Shellsort with three increments

A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments $(h,g,1)$. In particular, when $h=\Theta(n^{7/15})$ and $g=\Theta(h^{1/5})$, the average running time is $O(n^{23/15})$. The proof involves interesting properties of the inversions in random permutations that have been $h$-sorted and $g$-sorted

Average-Case Complexity of Shellsort

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is \Omega (pn^{1 + 1/p) for all $p \leq \log n$. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.Comment: 11 pages. Submitted to ICALP'9

Zig-zag Sort: A Simple Deterministic Data-Oblivious Sorting Algorithm Running in O(n log n) Time

We describe and analyze Zig-zag Sort--a deterministic data-oblivious sorting algorithm running in O(n log n) time that is arguably simpler than previously known algorithms with similar properties, which are based on the AKS sorting network. Because it is data-oblivious and deterministic, Zig-zag Sort can be implemented as a simple O(n log n)-size sorting network, thereby providing a solution to an open problem posed by Incerpi and Sedgewick in 1985. In addition, Zig-zag Sort is a variant of Shellsort, and is, in fact, the first deterministic Shellsort variant running in O(n log n) time. The existence of such an algorithm was posed as an open problem by Plaxton et al. in 1992 and also by Sedgewick in 1996. More relevant for today, however, is the fact that the existence of a simple data-oblivious deterministic sorting algorithm running in O(n log n) time simplifies the inner-loop computation in several proposed oblivious-RAM simulation methods (which utilize AKS sorting networks), and this, in turn, implies simplified mechanisms for privacy-preserving data outsourcing in several cloud computing applications. We provide both constructive and non-constructive implementations of Zig-zag Sort, based on the existence of a circuit known as an epsilon-halver, such that the constant factors in our constructive implementations are orders of magnitude smaller than those for constructive variants of the AKS sorting network, which are also based on the use of epsilon-halvers.Comment: Appearing in ACM Symp. on Theory of Computing (STOC) 201

Ada style guide (version 1.1)

Ada is a programming language of considerable expressive power. The Ada Language Reference Manual provides a thorough definition of the language. However, it does not offer sufficient guidance on the appropriate use of Ada's powerful features. For this reason, the Goddard Space Flight Center Ada User's Group has produced this style guide which addresses such program style issues. The guide covers three areas of Ada program style: the structural decomposition of a program; the coding and the use of specific Ada features; and the textural formatting of a program