## Lower Bounds for Shellsort (1997)

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Venue: | In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science |

Citations: | 14 - 5 self |

### BibTeX

@INPROCEEDINGS{Plaxton97lowerbounds,

author = {C. Greg Plaxton and Torsten Suel},

title = {Lower Bounds for Shellsort},

booktitle = {In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science},

year = {1997},

pages = {226--235}

}

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### Abstract

We show lower bounds on the worst-case 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 trade-off 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 78712--1188. 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 ...

### Citations

2327 |
The art of computer programming
- Knuth
- 1981
(Show Context)
Citation Context ...bn=4c ; : : : ; h 0 = 1. However, this choice of H leads to a worst case running time of (n 2 ) if n is a power of 2. Subsequently, several authors proposed modications to Shell's original sequence =-=[9, 5, 8]-=- in the hope of obtaining a better running time. Papernov and Stasevich [10] showed that the sequence of Hibbard [5], consisting of the increments of the form 2 k 1, achieves a running time of O(n 3... |

541 | Sorting networks and their applications
- Batcher
- 1968
(Show Context)
Citation Context ...sort sorting network of depth 0:6 lg 2 n based on increments of the form 2 i 3 j was given by Pratt [13]. Thus, his network came very close to the fastest known network at that time, due to Batcher =-=[2]-=-, with depth 0:5 lg 2 n. In 1983, Ajtai, Komlos, and Szemeredi [1] designed a sorting network of depth O(lg n); however, their construction suers from an irregular topology and a large constant h... |

214 | An O(n log n) sorting network - TAI, KOMLOS, et al. - 1983 |

115 |
Sorting in c log n parallel steps
- Ajtai, Komlós, et al.
- 1983
(Show Context)
Citation Context ...form 2 i 3 j was given by Pratt [13]. Thus, his network came very close to the fastest known network at that time, due to Batcher [2], with depth 0:5 lg 2 n. In 1983, Ajtai, Komlos, and Szemeredi =-=[1]-=- designed a sorting network of depth O(lg n); however, their construction suers from an irregular topology and a large constant hidden by the O-notation. This situation has motivated the search for O... |

99 | Average-case analysis of algorithms and data structures
- Flajolet, Vitter
- 1990
(Show Context)
Citation Context ...uation has motivated the search for O(lg n)-depth sorting networks with simpler topologies or a smaller multiplicative constant. Shellsort has been considered a potential candidate for such a network =-=[16]-=-, due to the rich variety of possible increment sequences and the lack of nontrivial general lower bounds. The lower bounds of Pratt and Weiss also apply to network size, but they only hold for very r... |

31 |
A high-speed sorting procedure
- Shell
- 1959
(Show Context)
Citation Context ...hor: Prof. Greg Plaxton, Department of Computer Science, University of Texas at Austin, Austin, Texas 78712{1188. 1 Introduction Shellsort is a classical sorting algorithm introduced by Shell in 1959 =-=[15]-=-. The algorithm is based on a sequence H = h 0 ; : : : ; h m1 of positive integers called an increment sequence. An inputsle A = A[0]; : : : ; A[n 1] of elements is sorted by performing an h j -sor... |

20 |
Improved upper bounds on shellsort
- Incerpi, Sedgewick
- 1985
(Show Context)
Citation Context ...ence is not popular because it has length (lg 2 n); implementations of Shellsort tend to use O(lg n)- length increment sequences because these result in better running times forsles of moderate size =-=[6]-=-. In addition, there is no hope of getting an O(n lg n)-time algorithm based on a sequence of length !(lg n). Pratt [13] also showed ans(n 3=2 ) lower bound for all nearly geometric sequences. Partly ... |

20 |
Shellsort and Sorting Networks
- Pratt
- 1971
(Show Context)
Citation Context ...mon feature of all of these sequences is that they are nearly geometric, meaning that they approximate a geometric sequence within an additive constant. An exception is the sequence designed by Pratt =-=[13]-=-, which consists of all increments of the form 2 i 3 j . This sequence gives a running time of O(n lg 2 n), which still represents the best asymptotic bound known for any increment sequence. In practi... |

11 |
A lower bound on the size of Shellsort sorting networks
- Cypher
- 1993
(Show Context)
Citation Context ...equences and the lack of nontrivial general lower bounds. The lower bounds of Pratt and Weiss also apply to network size, but they only hold for very restricted classes of increment sequences. Cypher =-=[3]-=- has established ans(n lg 2 n= lg lg n) lower bound for the size of Shellsort networks. However, his proof technique only works for monotone increment sequences, that is, sequences that are monotonica... |

9 |
An analysis of (h; k; 1)-Shellsort
- Yao
- 1980
(Show Context)
Citation Context ... given by Knuth [8], who determines an average case running time ofs(n 3=2 ) for Shell's original sequence. Increment sequences of the form (h; 1) and (h; k; 1) were investigated by Knuth [8] and Yao =-=[21]-=-, respectively. Weiss [18] conducted an extensive empirical study and conjectured that Shellsort will on average not perform signicantly better than in the worst case. Any general upper and lower bou... |

