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It is well-known that signed permutations can be sorted in polynomial time, while sorting unsigned ones is an NP -hard problem. Although sorting signed permutations by reversals is an “easy” computational problem, the study of the average reversal distance is justified because it gives rise to interesting combinatorial questions and also because in applications related with genome similarity analysis, exact solutions for this problem have been used to design approximate algorithms for its unsigned NP -hard version. Thus, the average number of reversals needed to sort signed permutations represents a good control mechanism for the quality of approximate solutions for the unsigned case. This paper analyzes the average number of reversals needed to sort signed permutations by exploring combinatorial properties of structures related with signed permutations, such as breakpoint graphs, and provides a recurrence formula for this average number. The recurrence formula is built based on the analysis of the probability that two nodes of the breakpoint graph belong to the same alternating cycle among all the breakpoint graphs related with permutations of length n . Through this analysis it is possible to compute the average reversal distance for signed variations of the identity permutation. Also, lower and upper bounds of the average reversal distance for signed permutations are provided. Additionally, based on computational data, it is shown how these bounds can be used in order to propose concrete upper and lower bounds.

We present a new sorting network on 24 channels, which uses only 12 layers, improving the previously best known bound by one layer. By monotonicity, this also implies improved sorting networks for 23 channels. This result was obtained by combining techniques for generating prefixes of sorting networks with propositional encodings.

Quicksort is one of the most popular sorting algorithms, it is based on a divide-and-conquer technique and has a wide acceptance as the fastest general-purpose sorting technique. Though it is successful in separating large partitions into small ones, quicksort runs slowly when it processes its small partitions, for which completing the sorting through using a different sorting algorithm is much plausible solution. This variant minimizes the overall execution time but it switches to a constant sorting algorithm at a constant cut-off point. To cope with this constancy problem, it has been suggested that a dynamic model which can choose the fastest sorting algorithm for the small partitions. The model includes continuation with quicksort so that the cut-off point is also more flexible. To implement this with an intelligent algorithm selection model, artificial neural networks are preferred due to their non-comparison, constant-time and low-cost architecture features. In spite of the fact that finding the best sorting algorithm by using a neural network causes some extra computational time, the gain in overall execution time is greater. As a result, a faster variant of quicksort has been implemented by using artificial neural network based algorithm selection approach. Experimental results of the proposed algorithm and the Several other fast sorting algorithms have been presented, compared and discussed. (C) 2012 Elsevier Inc. All rights reserved.

A cut-and-paste operation can be a reversal, a transposition, or a transreversal on circular or linear permutations. There are several approximation algorithms for sorting signed permutations by combinations of these operations. For sorting unsigned permutations, we only know an algorithm with performance ratio 3 and its improved version with performance ratio [email protected] allowing reversals and transpositions. In this paper, we present a 2.25-approximation algorithm for sorting unsigned circular permutations by cut-and-paste operations. A structure called tie is proposed to represent the alternating path of length 5. We can classify the ties into 6 types and find ways to remove the breakpoints for each type of ties, so that every cut-and-paste operation can reduce at least 43 breakpoints averagely. Our algorithm can be used to sort unsigned linear permutations and achieve the performance ratio 2.25 if another operation named revrev is allowed.

A product network defines a class of topologies that are very often used such as meshes, tori, and hypercubes, etc. This paper proposes a generalized algorithm for fault-tolerant parallel sorting in product networks. To tolerate r - 1 faulty nodes, an r-dimensional product network containing faulty nodes is partitioned into a number of subgraphs such that each subgraph contains at most one fault. Our generalized sorting algorithm is divided into two steps. First, a single-fault sorting operation is presented to correctly performed on each faulty subgraph containing one fault. Second, each subgraph is considered a supernode, and a fault-tolerant multiway merging operation is presented to recursively merge two sorted subsequences into one sorted sequence. Our generalized sorting algorithm can be applied to any product network only if the factor graph of the product graph can be embedding in a ring. Further, we also show the time complexity of our sorting operations on a grid, hypercube, and Petersen cube. Performance analysis illustrates that our generalized sorting scheme is a truly efficient fault-tolerant algorithm.

We present a parallel sorting algorithm and its proof which sorts a sequence of n elements in time O(log2n) with n/2 processors on an EREW-PRAM computational model. A sorting network directly implements the algorithm using O(n·log n)PEs. The algorithm is based on the elementary Compare-Exchange operation and has the advantage that it does not require a powerful computational model, uses the least amount of space for the sorting problem, has small constants and can be implemented directly on a sorting network. Furthermore, the architecture of the network is simple and makes no unrealistic technological assumptions.

A new parallel sorting algorithm suitable for MIMD multiprocessor is presented. The algorithm reduces memory and bus contention, which many parallel sorting algorithms suffer from, by using a regular sampling of the data to ensure good pivot selection. For n data elements to be sorted and p processors, when n ≥ p 3 the algorithm is shown to be asymptotically optimal. In theory, the algorithm is within a factor of 2 of achieving ideal load balancing. In practice, there is almost a perfect partitioning of work. On a variety of shared and distributed memory machines, the algorithm achieves better than half-linear speedups.

In many situations, a statistical method must be applied repeatedly, e.g., during a series of iterations. Procedures based on ranks and order statistics are often attractive but, if used in a repetitive context, may require sorting each of a sequence of data sets. In many cases, the sort performed on a given member of such sequences can be used to induce partial sorting in the next member. This can in turn lead to sharply lower computing costs: While sorting random input requires O(n log n) time, some types of nonrandom input can be sorted in just O(n) time. Two sorting algorithms designed for partly sorted input are presented, and their performance compared with that of some widely used sorting methods. The advantages of capitalizing on partly sorted input are demonstrated in a simulation study of an iterative algorithm for computing Jaeckel's robust estimator in simple regression.