Let S be a string of length n . Suppose we are in iteration k of Z-algorithm when run on S and assume k > r_{k-1}. As we are in case 1 we must compute Z[k] using explicit comparisons.

Case 2a of the Z-algorithm handles the scenario where l\le k\le r and Z[k-l+1] < r-k+1. This case requires explicit character comparisons to determine Z[k] value 

Let S be a string of length n, and, using 1-based indexing, let r_k denote the value of r at iteration k of the Z-algorithm when run on S. Then the sequence (r_1, r_2, \dots, r_n) is always non-decreasing, with r_{i-1} \le r_i \quad \forall\, 1 < i \le n, for any S.

Let S be a string of length n . In 1-based indexing, if Z[k] > 0 , for 2\le k \le n then S[k\dots k+Z[k] - 1] = S[1\dots Z[k]].

Let S be a string of length n . Then, using 1-based indexing, do we care about the value stored in Z[1] ?

The naive algorithm does the fewest number of comparisons when all characters in the pattern and text are the same.

Let  T be a text of length n and P a pattern of length m . The worst case time complexity of the naive algorithm is O(n+m) .