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We use the idea to design an algorithm to the DM G problem. A naive method is to consider matching the preﬁxes Pi,1 for all 1 ≤ i ≤ d, and then look for the suﬃxes subpatterns Pi,2 , 1 ≤ i ≤ d, after the appropriate gap and intersect the occurrences to report the dictionary patterns matchings. However, as some of the patterns may share subpatterns and some subpatterns may include other subpatterns, there may be several distinct subpatterns occurring at the same text location, each of diﬀerent length.

Wkαk is called a Lyndon decomposition if and only if w1 > w2 > . . > wk are Lyndon words [3]. The last factor wk is the minimal suﬃx of x [5] and from the deﬁnition we easily obtain that wkαk is the largest αk−1 is the Lyndon decompower of wk which is a suﬃx of x. Also, w1α1 w2α2 . . wk−1 position of the remaining preﬁx of x, which gives us the following corollary. Corollary 6. For any 1 ≤ τ ≤ log n a string T of length n can be stored in an O(n)-space data structure that enables to compute the Lyndon decomposition of any substring of T in O(kτ ) time, where k is the number of distinct factors in the decomposition.

146–155. Springer, Heidelberg (2006) 25. : On-line construction of suﬃx trees. Algorithmica 14(3), 249–260 (1995) 26. : Preserving order in a forest in less than logarithmic time. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 75–84 (1975) 27. Verint. Packet intrusion detection. Personal communication (2013) 28. : Linear pattern matching algorithm. In: Proc. 14 IEEE Symposium on Switching and Automata Theory, pp. 1–11 (1973) 29. : A faster algorithm for matching a set of patterns with variable length don’t cares.