Path Compression in Union Find in DSA C - Execution Trace

Concept Flow - Path Compression in Union Find

Start: Find(x)

↓

Is parent[x

No↓

Recursively find parent[x

↓

Set parent[x

↓

Return root

The find operation checks if a node is its own parent. If not, it recursively finds the root and updates the node's parent to the root to flatten the structure.

Execution Sample

DSA C

int find(int x) {
  if (parent[x] != x) {
    parent[x] = find(parent[x]);
  }
  return parent[x];
}

This code finds the root of x and compresses the path by making x point directly to the root.

Execution Table

Step	Operation	Node x	parent[x] before	Recursive Call	parent[x] after	Visual State
1	Call find(4)	4	3	find(3)	TBD	4->3->1->1
2	Call find(3)	3	1	find(1)	TBD	4->3->1->1
3	Call find(1)	1	1	No recursion	1	4->3->1->1
4	Set parent[3] = 1	3	1	Return 1	1	4->3->1->1
5	Set parent[4] = 1	4	3	Return 1	1	4->1->1->1
6	Return root for find(4)	4	1	Return 1	1	4->1->1->1
7	Final parent array	-	-	-	-	parent = [0,1,2,1,1]

💡 Recursion ends when node is its own parent; path compression updates intermediate parents to root.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 4	After Step 5	Final
parent[4]	3	3	3	3	1	1
parent[3]	1	1	1	1	1	1
parent[1]	1	1	1	1	1	1

Key Moments - 3 Insights

Why does find(4) call find(3) before returning?

Because parent[4] is 3, not 4 itself, so we must find the root of 3 first (see execution_table step 1 and 2).

How does path compression update parent pointers?

After finding the root, each node on the path sets its parent directly to the root (see steps 4 and 5 in execution_table).

Why does recursion stop when parent[x] == x?

Because that means x is the root of its set, so no further recursion is needed (step 3 in execution_table).

Visual Quiz - 3 Questions

Look at the execution_table, what is parent[4] after step 5?

A 3

B 4

C 1

D 0

💡 Hint

Check the 'parent[x] after' column for step 5 in execution_table.

At which step does the recursion stop because parent[x] == x?

A Step 2

B Step 3

C Step 1

D Step 5

💡 Hint

Look for 'No recursion' in the Recursive Call column in execution_table.

If parent[4] was initially 4 (itself), what would happen when find(4) is called?

A No recursion, find returns 4 immediately

B Recursion would continue deeper

C parent[4] would be updated to 1

D An error would occur

💡 Hint

Refer to the condition 'parent[x] == x' stopping recursion in concept_flow.

Concept Snapshot

Path Compression in Union Find:
- Used in find(x) to flatten tree.
- If parent[x] != x, recursively find root.
- Set parent[x] = root to speed future finds.
- Reduces tree height, improves efficiency.
- Key for almost O(1) amortized find operations.

Full Transcript

Path Compression in Union Find improves the find operation by making each node on the path point directly to the root. The find function checks if a node is its own parent; if not, it recursively finds the root and updates the node's parent to that root. This flattens the structure, making future finds faster. The execution trace shows find(4) calling find(3), which calls find(1), the root. Then parent pointers of 3 and 4 are updated to 1. This process stops recursion when a node is its own parent. The variable tracker shows how parent pointers change step-by-step. Key moments clarify why recursion happens and how path compression updates parents. The quiz tests understanding of parent updates and recursion stopping conditions.