5,558 research outputs found
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
Some Results on Superpatterns for Preferential Arrangements
A {\it superpattern} is a string of characters of length that contains as
a subsequence, and in a sense that depends on the context, all the smaller
strings of length 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 that contains all -permutations with high
probability.Comment: 13 page
Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns
We explore a new type of replacement of patterns in permutations, suggested
by James Propp, that does not preserve the length of permutations. In
particular, we focus on replacements between 123 and a pattern of two integer
elements. We apply these replacements in the classical sense; that is, the
elements being replaced need not be adjacent in position or value. Given each
replacement, the set of all permutations is partitioned into equivalence
classes consisting of permutations reachable from one another through a series
of bi-directional replacements. We break the eighteen replacements of interest
into four categories by the structure of their classes and fully characterize
all of their classes.Comment: 14 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
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