Problem Description

You're given a string word and an array of strings forbidden. The goal is to find the length of the longest substring within word that doesn't contain any sequence of characters present in forbidden. A substring is simply a contiguous sequence of characters within another string and it can even be an empty string.

To make a string "valid" in the context of this problem, none of its substrings can match any string present in forbidden. It's your task to figure out what's the maximum length of such a "valid" substring within the given word.

Intuition

The intuition behind the solution revolves around the concept of dynamic window sliding coupled with efficient substring checking. The brute force method of checking every possible substring would be too slow, especially since string word can be quite large. Instead, we need to find a quicker way of validating substrings against the forbidden list.

We create a sliding window that expands and contracts as needed while scanning the word from left to right. The set s allows for constant-time checks if a substring is in forbidden—this is much faster than repeatedly scanning a list.

Our approach is as follows:

• Start with two pointers i and j, initializing i to the beginning of the word and j to increment over each character.
• At every step, check the current window from j down to i to see if any substring is in forbidden. If it is, move i past the forbidden substring's start to make a valid window again.
• Keep track of the maximum valid window seen so far as ans.

One key optimization lies in the knowledge that the forbidden substrings have a maximum possible length. Thus the inner loop doesn't need to check further back than this maximum length from j, which prevents needless comparisons and significantly speeds up the algorithm.

This solution effectively balances between the thoroughness of checking all relevant substrings and the speed of skipping unnecessary checks, achieving the task within acceptable time complexity.

Learn more about Sliding Window patterns.

Solution Approach

In the provided solution, a set data structure is utilized to store the forbidden strings. The choice of a set is crucial because it provides O(1) lookup time complexity, which means checking if a substring is in forbidden is very fast, regardless of the size of forbidden.

Here is the step-by-step explanation of the code:

1. The set s is created from the forbidden array to take advantage of fast lookups.
2. Two pointers are introduced:
   • i is initialized to 0, pointing to the beginning of the current valid substring.
   • j is used to iterate through each character in word.
3. A variable ans is used to keep track of the length of the longest valid substring found so far.
4. We iterate over the string word with the help of j.
   • During each iteration, we look for any potential forbidden substrings within a window --- the substring from [k : j + 1], where k will iterate backwards from j but not further back than 10 places (owing to the maximum length of forbidden substrings) or before i, whichever is lesser.
   • The use of max(j - 10, i - 1) ensures that we don't check beyond the necessary window backwards from the current character.
5. If a forbidden substring is found, i is moved to one character past the start of the forbidden substring. This effectively skips the forbidden part, searching for the next valid starting point.
6. ans is updated to be the maximum of its current value or j - i + 1, which represents the length of the new valid substring from the current position j back to i.
7. The iteration continues until the end of word is reached, with ans being updated as necessary.
8. At the end, ans holds the length of the longest valid substring, and it is returned as the solution.

The key algorithmic pattern here is the sliding window, employed to dynamically adjust the substring being inspected. Effective use of data structuring and algorithmic patterns are what make this solution both elegant and efficient.

Example Walkthrough

Let's take an example to understand how this solution works. Suppose we have word = "abcdefgh" and forbidden = ["bc", "fg"]. We want to find the longest substring of word that doesn't contain any sequence of characters from forbidden.

We start by creating a set s containing forbidden strings for quick lookup: s = {"bc", "fg"}. We initialize two pointers: i at index 0, and j which will traverse through word.

The ans variable starts at 0 to keep track of the length of the longest valid substring found so far. Now, we begin iterating over word:

1. With j at 0, our window is ["a"], and since it doesn't match any forbidden strings, we move on.

2. With j at 1, our window is ["ab"], which is also not forbidden. We continue.

3. With j at 2, our window is ["abc"], and checking from the previous 10 characters (or from i if less than 10), we find a match with s because "bc" is in the window. We update i to one position after where "bc" started, so i becomes 2.

4. Now j moves to 3, and our new window starts from i (2) to j (3), which is ["cd"]. It is valid, so we continue.

5. With j at 4, our window is ["cde"], still valid.

6. With j moving up to 5, our window is now ["cdef"], which is still not in forbidden.

7. At j = 6, our window becomes ["cdefg"], which contains "fg". We update i to move past the start of "fg", making i = 6.

8. As j moves to 7, our window is ["h"], since i was just updated to 6. This window is valid.

Throughout the process, we update ans to be the maximum length of valid substrings we have found. The sequence "cde" was the longest valid substring before we encountered "fg", with a length of 3. Therefore, ans is 3.

In this case, the longest substring of word without any forbidden sequences would be "cde" which has a length of 3. This is the value we return.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code can be analyzed by going through the nested loops and operations within them. The outer loop runs for len(word) times, where 'word' is the input string and its length is denoted by 'n'. The inner loop at worst case runs for a maximum of 10 iterations since it looks back from the current position j only up to 10 characters because of max(j - 10, i - 1) condition.

Hence, in the worst-case scenario, for each character we might look at the next 10 characters to check if the substring is in the forbidden set. Since checking whether a substring is in a set is O(1) operation due to hashing, the worst case would consider these constant time checks up to 10 times for each character.

Combining both loops, the worst-case time complexity can therefore be simplified to O(n), where each character is checked against a constant number of potential forbidden substrings.

Space Complexity

The space complexity consists of the space required for the set 's' and a few integer variables. Since 's' contains the forbidden substrings, its space complexity will be O(m), where 'm' is the total length of all forbidden substrings combined.

The other variables, 'ans', 'i', and 'j' are constant space, adding O(1) to the total space complexity.

Therefore, the total space complexity of the code is O(m + 1) which simplifies to O(m), denoting the space taken up by the set of forbidden substrings.

Learn more about how to find time and space complexity quickly using problem constraints.