Self-Alignments in Words and Their Applications

Abstract Some quantities associated with periodicities in words are analyzed within the Bernoulli probabilistic model. In particular, the following problem is addressed. Assume that a string X is given, with symbols emitted randomly but independently according to some known distribution of probabilities. Then, for each pair ( W , Z ) of distinct suffixes of X , the expected length of the longest common prefix of W and Z is sought. The collection of these lengths, that are called here self-alignments , plays a crucial role in several algorithmic problems on words, such as building suffix trees or inverted files, detecting squares and other regularities, computing substring statistics, etc. The asymptotically best algorithms for these problems are quite complex and thus risk being unpractical. The present analysis of self-alignments and related measures suggests that, in a variety of cases, more straightforward algorithmic solutions may yield comparable or even better performances.

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