def minimize_hamming_distance_brute_force(source_array, target_array, allowed_swaps):
    number_of_elements = len(source_array)
    minimum_hamming_distance = number_of_elements

    def calculate_hamming_distance(first_array, second_array):
        hamming_distance = 0
        for index in range(number_of_elements):
            if first_array[index] != second_array[index]:
                hamming_distance += 1
        return hamming_distance

    def find_connected_components(edges, number_of_nodes):
        visited = [False] * number_of_nodes
        connected_components = []

        def depth_first_search(node, component):
            visited[node] = True
            component.append(node)
            for neighbor in adjacency_list[node]:
                if not visited[neighbor]:
                    depth_first_search(neighbor, component)

        adjacency_list = [[] for _ in range(number_of_nodes)]
        for first, second in edges:
            adjacency_list[first].append(second)
            adjacency_list[second].append(first)

        for node in range(number_of_nodes):
            if not visited[node]:
                component = []
                depth_first_search(node, component)
                connected_components.append(component)

        return connected_components

    connected_components = find_connected_components(allowed_swaps, number_of_elements)

    def generate_permutations(group, current_permutation, all_permutations):
        if not group:
            all_permutations.append(current_permutation)
            return

        for index in range(len(group)):
            remaining_elements = group[:index] + group[index+1:]
            generate_permutations(remaining_elements, current_permutation + [group[index]], all_permutations)

    # Enumerate all possible permutations for each connected component
    all_group_permutations = []
    for group in connected_components:
        all_permutations_for_group = []
        generate_permutations(list(range(len(group))), [], all_permutations_for_group)
        all_group_permutations.append((group, all_permutations_for_group))

    # Iterate through all possible combinations of permutations across groups
    number_of_groups = len(connected_components)
    group_permutation_indices = [0] * number_of_groups

    while True:
        temp_source_array = source_array[:]
        # Apply the current combination of permutations
        for group_index in range(number_of_groups):
            group, all_permutations = all_group_permutations[group_index]
            current_permutation = all_permutations[group_permutation_indices[group_index]]
            for index_in_permutation, original_index_in_group in enumerate(current_permutation):
                temp_source_array[group[index_in_permutation]] = source_array[group[original_index_in_group]]

        hamming_distance = calculate_hamming_distance(temp_source_array, target_array)
        minimum_hamming_distance = min(minimum_hamming_distance, hamming_distance)

        # Increment the permutation indices and check for termination
        group_index = number_of_groups - 1
        while group_index >= 0:
            group_permutation_indices[group_index] += 1
            if group_permutation_indices[group_index] == len(all_group_permutations[group_index][1]):
                group_permutation_indices[group_index] = 0
                group_index -= 1
            else:
                break
        else:
            break

    return minimum_hamming_distance

def minimize_hamming_distance(source_list, target_list, allowed_swaps):
    number_of_elements = len(source_list)
    parent = list(range(number_of_elements))

    def find(element_index):
        if parent[element_index] != element_index:
            parent[element_index] = find(parent[element_index])
        return parent[element_index]

    def union(element_index_x, element_index_y):
        root_x = find(element_index_x)
        root_y = find(element_index_y)
        if root_x != root_y:
            parent[root_x] = root_y

    # Group swappable positions using the Union-Find algorithm.
    for element_index_x, element_index_y in allowed_swaps:
        union(element_index_x, element_index_y)

    swap_groups = {}
    for element_index in range(number_of_elements):
        root = find(element_index)
        if root not in swap_groups:
            swap_groups[root] = []
        swap_groups[root].append(element_index)

    # Collect elements for each group from source and target lists.
    for group_root, group_indices in swap_groups.items():
        source_group_elements = [source_list[element_index] for element_index in group_indices]
        target_group_elements = [target_list[element_index] for element_index in group_indices]
        source_group_elements.sort()
        target_group_elements.sort()

        # Assign the sorted values back to the original positions in the lists.
        for element_index_within_group, original_element_index in enumerate(group_indices):

            source_list[original_element_index] = source_group_elements[element_index_within_group]
            target_list[original_element_index] = target_group_elements[element_index_within_group]

    # Count differences after optimal swaps.
    hamming_distance = 0
    for element_index in range(number_of_elements):

        # Counting the differences is the hamming distance
        if source_list[element_index] != target_list[element_index]:

            hamming_distance += 1

    return hamming_distance