Solution Explanation: 1258. Synonymous Sentences

This problem involves generating all possible synonymous sentences from a given sentence and a list of synonym pairs. The solution leverages a Union-Find data structure for efficient synonym group identification and Depth-First Search (DFS) for sentence generation.

1. Union-Find (Disjoint Set Union):

The core idea is to represent synonyms as connected components in a graph. Each word is a node, and synonym pairs create edges. The Union-Find data structure efficiently manages these components:

• find(x): Determines the root node (representative) of the component containing node x. This uses path compression for optimization.
• union(a, b): Merges the components containing nodes a and b.

2. Building the Union-Find Structure:

• We create a UnionFind object. The number of nodes is the number of unique words in the synonyms list.
• A dictionary (d in Python, Map in Java, unordered_map in C++) maps each unique word to its corresponding node index.
• We iterate through the synonyms list. For each pair [a, b], we use union(d[a], d[b]) to merge their components.

3. Grouping Synonyms:

After building the Union-Find structure, we need to group synonyms together. We iterate through the unique words and use find() to get the root node for each word. This root represents the entire synonym group. We store these groups in a dictionary or array (g in the code) for easy access.

4. Depth-First Search (DFS):

The DFS algorithm generates all possible sentences. It operates recursively:

• dfs(i): Processes the i-th word of the sentence.
• Base Case: If i reaches the end of the sentence, the current sentence is added to the result list.
• Recursive Step:
  • If the word is not a synonym, it's appended to the current sentence, and dfs is called for the next word.
  • If the word is a synonym, we find its synonym group using the Union-Find structure. We iterate through all synonyms in the group, append each synonym to the sentence, and recursively call dfs for the next word. After each recursive call, the last added synonym is removed to explore other possibilities.

5. Sorting and Returning:

Finally, the generated sentences are sorted lexicographically, and the sorted list is returned.

Time Complexity Analysis:

• Union-Find operations (find, union): Amortized O(α(n)) where α is the inverse Ackermann function, which grows extremely slowly and can be considered practically constant.
• Building the synonym groups: O(m), where m is the number of unique words.
• DFS: In the worst case, the number of possible sentences is exponential with respect to the number of synonym groups in the sentence. Let k be the number of words in the input sentence and g be the maximum size of a synonym group. The complexity in the worst case could be approximated as O(gk). However, this is a worst-case scenario. In practice, it might be significantly less.

Space Complexity Analysis:

• Union-Find data structure: O(m), where m is the number of unique words.
• Synonym groups: O(m).
• Recursive calls during DFS: This depends on the number of generated sentences which, as noted above, could be O(gk) in the worst case, but likely much less in practice.

Overall, the dominant factor is often the DFS traversal, which can be exponential in the worst case, but is more likely to be significantly lower in many real-world scenarios. The Union-Find structure helps keep the time complexity of finding and merging synonym groups relatively low.