Compute the triple match number gain from the swapping
    Arguments:
    mapping: current node mapping list
    node_id1: node 1 index in AMR 1
    mapping_id1: the node index in AMR 2 node 1 maps to (in the current mapping)
    node_id2: node 2 index in AMR 1
    mapping_id2: the node index in AMR 2 node 2 maps to (in the current mapping)
    weight_dict: weight dictionary
    match_num: the original matching triple number
    Returns:
    the gain number (might be negative)

def swap_gain(mapping, node_id1, mapping_id1, node_id2, mapping_id2, weight_dict, match_num):
    """
    Compute the triple match number gain from the swapping
    Arguments:
    mapping: current node mapping list
    node_id1: node 1 index in AMR 1
    mapping_id1: the node index in AMR 2 node 1 maps to (in the current mapping)
    node_id2: node 2 index in AMR 1
    mapping_id2: the node index in AMR 2 node 2 maps to (in the current mapping)
    weight_dict: weight dictionary
    match_num: the original matching triple number
    Returns:
    the gain number (might be negative)

    """
    new_mapping_list = mapping[:]
    # Before swapping, node_id1 maps to mapping_id1, and node_id2 maps to mapping_id2
    # After swapping, node_id1 maps to mapping_id2 and node_id2 maps to mapping_id1
    new_mapping_list[node_id1] = mapping_id2
    new_mapping_list[node_id2] = mapping_id1
    if tuple(new_mapping_list) in match_triple_dict:
        return match_triple_dict[tuple(new_mapping_list)] - match_num
    gain = 0
    new_mapping1 = (node_id1, mapping_id2)
    new_mapping2 = (node_id2, mapping_id1)
    old_mapping1 = (node_id1, mapping_id1)
    old_mapping2 = (node_id2, mapping_id2)
    if node_id1 > node_id2:
        new_mapping2 = (node_id1, mapping_id2)
        new_mapping1 = (node_id2, mapping_id1)
        old_mapping1 = (node_id2, mapping_id2)
        old_mapping2 = (node_id1, mapping_id1)
    if new_mapping1 in weight_dict:
        for key in weight_dict[new_mapping1]:
            if key == -1:
                gain += weight_dict[new_mapping1][-1]
            elif new_mapping_list[key[0]] == key[1]:
                gain += weight_dict[new_mapping1][key]
    if new_mapping2 in weight_dict:
        for key in weight_dict[new_mapping2]:
            if key == -1:
                gain += weight_dict[new_mapping2][-1]
            # to avoid duplicate
            elif key[0] == node_id1:
                continue
            elif new_mapping_list[key[0]] == key[1]:
                gain += weight_dict[new_mapping2][key]
    if old_mapping1 in weight_dict:
        for key in weight_dict[old_mapping1]:
            if key == -1:
                gain -= weight_dict[old_mapping1][-1]
            elif mapping[key[0]] == key[1]:
                gain -= weight_dict[old_mapping1][key]
    if old_mapping2 in weight_dict:
        for key in weight_dict[old_mapping2]:
            if key == -1:
                gain -= weight_dict[old_mapping2][-1]
            # to avoid duplicate
            elif key[0] == node_id1:
                continue
            elif mapping[key[0]] == key[1]:
                gain -= weight_dict[old_mapping2][key]
    match_triple_dict[tuple(new_mapping_list)] = match_num + gain
    return gain