## Randomized Shellsort: A simple oblivious sorting algorithm (2010)

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Venue: | In Proceedings 21st ACM-SIAM Symposium on Discrete Algorithms (SODA |

Citations: | 12 - 2 self |

### BibTeX

@INPROCEEDINGS{Goodrich10randomizedshellsort:,

author = {Michael T. Goodrich},

title = {Randomized Shellsort: A simple oblivious sorting algorithm},

booktitle = {In Proceedings 21st ACM-SIAM Symposium on Discrete Algorithms (SODA},

year = {2010}

}

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### Abstract

In this paper, we describe a randomized Shellsort algorithm. This algorithm is a simple, randomized, data-oblivious version of the Shellsort algorithm that always runs in O(n log n) time and succeeds in sorting any given input permutation with very high probability. Taken together, these properties imply applications in the design of new efficient privacypreserving computations based on the secure multi-party computation (SMC) paradigm. In addition, by a trivial conversion of this Monte Carlo algorithm to its Las Vegas equivalent, one gets the first version of Shellsort with a running time that is provably O(n log n) with very high probability. 1

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Citation Context ...timal but not data-oblivious, as are the PRAM sorting algorithms of Shavit et al. [47], Cole [11], Reif [41], and Goodrich and Kosaraju [21], as well as the multi-butterfly routing algorithm of Upfal =-=[49]-=- (which, strictly speaking, is not a sorting algorithm, but is nevertheless related). The shear-sort algorithm of Scherson and Sen [43] is simple and data-oblivious but not timeoptimal. The columnsort... |

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Citation Context ...ng of all products of powers of 2 and 3 less than n results in a worst-case running time of O(n log 2 n). Several other offset sequences have been studied (e.g., see the excellent survey of Sedgewick =-=[42]-=-), but none beat the asymptotic performance 1 This paper is dedicated to my parents, Ronald and Grace Goodrich, to commemorate their 50th wedding anniversary. of the Pratt sequence. Moreover, Plaxton ... |

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Citation Context ...ly a family of algorithms, since there are so many different offset sequences. The trick in implementing a traditional version of Shellsort, therefore, is coming up with a good offset sequence. Pratt =-=[36]-=- shows that using a sequence consisting of all products of powers of 2 and 3 less than n results in a worst-case running time of O(n log 2 n). Several other offset sequences have been studied (e.g., s... |

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Citation Context ...—which will work correctly, for example, with the Pratt offset sequence, and which seems to work well in practice for geometric offset sequences with ratios less than 1.33 [12]. Incerpi and Sedgewick =-=[21, 22]-=- study a version of Shellsort that replaces the insertion-sort by a shaker pass (see also [7, 46]). This is a left-to-right bubble-sort pass followed by a right-to-left bubble-sort pass and it also se... |

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Citation Context ...g 2 n). Several other offset sequences have been studied (e.g., see the excellent survey of Sedgewick [45]), but none beat the asymptotic performance of the Pratt sequence. Moreover, Plaxton and Suel =-=[38]-=- establish a lower bound of Ω(n log 2 n/(log log n) 2 ) for the worst-case running time of Shellsort with any input sequence (see also [13]) and Jiang et al. [26] establish a lower bound of Ω(pn 1+1/p... |

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Citation Context ...niversary. of the Pratt sequence. Moreover, Plaxton and Suel [35] establish a lower bound of Ω(n log 2 n/(log log n) 2 ) for the worst-case running time of Shellsort with any input sequence (see also =-=[11]-=-) and Jiang et al. [23] establish a lower bound of Ω(pn1+1/p ) for the average-case running time of Shellsort. Thus, the only way to achieve an O(n log n) averagetime bound for Shellsort is to use an ... |

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Citation Context ... sequence. Moreover, Plaxton and Suel [35] establish a lower bound of Ω(n log 2 n/(log log n) 2 ) for the worst-case running time of Shellsort with any input sequence (see also [11]) and Jiang et al. =-=[23]-=- establish a lower bound of Ω(pn1+1/p ) for the average-case running time of Shellsort. Thus, the only way to achieve an O(n log n) averagetime bound for Shellsort is to use an offset sequence of leng... |

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Citation Context ...e offset sequence and how it is used, there are other existing variations to Shellsort, which are based on replacing the insertion-sort in the inner loop with other actions. For instance, Dobosiewicz =-=[12]-=- proposes replacing the insertion-sort with a single lineartime bubble-sort pass—doing a left-to-right sequence of compare-exchanges between elements at offset-distances apart—which will work correctl... |

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Citation Context ...saraju [18]. The shear-sort algorithm of Scherson and Sen [40] is simple and data-oblivious but not time-optimal. The columnsort algorithm of Leighton [26] and the sorting method of Maggs and Vöcking =-=[28]-=- are asymptotically fast, but they both employ the AKS network; hence, they are not simple. Finally, note that well-known time-optimal sorting algorithms, such as radix-sort, quicksort, heapsort, and ... |

