Solution Explanation: Before and After Puzzle

This problem asks us to find all possible concatenations of phrases from a given list where the last word of one phrase matches the first word of another. The resulting concatenated phrases must be unique and lexicographically sorted.

Approach:

The solution uses a hash table (or set in some languages) to efficiently store and manage unique concatenated phrases. The process involves these steps:

1. Preprocessing: For each phrase, extract the first and last words. This is done to speed up the comparison process later. We store this information efficiently, possibly in an array of pairs.

2. Phrase Combination: Iterate through all possible pairs of phrases. For each pair, check if the last word of the first phrase equals the first word of the second phrase. If they match, concatenate the phrases (removing the redundant first word from the second phrase) and add the result to the hash table. The hash table ensures that only unique concatenated phrases are stored.

3. Sorting and Output: Convert the hash table (set) into a sorted array (list) and return it as the solution. The sorting ensures the lexicographical order required.

Time Complexity Analysis:

• Preprocessing: O(N*M), where N is the number of phrases and M is the average length of a phrase (due to splitting the phrases into words).

• Phrase Combination: O(N^2), where we compare each phrase with every other phrase. The concatenation and hash table insertion operations take O(M) in the worst case (when concatenating long phrases). Thus, the overall time complexity for this step can be considered O(N^2 * M).

• Sorting: O(K log K), where K is the number of unique concatenated phrases. In the worst case, K can be O(N^2).

Therefore, the overall time complexity is dominated by the phrase combination step, resulting in O(N^2 * M), where we can consider M to be relatively small compared to N in most cases.

Space Complexity Analysis:

• Preprocessing: O(N*M), to store the first and last words of each phrase.

• Phrase Combination: O(K), where K is the number of unique concatenated phrases. In the worst case, K could be O(N^2).

Therefore, the overall space complexity is O(N^2) in the worst case. The space used by the hash table dominates.

Code Examples:

The provided code examples in Python, Java, C++, Go, and TypeScript all follow this approach, demonstrating variations in syntax and data structures but the underlying algorithm is consistent. The choice of hash set (or equivalent) is crucial for efficiently handling unique phrases and avoiding duplicates. The sorting at the end is standard library functionality in all of the languages.