A prefix normal word is a binary word whose prefixes contain at least as many 1s as any of its factors of the same length. Introduced by Fici and Lipták in 2011 the notion of prefix normality is so far only defined for words over the binary alphabet. In this work we investigate a generalisation for finite words over arbitrary finite alphabets, namely weighted prefix normality. We prove that weighted prefix normality is more expressive than binary prefix normality. Furthermore, we investigate the existence of a weighted prefix normal form since weighted prefix normality comes with several new peculiarities that did not already occur in the binary case. We characterise these issues and finally present a standard technique to obtain a generalised prefix normal form for all words overarbitrary, finite alphabets.
Prefix normal words are binary words that have no factor with more $1$s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Lipták, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.
Prefix normal words are binary words in which each prefix has at least the same number of $\so$s as any factor of the same length. Firstly introduced by Fici and Lipták in 2011, the problem of determining the index of the prefix equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix no…
We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix . We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square pref…
We present a new class of binary words: the prefix normal words. They are defined by the property that for any given length $k$, no factor of length $k$ has more $a$'s than the prefix of the same length. These words arise in the context of indexing for jumbled pattern matching (a.k.a. permutation matching or Parikh vector matching), where the aim is to decide whether a string has a factor with a given multiplicity of characters, i.e., with a given Parikh vector. Using prefix normal words, we give the first non-trivial characterization of binary words having the same set of Parikh vectors of fa…
The prefix palindromic length $p_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. This function is surprisingly difficult to study; in particular, the conjecture that $p_{\mathbf{u}}(n)$ can be bounded only if $\mathbf{u}$ is ultimately periodic is open since 2013. A more recent conjecture concerns the prefix palindromic length of the period doubling word: it seems that it is not $2$-regular, and if it is true, this would give a rare if not unique example of a non-regular function of …
The prefix palindromic length $\mathrm{PPL}_{\mathbf{u}}(n)$ of an infinite word $\mathbf{u}$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $\mathbf{u}$. Since 2013, it is still unknown if $\mathrm{PPL}_{\mathbf{u}}(n)$ is unbounded for every aperiodic infinite word $\mathbf{u}$, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function $\mathrm{PPL}_{\mathbf{u}}(n)$ has been precisely computed is the Thue-Morse word $\mathbf{t}$. This word is $2$-autom…
We study the problems of finding a shortest synchronizing word and its length for a given prefix code. This is done in two different settings: when the code is defined by an arbitrary decoder recognizing its star and when the code is defined by its literal decoder (whose size is polynomially equivalent to the total length of all words in the code). For the first case for every $\varepsilon > 0$ we prove $n^{1 - \varepsilon}$-inapproximability for recognizable binary maximal prefix codes, $Θ(\log n)$-inapproximability for finite binary maximal prefix codes and $n^{\frac{1}{2} - \varepsilon}$-in…
A Note on the Longest Common Compatible Prefix Problem for Partial Words
对于部分字$w$，两个位置的最长公共兼容前缀$i,j$（表示为$lccp(i,j)$）是最大的$k$，使得$w[i,i+k-1]\uparrow w[j,j+k-1]$，其中$\uparrow$是部分字的兼容关系（它不是等价关系）。LCCP问题是以这样一种方式预处理部分单词，即任何关于该单词的查询$lccp(i,j)$都可以在$O(1)$时间内得到回答。这是一个自然推广的最长公共前缀（LCP）的问题，其中$O(n)$预处理时间和$O(1)$查询时间的解决方案存在的规则的话。最近F. Blanchet-Sadri and J. Lazarow（LATA 2013）.预处理时间为$O(nh+n)$，其中$h$是$w$中的“孔”数。该算法是为固定字母表上的部分单词而设计的，并且相当复杂。我们提出了一个简单的解决方案，这个问题有稍微好一点的运行时间，适用于任何线性排序字母表。我们的预处理是在时间$O(nμ+n)$，其中$μ$是$w$中的孔块的数量。我们的算法使用对齐算法和动态规划的思想。
A language over an alphabet $B = A \cup \overline{A}$ of opening ($A$) and closing ($\overline{A}$) brackets, is balanced if it is a subset of the Dyck language $D_B$ over $B$, and it is well-formed if all words are prefixes of words in $D_B$. We show that well-formedness of a context-free language is decidable in polynomial time, and that the longest common reduced suffix can be computed in polynomial time. With this at a hand we decide for the class 2-TWs of non-linear tree transducers with output alphabet $B^*$ whether or not the output language is balanced.
