6.9. Amortized Complexity

Try these first

1. 1.

   What does the union operation do in the Union-Find data structure?

   Hint

   Think about combining groups.

2. 2.

   What is the find operation used for?

   Hint

   Consider it like identifying membership.

3. 3.

   What operation combines two disjoint components into one?

   • Find
   • Union
   • Connect

   Hint

   Think of it as joining two groups.

4. 4.

   Is every union operation guaranteed to take O(1) time?

   • True
   • False

   Hint

   Consider the worst-case scenario for unoptimized operations.

5. 5.

   Design a contrived example where multiple union operations lead to one of the components experiencing maximum path length. Explain how path compression would have altered this scenario.

   Hint

   Consider how repeatedly merging one element into another affects their relations.

6. 6.

   Implement a basic version of the Union-Find structure in code. Discuss how your implementation optimizes the time complexity for a series of union operations.

   Hint

   Ensure to keep methods distinct for union and find, highlighting time complexities.