Word equations of paths

Learning Shortest Paths for Word Graphs

The vast amount of information on the Web drives the need for aggregation and summarisation techniques. We study event extraction as a text summarisation task using redundant sentences which is also known as sentence compression. Given a set of sentences describing the same event, we aim at generating a summarisation that is (i) a single sentence, (ii) simply structured and easily understandabl...

Improved Shortest Paths on the Word RAM

Thorup recently showed that single-source shortest-paths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0, . . . , 2 − 1} can be solved in O(n+m) time and space on a unit-cost random-access machine with a word length of w bits. His algorithm works by traversing a so-called component tree. Two new related results are provided here. First, and most importan...

Two-Variable Word Equations

We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Speci cally, we prove that there are only ve types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizat...

Quadrati Word Equations

Towards parametrizing word equations

Classically, in order to resolve an equation u ≈ v over a free monoid X∗, we reduce it by a suitable family F of substitutions to a family of equations uf ≈ vf , f ∈ F , each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get F in a handy parametrized form. The method we propose consists in parametrizing the path traces in t...