Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewicz-sort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.

Abstract: One of the fundamental issues in computer science is ordering a list of items. Although there is a huge number of sorting algorithms, sorting problem has attracted a great deal of research; because efficient sorting is important to optimize the use of other algorithms. This paper presents two new sorting algorithms, enhanced selection sort and enhanced bubble Sort algorithms. Enhanced selection sort is an enhancement on selection sort by making it slightly faster and stable sorting algorithm. Enhanced bubble sort is an enhancement on both bubble sort and selection sort algorithms with O(nlgn) complexity instead of O(n 2) for bubble sort and selection sort algorithms. The two new algorithms are analyzed, implemented, tested, and compared and the results were promising.

Implementations of map-reduce are being used to perform many operations on very large data. We explore alternative ways that a system could use the environment and capabilities of map-reduce implementations such as Hadoop, yet perform operations that are not identical to map-reduce. The centerpiece of this exploration is a computational model that captures the essentials of the environment in which systems like Hadoop operate. Files are unordered sets of tuples that can be read and/or written in parallel; processes are limited in the amount of input/output they can perform, and processors are available in essentially unlimited supply. We develop, in this model, an algorithm for sorting that has a worst-case running time better than the obvious implementations of parallel sorting. 1

We study sorting algorithms based on randomized roundrobin comparisons. Specifically, we study Spin-the-bottle 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, data-oblivious sorting algorithms, which are useful in privacy-preserving computations, but, as we show, Annealing sort is much more efficient. We show that there is an input permutation that causes Spin-the-bottle sort to require Ω(n 2 log n) expected time in order to succeed, and that in O(n 2 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. 1