7.3. Node Representation with Pointers

Try these first

1. 1.

   What are the two parts of a node in the union-find data structure?

   Hint

   Think about how each element relates to other components.

2. 2.

   Define path compression in simple terms.

   Hint

   Consider how it impacts node connections after a search.

3. 3.

   What does a node in the union-find structure represent?

   • A single element
   • A component
   • Both a and b

   Hint

   Remember the structure of a node.

4. 4.

   True or False: Path compression can only be applied once during a find operation.

   • True
   • False

   Hint

   Consider how nodes are adjusted during traversal.

5. 5.

   Given a set of elements {1, 2, 3, 4, 5}, illustrate the union-find operations after performing unions (1,2) and (3,4). Followed by a union (2,4). Construct the final structure.

   Hint

   Start by visualizing the connections after each union.

6. 6.

   Explain how amortized analysis is done for the union-find structure. What can we conclude about time complexities per operation?

   Hint

   Consider approximating over multiple sequences rather than single operations.