Final answer:
To add a Print-Set operation in a disjoint-set forest, a linked list attribute should be added to each node. The list contains all set members, and merges during UNION operations through pointer updates, enabling O(n) Print-Set without affecting other operation complexities.
Step-by-step explanation:
In order to add a Print-Set operation to a disjoint-set forest that prints all the members of a node's set efficiently, we can modify the data structure by adding a new attribute to each node. This attribute will be a linked list or some other linear time accessible and traversable collection that contains all the nodes that are members of the set represented by the set representative. When a UNION operation is performed, we would merge the lists from the two sets together. This ensures that each set is represented by a single list that can be traversed in O(n) time, where n is the number of members in the set.
To maintain the asymptotic running times of the other operations (Make-Set, Union, Find-Set), the merge operation for the linked lists should be done in a way that does not increase their time complexity. Typically, this would be a pointer update, requiring constant time. Therefore, during the UNION operation, which takes O(log n) time, we'll simply update the pointers of the nodes in the smaller set to point to the new set representative (root of the merged tree), and concatenate the two lists. This ensures that the added operation does not change the existing time complexities.
As a result, Print-Set(x) can be performed in O(m) time where m is the number of nodes in the list, and the complexities of Make-Set, Union, and Find-Set remain unchanged, achieving the desired functionality efficiently.