You are given a string, s, and an array of strings, word_dict, representing a dictionary. Your task is to add spaces to s to break it up into a sequence of valid words from word_dict. We are required to return an array of all possible sequences of words (sentences). The order in which the sentences are listed is not significant.

Note: The same dictionary word may be reused multiple times in the segmentation.

Constraints:

• $1 \leq$ s.length $\leq 20$

• $1 \leq$ word_dict.length $\leq 1000$

• $1 \leq$ word_dict[i].length $\leq 10$

• s and word_dict[i] consist of only lowercase English letters.

• All the strings of word_dict are unique.

So far, you’ve probably brainstormed some approaches and have an idea of how to solve this problem. Let’s explore some of these approaches and figure out which one to follow based on considerations such as time complexity and any implementation constraints.

The naive approach to solve this problem is to use a traditional recursive strategy in which we take each prefix of the input string, s, and compare it to each word in the dictionary. If it matches, we take the string’s suffix and repeat the process.

Here is how the algorithm works:

• Base case: If the string is empty, there are no characters in the string that are left to process, so there’ll be no sentences that can be formed. Hence, we return an empty array.

• Otherwise, the string will not be empty, so we’ll iterate every word of the dictionary and check whether or not the string starts with the current dictionary word. This ensures that only valid word combinations are considered:

  • If it doesn’t start with the current dictionary word, no valid combinations can be formed from this word, so we move on to the next dictionary word.

  • If it does start with the current dictionary word, we have two options:

    • If the length of the current dictionary word is equal to the length of the string, it means the entire string can be formed from the current dictionary word. In this case, the string s is directly added to the result without any further processing.

    • Recursive case: Otherwise, the length of the current dictionary word will be less than the length of the string. This means that the string can be broken down further. Therefore, we make a recursive call to evaluate the remaining portion (suffix) of the string.

  • We’ll then concatenate the prefix and the result of the suffix computed by the recursive call above and store it in the result.

• After all possible combinations have been explored, we return the result.

The time complexity of this solution is $O(k^n⋅m)$ ...