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What is the worst-case time complexity for a sequence of m union and find operat

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Question: What is the worst-case time complexity for a sequence of m union and find operations in Disjoint Set Union with path compression and union by rank?

Options:

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

Correct Answer: O(m α(n))

Solution: 

The worst-case time complexity for a sequence of m union and find operations in Disjoint Set Union with path compression and union by rank is O(m α(n)).

What is the worst-case time complexity for a sequence of m union and find operat

Practice Questions

Q1
What is the worst-case time complexity for a sequence of m union and find operations in Disjoint Set Union with path compression and union by rank?
  1. O(m)
  2. O(m log n)
  3. O(m α(n))
  4. O(n)

Questions & Step-by-Step Solutions

What is the worst-case time complexity for a sequence of m union and find operations in Disjoint Set Union with path compression and union by rank?
  • Step 1: Understand what 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 two main operations in DSU: 'union' (to combine two sets) and 'find' (to determine which set a particular element belongs to).
  • Step 3: Recognize that 'path compression' is an optimization technique used in the 'find' operation to make future queries faster by flattening the structure of the tree whenever 'find' is called.
  • Step 4: Understand that 'union by rank' is another optimization that ensures the smaller tree is always added under the root of the larger tree, keeping the overall tree height small.
  • Step 5: Realize that the combination of path compression and union by rank makes the operations very efficient.
  • Step 6: Learn about the function α(n), known as the inverse Ackermann function, which grows very slowly and is almost constant for all practical values of n.
  • Step 7: Conclude that the worst-case time complexity for m union and find operations in this optimized DSU is O(m α(n)), meaning it is very efficient even for large inputs.
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