Solution Explanation:

This problem asks whether two sentences are similar based on a given set of similar word pairs. The similarity is transitive, meaning if word A is similar to B, and B is similar to C, then A is similar to C. This suggests using a Union-Find (Disjoint-Set Union) data structure to efficiently track the similarity groups.

Algorithm:

1. Initialization: Create a Union-Find data structure. Each word will initially represent its own set. The parent array p is used to track the root of each set.

2. Build Union-Find: Iterate through the similarPairs. For each pair (x, y), find the root of the sets containing x and y using the find function. If they belong to different sets, unite them using union operation.

3. Compare Sentences: Iterate through sentence1 and sentence2 simultaneously.

   • If sentence1[i] and sentence2[i] are identical, proceed to the next word pair.
   • If they are different:
     • Check if either word is not in the words map (meaning it's not part of any similarity group). If so, they are not similar, return false.
     • Find the roots of their respective sets using the find function. If the roots are different, the words are not similar, return false.

4. Return True: If all word pairs are similar, return true.

Data Structures:

• Union-Find (Disjoint Set Union): This efficient data structure manages sets of elements and allows for quick union and find operations. The parent array p stores the parent of each element. find(x) recursively finds the root of the set containing x.
• Hash Map (words): Maps words to their indices in the Union-Find structure for efficient lookup. This mapping is important because it allows for efficient tracking of word relationships and avoiding redundant computations.

Time Complexity Analysis:

• Building the Union-Find structure takes O(M*α(M)) time where M is the number of similarPairs, and α is the Inverse Ackermann function which grows extremely slowly, often considered constant for all practical purposes.
• Comparing the sentences takes O(N) time, where N is the length of the sentences. The find operation in the Union-Find is also amortized O(α(M)).
• Therefore, the overall time complexity is dominated by the comparison, making it O(N + M*α(M)), which is essentially O(N + M) for practical inputs.

Space Complexity Analysis:

• The Union-Find structure uses O(M) space.
• The words hash map uses O(M) space.
• Thus, the total space complexity is O(M).

Code Examples (Python, Java, C++, Go):

The provided code snippets in various languages implement this algorithm effectively. The find function is the core of the Union-Find, performing path compression for optimization. The words map ensures efficient access to the Union-Find representation of each word. The main loop compares words pair-wise and leverages the Union-Find to quickly determine similarity.