Some Results on Superpatterns for Preferential Arrangements

A {\it superpattern} is a string of characters of length $n$ that contains as a subsequence, and in a sense that depends on the context, all the smaller strings of length $k$ in a certain class. We prove structural and probabilistic results on superpatterns for {\em preferential arrangements}, including (i) a theorem that demonstrates that a string is a superpattern for all preferential arrangements if and only if it is a superpattern for all permutations; and (ii) a result that is reminiscent of a still unresolved conjecture of Alon on the smallest permutation on $[n]$ that contains all $k$-permutations with high probability.Comment: 13 page

Image Characterization and Classification by Physical Complexity

We present a method for estimating the complexity of an image based on Bennett's concept of logical depth. Bennett identified logical depth as the appropriate measure of organized complexity, and hence as being better suited to the evaluation of the complexity of objects in the physical world. Its use results in a different, and in some sense a finer characterization than is obtained through the application of the concept of Kolmogorov complexity alone. We use this measure to classify images by their information content. The method provides a means for classifying and evaluating the complexity of objects by way of their visual representations. To the authors' knowledge, the method and application inspired by the concept of logical depth presented herein are being proposed and implemented for the first time.Comment: 30 pages, 21 figure

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

Shortest prefix strings containing all subset permutations

AbstractWhat is the length of the shortest string consisting of elements of {1,…n} that contains as subsequences all permutations of any k-element subset? Many authors have considered the special case where k=n. We instead consider an incremental variation on this problem first proposed by Koutas and Hu. For a fixed value of n they ask for a string such that for all values of k⩽n, the prefix containing all permutations of any k-element subset as subsequences is as short as possible. The problem can also be viewed as follows:For k=1 one needs n distinct digits to find each of the n possible permutations. In going from k to k+1, one starts with a string containing all k-element permutations as subsequences, and one adds as few digits as possible to the end of the string so that the new string contains all (k+1)-element permutations.We give a new construction that gives shorter strings than the best previous construction. We then prove a weak form of lower bound for the number of digits added in successive suffixes. The lower bound proof leads to a construction that matches the bound exactly. The length of a shortest prefix string is k(n−2)+[13(k+1)]+3, for k > 2.The lengths for k=1, 2 are n and 2n−1. This proves the natural conjecture that requiring the strings to be prefixes strictly increases the length of the strings required for all but the smallest values of k