What is the worst-case time complexity of the 'Union' operation in Disjoint Set

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Question: What is the worst-case time complexity of the \'Union\' operation in Disjoint Set Union with union by rank?

Options:

Correct Answer: O(α(n))

Solution:

The worst-case time complexity of the \'Union\' operation in Disjoint Set Union with union by rank is O(α(n)).

What is the worst-case time complexity of the 'Union' operation in Disjoint Set Union with union by rank?

What is the worst-case time complexity of the 'Union' operation in Disjoint Set Union with union by rank?

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 'Union by rank' is a technique used to keep the tree representing the subsets balanced. It attaches the smaller tree under the root of the larger tree.
Step 4: Realize that the time complexity of the 'Union' operation depends on how deep the trees are. A deeper tree means more time to find the root.
Step 5: Understand that the 'Union' operation, when combined with another technique called 'Path Compression', results in very efficient operations.
Step 6: The function α(n) is the inverse Ackermann function, which grows very slowly. For all practical purposes, it can be considered a constant for reasonable values of n.
Step 7: Conclude that the worst-case time complexity of the 'Union' operation in DSU with union by rank is O(α(n)).

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