6.2. Union and Find Operations

Try these first

1. 1.

   What are the two primary operations of the Union-Find data structure?

   Hint

   Think about what these operations do to a set.

2. 2.

   What is the worst-case time complexity for the naive union operation?

   Hint

   Consider the number of elements you may have to visit in a union operation.

3. 3.

   What does the find operation in the Union-Find structure do?

   • Merges two components
   • Determines the component an element belongs to
   • Creates a new set

   Hint

   Think about the purpose of checking components.

4. 4.

   True or False: The union by size technique merges the larger tree into the smaller tree.

   • True
   • False

   Hint

   Consider what it means for trees to be balanced.

5. 5.

   Given a set of cities represented as nodes and roads as edges connecting them, explain how you would implement a Union-Find structure to detect cycles when planning a new road network.

   Hint

   Consider the implications of connecting components when checking for cycles.

6. 6.

   Implement a Union-Find algorithm with path compression and union by size. Analyze its complexity over a series of operations.

   Hint

   Look at how merging impacts future operations and their costs.