Find All Good Strings

The problem involves counting good strings of length n that are lexicographically between s1 and s2 and don't contain the substring evil. Using dynamic programming with four states (position, evil substring match, and equality to s1/s2) optimizes the solution to count valid strings without explicitly generating all possibilities.

Problem Statement

Given two strings s1 and s2 of length n and another string evil, you need to return the number of valid strings of length n that meet the following conditions: the string is lexicographically greater than or equal to s1, lexicographically smaller than or equal to s2, and does not contain evil as a substring. Since the answer could be large, return the result modulo 10^9 + 7.

For instance, with n = 2, s1 = "aa", s2 = "da", and evil = "b", the result is 51. This is because 25 good strings start with 'a' ("aa", "ac", etc.), 25 start with 'c' ("ca", "cc", etc.), and 1 starts with 'd' ("da").

Examples

Example 1

Input: n = 2, s1 = "aa", s2 = "da", evil = "b"

Output: 51

There are 25 good strings starting with 'a': "aa","ac","ad",...,"az". Then there are 25 good strings starting with 'c': "ca","cc","cd",...,"cz" and finally there is one good string starting with 'd': "da".

Example 2

Input: n = 8, s1 = "leetcode", s2 = "leetgoes", evil = "leet"

Output: 0

All strings greater than or equal to s1 and smaller than or equal to s2 start with the prefix "leet", therefore, there is not any good string.

Example 3

Input: n = 2, s1 = "gx", s2 = "gz", evil = "x"

Output: 2

Example details omitted.

Constraints

• s1.length == n
• s2.length == n
• s1 <= s2
• 1 <= n <= 500
• 1 <= evil.length <= 50
• All strings consist of lowercase English letters.

Solution Approach

State Transition Dynamic Programming

This problem can be solved using dynamic programming with four states: position (pos), the current position in the evil substring (posEvil), whether the string is equal to s1 (equalToS1), and whether it is equal to s2 (equalToS2). Transition through these states while ensuring that no substring equals evil.

Modulo Arithmetic for Large Numbers

Since the number of good strings can be very large, the result should be returned modulo 10^9 + 7. During the dynamic programming transitions, ensure each result is computed modulo this value to avoid overflow and unnecessary large numbers.

Efficient String Matching

The key to solving this problem is efficient string matching for the evil substring. As you iterate through potential strings, make sure to avoid forming any string that contains the evil substring using an automaton-based approach for substring matching.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity of this problem depend on the number of states and transitions in the dynamic programming approach. Since we are using a 4-state DP and each state depends on previous ones, the time complexity is O(n * evil_length * 2), where n is the string length and evil_length is the length of the evil substring. The space complexity is O(n * evil_length * 2) due to the storage of DP states.

What Interviewers Usually Probe

• The candidate demonstrates an understanding of dynamic programming with multiple states.
• The candidate is able to handle large number computations efficiently with modulo arithmetic.
• The candidate should show knowledge of how to avoid building strings directly by using automaton or substring matching techniques.

Common Pitfalls or Variants

Common pitfalls

• Failing to account for all four states in the dynamic programming transition, leading to incorrect string counting.
• Not using modulo arithmetic correctly to handle large numbers, which can result in integer overflow or incorrect answers.
• Overcomplicating the solution by trying to generate all possible strings, rather than leveraging DP to solve the problem more efficiently.

Follow-up variants

• Modify the problem to allow any substring rather than just evil, and solve it with the same DP approach.
• Extend the problem to support lexicographical range queries across multiple strings with different evil substrings.
• Increase the string length (n) and analyze how the solution scales in terms of performance and complexity.

FAQ

What is the primary approach to solve the Find All Good Strings problem?

The problem is solved using dynamic programming with four states that track the current position in the string, the matching progress with the evil substring, and the equality constraints with s1 and s2.

How does dynamic programming help in solving this problem?

Dynamic programming optimizes the solution by breaking the problem into subproblems, where each state transition counts the number of valid strings without generating all possible combinations.

What is the modulo operation for in this problem?

The modulo operation is used to avoid overflow and ensure that the result is within the range of 10^9 + 7, which is a common large prime number used in competitive programming.

Why is it important to avoid generating all possible strings?

Generating all possible strings would be computationally expensive, especially with a large n. Using dynamic programming allows you to efficiently count valid strings without explicitly constructing them.

Can you explain the role of the evil substring in this problem?

The evil substring must be avoided. The dynamic programming approach ensures that no valid string contains the evil substring by tracking the matching progress of evil and ensuring no invalid substring is formed.