Related papers: On Maximal Unbordered Factors
Let's denote a complete $m$-ary rooted tree graph of height $n$ as $G$. In scope of this paper we prove the certain relations between the properties of $G$ and the expectation and variance of the distribution of lengths of strings,…
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the 3SUM conjecture) of several problems related to finding Abelian square and additive square factors in a string. In particular, we conclude conditional optimality…
Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum…
We obtain results on the limiting distribution of the six-length of a random functional graph, also called a functional digraph or random mapping, with given in-degree sequence. The six-length of a vertex $v\in V$ is defined from the…
Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a…
We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an…
A subset of the Hamming cube over $n$-letter alphabet is said to be $d$-maximal if its diameter is $d$, and adding any point increases the diameter. Our main result shows that each $d$-maximal set is either of size at most $(n+o(n))^d$ or…
In the general AntiFactor problem, a graph $G$ is given with a set $X_v\subseteq \mathbb{N}$ of forbidden degrees for every vertex $v$ and the task is to find a set $S$ of edges such that the degree of $v$ in $S$ is not in the set $X_v$.…
An \itbf{indeterminate string} (or, more simply, just a \itbf{string}) $\s{x} = \s{x}[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$. We say that $\s{x}[i_1]$ and $\s{x}[i_2]$ \itbf{match} (written $\s{x}[i_1]…
We prove that the maximum determinant of an $n \times n $ matrix, with entries in $\{0,1\}$ and at most $n+k$ non-zero entries, is at most $2^{k/3}$, which is best possible when $k$ is a multiple of 3. This result solves a conjecture of…
We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest…
This paper presents lossless prefix codes optimized with respect to a pay-off criterion consisting of a convex combination of maximum codeword length and average codeword length. The optimal codeword lengths obtained are based on a new…
Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an…
In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if $\beta (G)$ is the metric dimension of a maximal outerplanar graph $G$ of order $n$, we prove that $2\le \beta (G) \le \lceil…
In this paper, we present a computational search for best-known merit factors of longer binary sequences with an odd length. Finding low autocorrelation binary sequences with optimal or suboptimal merit factors is a very difficult…
We consider the problem of finding repetitive structures and inherent patterns in a given string $\s{s}$ of length $n$ over a finite totally ordered alphabet. A border $\s{u}$ of a string $\s{s}$ is both a prefix and a suffix of $\s{s}$…
Given a string $S$ of length $n$, the classic string indexing problem is to preprocess $S$ into a compact data structure that supports efficient subsequent pattern queries. In the \emph{deterministic} variant the goal is to solve the string…
The critical exponent $E(\mathbf u)$ of an infinite sequence $\mathbf u$ over a finite alphabet expresses the maximal repetition of a factor in $\mathbf u$. By the famous Dejean's theorem, $E(\mathbf u) \geq 1+\frac1{d-1}$ for every $d$-ary…
We estimate the maximum ratio between the $\sigma_t$- and $\sigma$-irregularity for graphs and trees of order $n$, which are respectively bounded by $\Theta(n^{5/2})$ and $n-2$. This answers a question and a conjecture by Filipovski et al.…
We show that the size of maximum cut in a planar graph with $m$ edges is at least $2m/3$. We also show that maximal planar graphs saturate this bound.