7 |
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Citation Context ...ell in practice for geometric offset sequences with ratios less than 1.33 [12]. Incerpi and Sedgewick [21, 22] study a version of Shellsort that replaces the insertion-sort by a shaker pass (see also =-=[7, 46]-=-). This is a left-to-right bubble-sort pass followed by a right-to-left bubble-sort pass and it also seems to do better in practice for geometric offset sequences [22]. Yet another modification of She... |

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5 |
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Citation Context ...ell in practice for geometric offset sequences with ratios less than 1.33 [12]. Incerpi and Sedgewick [21, 22] study a version of Shellsort that replaces the insertion-sort by a shaker pass (see also =-=[7, 46]-=-). This is a left-to-right bubble-sort pass followed by a right-to-left bubble-sort pass and it also seems to do better in practice for geometric offset sequences [22]. Yet another modification of She... |

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Citation Context ...al data-oblivious sequential sorting algorithms. For example, the online routing method of Arora et al. [2] is time-optimal but not data-oblivious, as are the PRAM sorting algorithms of Shavit et al. =-=[44]-=-, Cole [9], Reif [38], and Goodrich and Kosaraju [18]. The shear-sort algorithm of Scherson and Sen [40] is simple and data-oblivious but not time-optimal. The columnsort algorithm of Leighton [26] an... |

4 | and J.Gill, “Sorting n Objects with a k-sorter
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Citation Context ...orithm itself, but it is nevertheless helpful in proving a high-probability bound. Define an n-element array, A, to be m-near-sorted if all but m of the n elements in A are in sorted order. A psorter =-=[3, 4]-=- is a deterministic sorting algorithm that can sort a subarray of size p as an atomic action. Suppose S is a dataoblivious (deterministic) 2m-sorter that runs in T (m) time. Define an S-shaker pass ov... |

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Citation Context ...its offset sequence. They don’t prove an O(n log n) average-time bound for this version, but they do provide some promising empirical data to support an average running time near O(n log n); see also =-=[31]-=-. 1.1 Data-Oblivious Sorting In addition to its simplicity, one of the interesting properties of Shellsort is that many of its variants are data-oblivious. Specifically, if we view compare-exchange op... |

4 | Asymptotic complexity from experiments? A case study for randomized algorithms
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Citation Context ...y of these variants of Shellsort that proves they have 1262 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.an expected running time of O(n log n). Sanders and Fleischer =-=[39]-=- describe an algorithm they call “randomized Shellsort,” which is a data-dependent Shellsort algorithm as in the above pseudo-code description, except that it uses products of random numbers as its of... |

3 |
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Citation Context ... if all of v’s descendents are associated with low regions. Likewise, a region, Ai, corresponding to an internal node v of B, is a high region if 2 A similar lemma is provided by Blackston and Ranade =-=[6]-=-, but they omit the proof. 1269 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.low regions 1 2 5 4 3 1 2 1 1 2 high regions 1 Because of our need for a high probability ... |

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Citation Context ...cols allow the parties to compute the value of f on their collective input values without revealing any of their specific data values (other than what can inferred from the output function, f, itself =-=[17]-=-). One of the main tools for building SMC protocols is to encode the function f as a circuit and then simulate an evaluation of this circuit using digitally-masked values, as in the Fairplay system [5... |

3 | PDM sorting algorithms that take a small number of passes
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Citation Context ...h failure probability at most ɛ, then it sorts any arbitrary array of size n with failure probability at most ɛ(n + 1). Proof: The lemma 2 follows from the proof of Theorem 3.3 by Rajasekaran and Sen =-=[37]-=-, which itself is based on the justification of Knuth [24] for the deterministic version of the zero-one principle for sorting networks. The essential fact is that an arbitrary n-element input array, ... |

3 |
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Citation Context ...e can achieve a sorting network with O(n log n) compare-exchange gates in the worst case, but their method is quite complicated and has a very large constant factor, even with subsequent improvements =-=[34, 43]-=-. Leighton and Plaxton [27] describe a randomized method for building a data-oblivious sorting network that uses O(n log n) compare-exchange gates and sorts any given input array with very high probab... |

2 |
Sorting with efficient use of special-purpose sorters
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Citation Context ...orithm itself, but it is nevertheless helpful in proving a high-probability bound. Define an n-element array, A, to be m-near-sorted if all but m of the n elements in A are in sorted order. A psorter =-=[3, 4]-=- is a deterministic sorting algorithm that can sort a subarray of size p as an atomic action. Suppose S is a dataoblivious (deterministic) 2m-sorter that runs in T (m) time. Define an S-shaker pass ov... |

2 | Concentration of measure for randomized algorithms: Techniques and analysis
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Citation Context ...ernoff bound (e.g., see [32,33]) to probabilistically bound the degree to which f differs from its expectation. We can nevertheless derive such a bound using the method of bounded average differences =-=[15]-=-. Imagine that we are computing f incrementally as the Xi,j are revealed by an algorithm performing the c region compare-exchange operations. Suppose further that the Xi,j’s are revealed lexicographic... |