Longest common extension

Given a word w of length n and i, j ∈ [n], the longest common extension is the longest substring starting at both i and j. In this note we estimate the average length of the longest common extension over all words w and all pairs (i, j), as well as the typical maximum length of the longest common extension. We also consider a variant of this problem, due to Blanchet-Sadri and Lazarow, in which the word is allowed to contain ‘holes’, which are special symbols functioning as ‘jokers’, i.e. are considered to be equal to any character. In particular, we estimate the average longest common extension over all words w with a small number of holes, extending a result by Blanchet-Sadri, Harred and Lazarow, and prove a similar result for words with holes appearing randomly