Figure 1

A ( L , d , 2)-repeat and a parallelogram. An example a (L, d, 2)-repeat with L = 11, d = 2. Diagonals 30, 31, and 32 are shown. Among them, 30 and 32 have distance 2, while 30 and 31 (as well as 31 and 32) are consecutive. Assuming that q = 2, a q-hit is represented by a thicker diagonal of length 2 plus a small black circle representing its pair of coordinates. The q-hit (19, 49) refers to the q-gram TA, and 19 (resp. 49) is its first (resp. second) projection. The q-hit (17, 49) refers to the same q-gram TA but has a different first projection. The words inside the grey boxes are two distinct fragments of the same sequence s, namely s[10, 20] and s [42, 52]; they have length 11, and their edit distance is 2. We obtain one word from the other by deleting s[13] – hence no q-hits in positions 12, 13 – and by inserting s [48] – no q-hits in positions 47, 48. We have p = 6 and the set $\mathcal{S}$ of 7 q-hits in diagonals 31 and 32 satisfies the properties (1), (2), (3), (4) and (5). If we add the q-hit in diagonal 30 in order to obtain a new set ${\mathcal{S}}^{\prime }$, properties (1), (2) still hold, but properties (3), (4) and (5) are no longer satisfied.