Related papers: The Number of Distinct Subsequences of a Random Bi…
When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct…
This note provides very simple, efficient algorithms for computing the number of distinct longest common subsequences of two input strings and for computing the number of LCS embeddings.
The discrepancy of a binary string is the maximum (absolute) difference between the number of ones and the number of zeroes over all possible substrings of the given binary string. In this note we determine the minimal discrepancy that a…
We give three different computations of the total number of runs of length $i$ in binary $n$-strings, and we discuss the connection of this problem with the compositions of $n$.
We define the ``shift-match number'' for a binary string and we compute the probability of occurrence of a given string as a subsequence in longer strings in terms of its shift-match number. We thus prove that the string matching…
We consider the complexities of substitutive sequences over a binary alphabet. By studying various types of special words, we show that, knowing some initial values, its complexity can be completely formulated via a recurrence formula…
In the deletion channel, an important problem is to determine the number of subsequences derived from a string $U$ of length $n$ when subjected to $t$ deletions. It is well-known that the number of subsequences in the setting exhibits a…
We consider the number of occurrences of subwords (non-consecutive sub-sequences) in a given word. We first define the notion of subword entropy of a given word that measures the maximal number of occurrences among all possible subwords. We…
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it…
Random residue sequences (RR) may be used in many random number applications including those related to multiple access in communications. This paper investigates variations on an algorithm to generate RR sequences that was proposed earlier…
Motivated by the study of deletion channels, this work presents improved bounds on the number of subsequences obtained from a binary sting X of length n under t deletions. It is known that the number of subsequences in this setting strongly…
We relate binary words with a given number of subsequences to continued fractions of rational numbers with a given denominator. We deduce that there are binary strings of length $O(\log n \log \log n)$ with exactly $n$ subsequences; this…
We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected…
There are many randomness notions. On the classical account, many of them are about whether a given infinite binary sequence is random for some given probability. If so, this probability turns out to be the same for all these notions, so…
Consider two independent random strings having same length and taking values uniformly in a common finite alphabet. We study the order of the variance of the length of the longest common subsequences (LCS) of these strings when long blocks,…
There are two methods for counting the number of occurrences of a string in another large string. One is to count the number of places where the string is found. The other is to determine how many pieces of string can be extracted without…
Finding the underlying probability distributions of a set of observed sequences under the constraint that each sequence is generated i.i.d by a distinct distribution is considered. The number of distributions, and hence the number of…
The distance of a binary operation from being associative can be "measured" by its associative spectrum, an appropriate sequence of positive integers. Particular instances and general properties of associative spectra are studied.
In this paper, the construction of finite-length binary sequences whose nonlinear complexity is not less than half of the length is investigated. By characterizing the structure of the sequences, an algorithm is proposed to generate all…
The distributions of the $m$-th longest runs of multivariate random sequences are considered. For random sequences made up of $k$ kinds of letters, the lengths of the runs are sorted in two ways to give two definitions of run length…