On Prefix Normal Words

Abstract

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 factors. We prove that the language of prefix normal words is not context-free and is strictly contained in the language of pre-necklaces, which are prefixes of powers of Lyndon words. We discuss further properties and state open problems.

On Collapsing Prefix Normal Words

“…Prefix normal words, introduced in [11] are a refinement of the abelian equivalence classes for binary alphabets {0, 1}. A word w is called prefix normal if the prefix of w of any length has at least the amount of 1s as any of w's factors of the same length.…”

“…Burcsi et al have shown in [4] that this relation is indeed an equivalence relation and moreover that each class contains exactly one uniquely determined prefix normal word -the prefix normal form. From a combinatorial point of view, prefix normal words are also of interest since they are connected to Lyndon words, in the sense that every prefix normal word is a pre-necklace [11].…”

“…Additionally, as shown in [11], the indexed jumbled pattern matching problem (see e.g. [1,2,14]) is connected to prefix normal forms: if the prefix normal forms are given, the indexed jumbled pattern matching problem can be solved in linear time O(n) of the word length n. The best known algorithm for this problem has a run-time of O(n 1.864 )(see [6]).…”

“…An algorithm for the computation of all prefix normal words of length n in amortized run-time O(n) per word is given in [3]. There also the lower bound of 2 ⌊ n 2 ⌋ is given and in [11] the authors showed 2 n−4 √ n log(n) ≤ ℓ ≤ 2 n−lg(n)−1 for the number of prefix normal words ℓ of the given length n. A closed formula for the number of prefix normal words is still unknown. In [22] the number of prefix normal words of length n (A194850), a list of binary prefix normal words (A238109), and the maximum size of a class of binary words of length n having the same prefix normal form (A238110), can be found.…”

“…In this work we investigate two conspicuities mentioned in [11,3]: palindromes and extension-critical words. As shown in [3] prefix normal palindromes play a special role since they are not prefix equivalent to any different word.…”