def roadsAndLibraries2(n, c_lib, c_road, cities):
    # cost 1: c_lib * n
    if c_lib <= c_road:
        return c_lib * n
        
    # cost 2: c_lib * k + c_road * (n - k), where k is num of connected components
    # Initialize parent array for disjoint-set data structure
    parent = list(range(n+1))
    # Initialize rank array for union by rank optimization
    rank = [0] * (n + 1)
    
    # Find operation to recursively find the representative of a set
    def find_parent(city):
        if parent[city] == city:
            return city
        # Path compression: update parent pointers to point directly to the root
        parent[city] = find_parent(parent[city])
        return parent[city]
        
    # Union operation to merge two sets
    def union(city1, city2):
        a, b = find_parent(city1), find_parent(city2)
        if a == b:
            return
        # Union by rank: merge smaller tree into larger tree
        if rank[a] > rank[b]:
            parent[b] = a
        elif rank[a] < rank[b]:
            parent[a] = b
        else:
            parent[b] = a
            rank[a] += 1
    
    # Merge sets based on given roads
    for city1, city2 in cities:
        union(city1, city2)
        
    # Count the number of connected components (sets with representatives)
    k = sum(1 for city in range(1, n+1) if parent[city] == city)
    
    # Calculate cost based on the number of connected components
    return c_lib * k + c_road * (n - k)