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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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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 ...
Randomized Shellsort: A simple oblivious sorting algorithm
 In Proceedings 21st ACMSIAM Symposium on Discrete Algorithms (SODA
, 2010
"... In this paper, we describe a randomized Shellsort algorithm. This algorithm is a simple, randomized, dataoblivious 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 ..."
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In this paper, we describe a randomized Shellsort algorithm. This algorithm is a simple, randomized, dataoblivious 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 multiparty 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
A Lower Bound for Sorting Networks Based on the Shuffle Permutation
 Mathematical Systems Theory
, 1994
"... We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of ninput sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)depth bitonic sor ..."
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We prove an \Omega\Gamma/1 2 n= lg lg n) lower bound for the depth of ninput sorting networks based on the shuffle permutation. The best previously known lower bound was the trivial \Omega\Gammaiv n) bound, while the best upper bound is given by Batcher's \Theta(lg 2 n)depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest. 1 Introduction A variety of different classes of sorting networks has been described in the literature. Of particular interest here are the socalled AKS network [1] discovered by Ajtai, Koml'os and Szemer'edi, and the sorting networks proposed by Batcher [2]. The AKS network is the only known sorting network with O(lg n) depth. However, the topology of the network is highly irregular, and the multiplicative constant hidden by the Onotation is impractically large [1, 11]. On the other hand, the networks proposed by Batcher have a relatively simple interconnection structure and a small constant...
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 p ..."
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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.
A SuperLogarithmic Lower Bound for Hypercubic Sorting Networks
 in Proceedings of the 21st International Colloquium on Automata, Languages, and Programming
, 1994
"... Hypercubic sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput hypercubic sorting networks with depth 2 O( p lg lg n) lg n have been discovered. These networks are the only known sorti ..."
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Hypercubic sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput hypercubic sorting networks with depth 2 O( p lg lg n) lg n have been discovered. These networks are the only known sorting networks of depth o(lg 2 n) that are not based on expanders, and their existence raises the question of whether a depth of O(lg n) can be achieved by any hypercubic sorting network. In this paper, we resolve this question by establishing an\Omega \Gamma lg n lg lg n lg lg lg n \Delta lower bound on the depth of any ninput hypercubic sorting network. Our lower bound can be extended to certain restricted classes of nonoblivious sorting algorithms on hypercubic machines. 1 Introduction A variety of different classes of sorting networks have been described in the literature. Of particular interest here are the socalled AKS network [1] discovered by Ajtai, Koml'os, and Szemer...
Spinthebottle Sort and Annealing Sort: Oblivious Sorting via Roundrobin Random Comparisons
"... We study sorting algorithms based on randomized roundrobin comparisons. Specifically, we study Spinthebottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, dat ..."
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We study sorting algorithms based on randomized roundrobin comparisons. Specifically, we study Spinthebottle sort, where comparisons are unrestricted, and Annealing sort, where comparisons are restricted to a distance bounded by a temperature parameter. Both algorithms are simple, randomized, dataoblivious sorting algorithms, which are useful in privacypreserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spinthebottle sort to require Ω(n² log n) expected time in order to succeed, and that in O(n² log n) time this algorithm succeeds with high probability for any input. We also show there is an specification of Annealing sort that runs in O(n log n) time and succeeds with very high probability.
Parallel sorting on ILLIAC array processor, in
 Proc. of the 7th Conference on 7th WSEAS International Conference on Systems Theory and Scientific Computation
, 2007
"... Abstract: Nowadays, we need to speed up solving computer problems such as sorting. Because of limitations in processor's speed, using parallel algorithms is inevitable. In parallel algorithms, because of cost limitations and architecture complexity, it's not suitable to increase the number ..."
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Abstract: Nowadays, we need to speed up solving computer problems such as sorting. Because of limitations in processor's speed, using parallel algorithms is inevitable. In parallel algorithms, because of cost limitations and architecture complexity, it's not suitable to increase the number of processors to speed up. This is also infeasible where we have plenty of data. In this paper we issue a new sorting algorithm. The algorithm is implemented on ILLIAC architecture. The aim of issued algorithm is to reduce total cost of sorting with a trade off between the number of processors and execution time.
A SuperLogarithmic Lower Bound for ShuffleUnshuffle Sorting Networks
, 1994
"... Shuffleunshuffle sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput shuffleunshuffle sorting networks with depth2O(plglgn)lgn have been discovered. These networks are the only known sorting ..."
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Shuffleunshuffle sorting networks are a class of comparator networks whose structure maps efficiently to the hypercube and any of its bounded degree variants. Recently, ninput shuffleunshuffle sorting networks with depth2O(plglgn)lgn have been discovered. These networks are the only known sorting networks of depth o(lg2n) that are not based on expanders, and their existence raises the question of whether a depth of O(lgn) can be achieved by any shuffleunshuffle sorting network. In this paper, we resolve this question by establishing lglglgnlower bound on the depth of anyninput shuffleunshuffle sorting network. Our lower bound can be extended to certain restricted classes of nonoblivious sorting algorithms on hypercubic machines.