7 |
An ecient variation of Bubble Sort
- Dobosiewicz
- 1980
(Show Context)
Citation Context ...eralized to a fairly large class of \Shellsort-like" algorithms, including the Shaker Sort algorithm of Incerpi and Sedgewick [7, 19] as well as other algorithms proposed by Knuth [8] and Dobosiewicz =-=[4]-=-. Poonen [12] has formally dened a class of such algorithms, called Shellsort-type algorithms, and has shown how to extend his lower bound to this class. We will not elaborate further on such possibl... |

7 |
An empirical study of minimal storage sorting
- Hibbard
- 1963
(Show Context)
Citation Context ...bn=4c ; : : : ; h 0 = 1. However, this choice of H leads to a worst case running time of (n 2 ) if n is a power of 2. Subsequently, several authors proposed modications to Shell's original sequence =-=[9, 5, 8]-=- in the hope of obtaining a better running time. Papernov and Stasevich [10] showed that the sequence of Hibbard [5], consisting of the increments of the form 2 k 1, achieves a running time of O(n 3... |

7 |
Bad cases for Shaker-sort
- Weiss, Sedgewick
- 1988
(Show Context)
Citation Context ...olds for arbitrary increment sequences. Our lower bound can be further generalized to a fairly large class of \Shellsort-like" algorithms, including the Shaker Sort algorithm of Incerpi and Sedgewick =-=[7, 19]-=- as well as other algorithms proposed by Knuth [8] and Dobosiewicz [4]. Poonen [12] has formally dened a class of such algorithms, called Shellsort-type algorithms, and has shown how to extend his lo... |

6 |
Practical variations of Shellsort
- Incerpi, Sedgewick
- 1987
(Show Context)
Citation Context ...olds for arbitrary increment sequences. Our lower bound can be further generalized to a fairly large class of \Shellsort-like" algorithms, including the Shaker Sort algorithm of Incerpi and Sedgewick =-=[7, 19]-=- as well as other algorithms proposed by Knuth [8] and Dobosiewicz [4]. Poonen [12] has formally dened a class of such algorithms, called Shellsort-type algorithms, and has shown how to extend his lo... |

5 | The worst case in Shellsort and related algorithms
- Poonen
- 1993
(Show Context)
Citation Context ...h this captures a very general class of sequences, it does not rule out the possibility of an O(lg n)-depth network based on some nonmonotone sequence. Recently, and independent of this paper, Poonen =-=[12]-=- has shown a lower bound of (n lg 2 n=(lg lg n) 2 ) that holds for arbitrary Shellsort algorithms. His lower bound has the form of a trade-o between the running time of a Shellsort algorithm and the ... |

4 |
A high-speed sorting procedure
- Lazarus, Frank
- 1960
(Show Context)
Citation Context ...bn=4c ; : : : ; h 0 = 1. However, this choice of H leads to a worst case running time of (n 2 ) if n is a power of 2. Subsequently, several authors proposed modications to Shell's original sequence =-=[9, 5, 8]-=- in the hope of obtaining a better running time. Papernov and Stasevich [10] showed that the sequence of Hibbard [5], consisting of the increments of the form 2 k 1, achieves a running time of O(n 3... |

4 |
A method for information sorting in computer memories
- Papernov, Stasevich
- 1965
(Show Context)
Citation Context ...time of (n 2 ) if n is a power of 2. Subsequently, several authors proposed modications to Shell's original sequence [9, 5, 8] in the hope of obtaining a better running time. Papernov and Stasevich =-=[10]-=- showed that the sequence of Hibbard [5], consisting of the increments of the form 2 k 1, achieves a running time of O(n 3=2 ). A common feature of all of these sequences is that they are nearly geo... |

4 |
Tight lower bounds for Shellsort
- Weiss, Sedgewick
- 1990
(Show Context)
Citation Context ...ation of Pratt's sequence. The sequences proposed by Incerpi and Sedgewick are all within a constant factor of a geometric sequence, that is, they satisfy h j = ( j ) for some constant > 0. Weiss =-=[17, 20]-=- showed that all sequences of this type take times(n 1+= p lgn ), but his proof assumed an as yet unproven conjecture on the number of inversions in the Frobenius pattern. Based on this 2 so-called I... |

4 |
Empirical study of the expected running time of Shellsort
- Weiss
- 1991
(Show Context)
Citation Context ...etermines an average case running time ofs(n 3=2 ) for Shell's original sequence. Increment sequences of the form (h; 1) and (h; k; 1) were investigated by Knuth [8] and Yao [21], respectively. Weiss =-=[18]-=- conducted an extensive empirical study and conjectured that Shellsort will on average not perform signicantly better than in the worst case. Any general upper and lower bound for the average case wo... |