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What is the time complexity of the 'Union' operation in an optimized Disjoint Se

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Question: What is the time complexity of the \'Union\' operation in an optimized Disjoint Set Union with path compression and union by rank?

Options:

  1. O(1)
  2. O(log n)
  3. O(n)
  4. O(α(n))

Correct Answer: O(α(n))

Solution: 

The time complexity of the \'Union\' operation in an optimized Disjoint Set Union with path compression and union by rank is O(α(n)), where α is the inverse Ackermann function.

What is the time complexity of the 'Union' operation in an optimized Disjoint Se

Practice Questions

Q1
What is the time complexity of the 'Union' operation in an optimized Disjoint Set Union with path compression and union by rank?
  1. O(1)
  2. O(log n)
  3. O(n)
  4. O(α(n))

Questions & Step-by-Step Solutions

What is the time complexity of the 'Union' operation in an optimized Disjoint Set Union with path compression and union by rank?
Correct Answer: O(α(n))
  • Step 1: Understand what a Disjoint Set Union (DSU) is. It is a data structure that keeps track of a partition of a set into disjoint subsets.
  • Step 2: Learn about the 'Union' operation. This operation combines two subsets into a single subset.
  • Step 3: Know that in an optimized DSU, we use two techniques: path compression and union by rank.
  • Step 4: Path compression helps to flatten the structure of the tree whenever 'find' is called, making future operations faster.
  • Step 5: Union by rank ensures that the smaller tree is always added under the root of the larger tree, keeping the overall tree height small.
  • Step 6: The combination of these two techniques makes the operations very efficient.
  • Step 7: The time complexity of the 'Union' operation is not constant but is very close to constant time, specifically O(α(n)), where α is the inverse Ackermann function.
  • Step 8: The inverse Ackermann function grows very slowly, so for all practical purposes, O(α(n)) is almost constant time.
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