Find All Good Strings - Solution & Explanation

This approach involves the use of dynamic programming combined with an automaton to identify occurrences of the 'evil' substring efficiently. The fundamental idea is to construct an automaton that helps in checking the presence of the evil substring while generating potential 'good' strings between s1 and s2.

The strategy is to use a 3D DP array: dp[pos][evil_length][bounded], where:

• pos denotes the current position in the string being considered.
• evil_length is the length of the substring in the automaton matching the of the evil string.
• bounded is a flag indicating whether the current string prefix is tightly bounded by s1 and s2.

The automaton is utilized to efficiently transition between states based on the current character appended, reducing the time complexity associated with substring checking.

The above Python solution uses a KMP-based automaton to efficiently process and transition states to check for occurrences of the evil substring within potential good strings. The state transitions are precomputed in the KMPAutomaton class, allowing for quick look-up and decision making during the DP recursion. The DP function dfs manages the recursive exploration of possible good strings, accounting for boundaries implied by s1 and s2, and ensuring that the evil substring does not appear.

We can use dynamic programming to solve this problem by considering three conditions for each position in the string: whether it should be bounded by s1, s2, and whether it currently contains the 'evil' substring. We can define a dp[pos][is_bound_s1][is_bound_s2][evil_len] table that keeps track of the number of good strings of length n that start from pos, have or not reached the boundaries s1 and s2 constraints, and the number of characters currently matching with the leading characters of the 'evil' string.

This C code utilizes dynamic programming and recursion to calculate the number of 'good' strings. We create a 4D dp array to store probabilities of acquiring 'good' strings up to a particular index with respect to s1 and s2 boundaries and the match length with 'evil'. The logic iteratively checks each character between given bounds and determines the next state recursively. The final result is computed efficiently using memoization.

We can optimize the solution by employing the KMP algorithm to check for the 'evil' substring. While iterating through potential strings, instead of directly looking for substrings, use the KMP partial match table to efficiently find matching 'evil' patterns. This approach also integrates with dynamic programming to manage state transitions.

This version of the C solution enhances the search operation by incorporating a KMP-based algorithm to ascertain the presence of 'evil'. It builds an LPS (Longest Prefix Suffix) array for 'evil' to guide the substring search while iterating character choices. This enables efficient matching in dynamically generating strings.