Maximal pair

Given a string S of length n, a maximal pair is a tuple (p₁,p₂,l), such that S[p₁..p₁ + l − 1] = S[p₂..p₂ + l − 1], but $S[p_1-1] \neq S[p_2-1]$ and $S[p_1+l] \neq S[p_2+l]$. A maximal repeat is a string represented by such tuple. A supermaximal repeat is a maximal repeat never occurring as a proper substring of another maximal repeat. Both maximal pairs, maximal repeats and supermaximal repeats can be found in Θ(n + z) time using a suffix tree ^[1], if there are z such structures.

Example

12345678901234
xabcyabcwabcyz

(2,6,3) and (6,10,3) are maximal pairs, but (2,10,3) is not, as y follows both substrings. abc and abcy are maximal repeats, but only abcy is a supermaximal repeat.

References

1. ^ Gusfield, Dan (1999) [1997]. Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology. USA: Cambridge University Press. p. 143. ISBN 0-521-58519-8.