish an almost tight trade-off between the running time of a Shellsort algorithm and the length of the underlying increment se- quence.”  ... 

Improved lower bounds for Shellsort

C.G. Plaxton, B. Poonen, T. Suel
1992 Proceedings., 33rd Annual Symposium on Foundations of Computer Science  
We give improved lower bounds for Shellsort based on a new and relatively simple proof idea. The lower bounds obtained are both stronger and more general than the previously known bounds.  ...  In particular, they hold for nonmonotone increment sequences and adaptive Shellsort algorithms, as well as for some recently proposed variations of Shellsort.  ...  Open Questions The primary remaining challenge in the study of Shellsort seems to be the virtual nonexistence of both upper and lower bounds for average case complexity.  ... 
doi:10.1109/sfcs.1992.267769 dblp:conf/focs/PlaxtonPS92 fatcat:su5a4luizzghtlqgbn66g3ashu

The Worst Case in Shellsort and Related Algorithms

B. Poonen
1993 Journal of Algorithms  
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  ...  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.  ...  Lower Bounds on the Worst Case We are now ready to prove a lower bound corresponding to the upper bound of Theorem 2.  ... 
doi:10.1006/jagm.1993.1032 fatcat:4ap77q3xq5curl3ewj57yrqu5y
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