Solution Explanation for Find All Good Strings

This problem requires finding the number of strings of length n that are lexicographically between s1 and s2 (inclusive) and do not contain the substring evil. A naive approach would be to generate all strings and check each one, but this is computationally infeasible for larger n. A more efficient solution uses dynamic programming.

Approach:

The core idea is to build a dynamic programming solution that avoids explicitly generating all possible strings. We'll use the following concepts:

1. KMP (Knuth-Morris-Pratt) Algorithm: We need an efficient way to check if a string contains evil as a substring. The KMP algorithm is perfect for this, providing O(m) time complexity (where m is the length of evil).

2. Dynamic Programming: We'll build a DP table to store the counts of good strings. The state will represent the current length of the string being built and the current suffix (to check for evil).

States and Transitions:

Let dp[i][j] represent the count of good strings of length i ending with the suffix j of evil. j can range from 0 (no matching suffix of evil) to len(evil) -1 (full match of evil found).

• Base Case: dp[0][0] = 1 (empty string is a valid starting point).
• Transitions: For each character c in 'a' to 'z', we can extend a string of length i-1 to a string of length i. We need to check:
  • Does adding c lead to the creation of substring evil? This is determined by the KMP failure function (next array).
  • Is the resulting string lexicographically within the [s1, s2] range?

Algorithm:

1. Preprocessing (KMP): Calculate the KMP next array for the evil string. This array helps efficiently find evil substrings.

2. DP Table Initialization: Initialize the dp table to 0. Set dp[0][0] = 1.

3. Iteration: Iterate through string lengths (i from 1 to n). For each length, iterate through possible suffix matches (j from 0 to len(evil) - 1). Iterate through all possible characters ('a' to 'z').

4. Character Addition: For each character c:

   • Calculate the new suffix match index next_j using the KMP next array.
   • Check if adding c keeps the string within the bounds [s1, s2].
   • If the new string is good, update dp[i][next_j] += dp[i - 1][j]

5. Final Result: The total count of good strings will be the sum of all dp[n][j] where j is 0 (i.e., no evil suffix).

Code (Python):

def kmp_preprocess(evil):
    next = [0] * len(evil)
    j = 0
    for i in range(1, len(evil)):
        while j > 0 and evil[i] != evil[j]:
            j = next[j - 1]
        if evil[i] == evil[j]:
            j += 1
        next[i] = j
    return next
 
def findGoodStrings(n, s1, s2, evil):
    mod = 10**9 + 7
    next = kmp_preprocess(evil)
    dp = [[0] * len(evil) for _ in range(n + 1)]
    dp[0][0] = 1
    for i in range(1, n + 1):
        for j in range(len(evil)):
            for c in range(ord('a'), ord('z') + 1):
                char = chr(c)
                next_j = j
                while next_j > 0 and char != evil[next_j]:
                    next_j = next[next_j - 1]
                if char == evil[next_j]:
                    next_j += 1
                
                cur_str = s1[:i-1] + char
                if i == n:
                  cur_str = s1[:i-1] + char
                  if (cur_str >= s1 and cur_str <= s2 and next_j == 0 ):
                    dp[i][next_j] = (dp[i][next_j] + dp[i - 1][j]) % mod
                elif next_j == 0 and cur_str >= s1 and cur_str <= s2:
                    dp[i][next_j] = (dp[i][next_j] + dp[i - 1][j]) % mod
 
 
    ans = 0
    for j in range(len(evil)):
        ans = (ans + dp[n][j]) % mod
    return ans
 

Time Complexity: O(n * 26 * len(evil)) which simplifies to O(n * len(evil)). The dominating factor is the DP iteration.

Space Complexity: O(n * len(evil)) for the DP table. The KMP next array is O(len(evil)).

This detailed explanation, along with the Python code, provides a complete solution for the "Find All Good Strings" problem. Adaptations to other languages (Java, C++, Go) would follow a similar structure, primarily altering syntax and potentially utilizing built-in KMP implementations if available.