Related papers: Maximal Unbordered Factors of Random Strings
An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up…
In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result…
A maximum priority matching is a matching in an undirected graph that maximizes a priority score defined with respect to given vertex priorities. An earlier paper showed how to find maximum priority matchings in unweighted graphs. This…
A \itbf{cover} of a string $x = x[1..n]$ is a proper substring $u$ of $x$ such that $x$ can be constructed from possibly overlapping instances of $u$. A recent paper \cite{FIKPPST13} relaxes this definition --- an \itbf{enhanced cover} $u$…
A set X of partial words over a finite alphabet A is called unavoidable if every two-sided infinite word over A has a factor compatible with an element of X. Unlike the case of a set of words without holes, the problem of deciding whether…
Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study…
We investigate the variance of the length of the longest common subsequences of two independent random words of size $n$, where the letters of one word are i.i.d. uniformly drawn from $\{\alpha_1, \alpha_2, \cdots, \alpha_m\}$, while the…
An order-preserving square in a string is a fragment of the form $uv$ where $u\neq v$ and $u$ is order-isomorphic to $v$. We show that a string $w$ of length $n$ over an alphabet of size $\sigma$ contains $\mathcal{O}(\sigma n)$…
For $\alpha\geq 1$, an $\alpha$-gapped repeat in a word $w$ is a factor $uvu$ of $w$ such that $|uv|\leq \alpha |u|$; the two factors $u$ in such a repeat are called arms, while the factor $v$ is called gap. Such a repeat is called maximal…
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…
In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are given two strings $X$ and $Y$ of total length $n$, and we are asked to find a pair of maximal-length factors, one of $X$ and the other of $Y$, such that their…
A factorization $f_1, \ldots, f_m$ of a string $w$ of length $n$ is called a repetition factorization of $w$ if $f_i$ is a repetition, i.e., $f_i$ is a form of $x^kx'$, where $x$ is a non-empty string, $x'$ is a (possibly-empty) proper…
We describe an algorithm computing an optimal prefix free code for $n$ unsorted positive weights in time within $O(n(1+\lg \alpha))\subseteq O(n\lg n)$, where the alternation $\alpha\in[1..n-1]$ measures the amount of sorting required by…
An absent factor of a string $w$ is a string $u$ which does not occur as a contiguous substring (a.k.a. factor) inside $w$. We extend this well-studied notion and define absent subsequences: a string $u$ is an absent subsequence of a string…
Let $H_n$ be a graph on $n$ vertices and let $\ber{H_n}$ denote the complement of $H_n$. Suppose that $\Delta = \Delta(n)$ is the maximum degree of $\ber{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of…
Lempel-Ziv (LZ77) factorization is a fundamental problem in string processing: Greedily partition a given string $T$ from left to right into blocks (called phrases) so that each phrase is either the leftmost occurrence of a letter or the…
A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected…
We show a new lower bound for the maximum number of runs in a string. We prove that for any e > 0, (a -- e)n is an asymptotic lower bound, where a = 56733/60064 = 0.944542. It is superior to the previous bound 0.927 given by Franek et al.…
Recent work has pinned down the existentially optimal size bounds for vertex fault-tolerant spanners: for any positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges resilient to $f$ vertex…
In this note we put forward a conjecture on the average optimal length for bipartite matching with a finite number of elements where the different lengths are independent one from the others and have an exponential distribution.