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

2015
*
arXiv
*
pre-print

We prove

arXiv:cs/9906008v2
fatcat:5evz5fy4rfg27ilbqdxasxxea4
*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*. ...##
###
A lower bound on the average-case complexity of shellsort

2000
*
Journal of the ACM
*

We demonstrate an ⍀( pn 1ϩ1/p)

doi:10.1145/355483.355488
fatcat:6h2yttuaqndo5lrx54utxdveri
*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 ...##
###
On The Average-Case Complexity of Shellsort
[article]

2017
*
arXiv
*
pre-print

We prove

arXiv:1501.06461v4
fatcat:boaew3jajngahj6h43afiiqicq
*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. ...##
###
Average-Case Complexity of Shellsort (Preliminary Version)
[chapter]

1999
*
Lecture Notes in Computer Science
*

We prove

doi:10.1007/3-540-48523-6_42
fatcat:seetfrrgunhx7m7clhdqstqkda
*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*. ...##
###
On the average-case complexity of Shellsort

2017
*
Random structures & algorithms (Print)
*

We prove

doi:10.1002/rsa.20737
fatcat:6pudhgpuijcflpxhkx542mb5nu
*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. ...##
###
Average-Case Complexity of Shellsort
[article]

1999
*
arXiv
*
pre-print

We prove

arXiv:cs/9901010v1
fatcat:3hsatxce2jhbvcwmkedlbkd5um
*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. ...##
###
Analysis of Shellsort and related algorithms
[chapter]

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. ...

##
###
The Incompressibility Method
[chapter]

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*. ...

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

2009
*
arXiv
*
pre-print

Especially in

arXiv:0905.4452v1
fatcat:amug6iugczdb3b65kxjrxuemwu
*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. ...##
###
ANALYSIS OF SHELLSORT ALGORITHMS

2019
*
International Journal of Advanced Research in Computer Science
*

In this paper, we have analyzed

doi:10.26483/ijarcs.v10i3.6433
fatcat:b3wj246iz5deleelxtmbv7lyse
*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. ...##
###
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

1997
*
Journal of Algorithms
*

We show

doi:10.1006/jagm.1996.0825
fatcat:uuhstzejondplopfecok3topke
*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*. ...##
###
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

1992
*
Proceedings., 33rd Annual Symposium on Foundations of Computer Science
*

We give improved

doi:10.1109/sfcs.1992.267769
dblp:conf/focs/PlaxtonPS92
fatcat:su5a4luizzghtlqgbn66g3ashu
*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*. ...##
###
The Worst Case in Shellsort and Related Algorithms

1993
*
Journal of Algorithms
*

We show that sorting

doi:10.1006/jagm.1993.1032
fatcat:4ap77q3xq5curl3ewj57yrqu5y
*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. ...
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