Solution Explanation:

This problem asks to determine if a path exists between two nodes in a graph, with the constraint that all edges along the path have a weight strictly less than a given limit. The solution utilizes a persistent union-find data structure to efficiently answer multiple queries.

Persistent Union-Find:

A standard union-find data structure efficiently tracks connected components. However, in this problem, we need to track connectivity for different edge weight limits. A persistent union-find achieves this. It maintains a history of union operations. Each time a union operation occurs with a specific weight limit, the parent pointers and ranks are updated, but previous states are preserved. This way, querying for connectivity under a specific weight limit can be done by reverting to the state after incorporating all edges with weights less than that limit.

Algorithm:

1. Initialization:

   • Create a PersistentUnionFind object with the number of nodes in the graph.
   • Sort the edgeList by edge weights (distances). This is crucial because we process edges in ascending order of their weights to maintain the connectivity history.

2. Union Operations:

   • Iterate through the sorted edgeList. For each edge [u, v, dis], perform a union(u, v, dis) operation in the PersistentUnionFind. The third argument (dis) indicates that this union is only valid for queries with limits greater than or equal to dis.

3. Query:

   • The query(p, q, limit) function uses the find() method of the PersistentUnionFind to determine if nodes p and q belong to the same connected component considering only edges with weights strictly less than limit. The find() method uses the stored version array to find the appropriate parent for a given node within the specified limit.

Time Complexity Analysis:

• Initialization: Sorting edgeList takes O(E log E) time, where E is the number of edges. The subsequent union operations are O(E α(E)), where α is the inverse Ackermann function, which grows extremely slowly and can be considered practically constant for all realistic input sizes. Therefore, the overall initialization is dominated by sorting, making it O(E log E).

• Query: The find() operation in the persistent union-find has a time complexity of O(log N), where N is the number of nodes, due to path compression. Each query performs two find() operations; therefore the query complexity is O(log N).

Space Complexity Analysis:

• The space complexity is dominated by the PersistentUnionFind data structure. It uses O(N) space for parent, rank, and version arrays. Hence, the overall space complexity is O(N).

Code Explanation (Python):

The Python code demonstrates the PersistentUnionFind class and the DistanceLimitedPathsExist class. The PersistentUnionFind class implements the persistent union-find data structure with path compression. The DistanceLimitedPathsExist class uses this data structure to efficiently answer path queries. The inf variable represents a large enough value to act as infinity within the context of the problem.

The other language examples follow the same logic and structure, adapting the syntax and data structures accordingly. The choice of using INT_MAX or Infinity depends on the language's ability to handle sufficiently large values for the version array.