Give the state of the disjoint-sets data structure after the...

Give the state of the disjoint-sets data structure after the following sequence of operations, starting from singleton sets $\{1\}, \ldots,\{8\} .$ Use path compression. In case of ties, always make the lower numbered root point to the higher numbered one. union $(1,2),$ union $(3,4),$ union $(5,6),$ union $(7,8),$ union $(1,4),$ union $(6,7),$ union $(4,5),$ find (1)

Give the state of the disjoint-sets data structure after the following sequence of operations, starting from singleton sets $\{1\}, \ldots,\{8\} .$ Use path compression. In case of ties, always make the lower numbered root point to the higher numbered one.
union $(1,2),$ union $(3,4),$ union $(5,6),$ union $(7,8),$ union $(1,4),$ union $(6,7),$ union $(4,5),$ find (1)

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Key Concepts

Tie-Breaking Rule

When performing union operations, tie-breaking rules are used to decide which representative should become the new root, particularly when both candidates have similar attributes or rankings. These rules maintain consistency and determinism in the structure.

Find Operation

The find operation identifies the root or representative of the set that an element belongs to. It is crucial for determining connectivity between elements and works efficiently in combination with union operations.

Path Compression

An optimization strategy applied during the find operation, path compression flattens the tree structure by making each node on the path point directly to the root, thereby decreasing the time complexity of future find operations.

Disjoint-Set Data Structure

A data structure that maintains a collection of non-overlapping sets and supports efficient operations to determine which set an element belongs to and to merge two sets. It is commonly represented using trees where each tree represents a set with a designated root.

Union Operation

This operation combines two distinct sets into a single set by linking their representative roots. It is a fundamental mechanism in managing dynamic connectivity and helps in merging sets as needed.

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