The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed Acyclic Word Graphs. The first algorithm decides in linear time whether a set of positions on the word is an attractor of the word. The second algorithm generates an attractor for a given word in a greedy manner. Although the problem of generating a smallest attractor is NP-hard, {the algorithm is efficient and produces very small attractors and even the smallest ones for several well-known infinite families of words.

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