Path Compression in Union Find in DSA Javascript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main purpose of path compression in Union Find?

Path compression helps to speed up the find operation by making nodes point directly to the root, flattening the structure for faster future queries.

beginner

In Union Find, what does the find function return after path compression?

It returns the root parent of the node, and updates all nodes along the path to point directly to this root.

intermediate

Why is path compression important for the efficiency of Union Find?

Because it reduces the tree height, making future find operations almost constant time, improving overall performance.

intermediate

Show the effect of path compression on the parent array after calling find(4) if the initial parent array is [0, 0, 1, 2, 3].

After find(4), the parent array becomes [0, 0, 0, 0, 0] because nodes 1, 2, 3, and 4 along the path are updated to point directly to root 0.

beginner

Write the key line of code in JavaScript that implements path compression inside the find function.

The key line is: parent[node] = find(parent[node]); which updates the parent of the node to its root recursively.

What does path compression do in Union Find?

A Makes every node point directly to the root parent

B Merges two sets without changing parents

C Deletes nodes from the structure

D Creates new nodes for each find operation

Path compression updates nodes to point directly to the root, speeding up future find operations.

Which operation benefits most from path compression?

A Find

B Delete

C Insert

D Union

The find operation becomes faster because path compression flattens the tree.

What is the time complexity of find with path compression over many operations?

A O(n)

B O(log n)

C Amortized almost O(1)

D O(n^2)

Path compression makes find operations almost constant time on average.

In the parent array [0, 0, 1, 2, 3], after find(4) with path compression, what is parent[4]?

A 3

B 2

C 1

D 0

Path compression updates parent[4] directly to the root 0.

Which line best shows path compression in JavaScript?

A parent[node] = node;

B parent[node] = find(parent[node]);

C parent[node] = parent[parent[node]];

D parent[node] = null;

This line recursively sets the parent to the root, compressing the path.

Explain how path compression works in the Union Find data structure and why it improves performance.

Think about what happens to the nodes on the path when you call find.

Describe the changes in the parent array before and after applying path compression during a find operation.

Consider a chain of parents and how it becomes shorter.

Practice

1. What is the main purpose of path compression in the Union Find data structure?

easy

A. To speed up the find operation by flattening the tree structure

B. To merge two sets without checking their roots

C. To initialize the parent array with default values

D. To count the number of connected components

Solution

1. Step 1: Understand the role of path compression

   Path compression is used during the find operation to make future queries faster by making nodes point directly to the root.

2. Step 2: Identify the effect on the tree structure

   This flattens the tree, reducing the height and speeding up subsequent find calls.

3. Final Answer:

   To speed up the find operation by flattening the tree structure -> Option A

4. Quick Check:

   Path compression = flatten tree = faster find [OK]

Hint: Path compression flattens trees to speed up find [OK]

Common Mistakes:

• Confusing path compression with union by rank
• Thinking it merges sets without root checks
• Believing it initializes data structures

2. Which of the following is the correct syntax for the find function with path compression in JavaScript?

easy

A. function find(x) { if (parent[x] === x) return x; else return find(parent[x]); }

B. function find(x) { if (parent[x] !== x) parent[x] = find(parent[x]); return parent[x]; }

C. function find(x) { while (parent[x] !== x) x = parent[x]; return x; }

D. function find(x) { parent[x] = x; return x; }

Solution

1. Step 1: Identify path compression in the code

   function find(x) { if (parent[x] !== x) parent[x] = find(parent[x]); return parent[x]; } updates parent[x] to the root during recursion, which is path compression.

2. Step 2: Compare other options

   function find(x) { while (parent[x] !== x) x = parent[x]; return x; } uses iteration but no path compression. function find(x) { if (parent[x] === x) return x; else return find(parent[x]); } uses recursion but no assignment to parent[x]. function find(x) { parent[x] = x; return x; } resets parent incorrectly.

3. Final Answer:

   function find(x) { if (parent[x] !== x) parent[x] = find(parent[x]); return parent[x]; } -> Option B

4. Quick Check:

   Path compression requires parent update = function find(x) { if (parent[x] !== x) parent[x] = find(parent[x]); return parent[x]; } [OK]

Hint: Look for parent update inside find for path compression [OK]

Common Mistakes:

• Missing parent update for path compression
• Using iteration without compression
• Resetting parent incorrectly

3. Given the following code snippet, what will be the output of console.log(parent) after calling find(3)?

let parent = [0, 0, 1, 2, 4];
function find(x) {
  if (parent[x] !== x) parent[x] = find(parent[x]);
  return parent[x];
}
find(3);
console.log(parent);

medium

A. [0, 0, 0, 0, 4]

B. [0, 0, 1, 2, 4]

C. [0, 0, 1, 1, 4]

D. [0, 1, 1, 2, 4]

Solution

1. Step 1: Trace the find(3) call

   find(3) checks parent[3] = 2, not 3, so calls find(2). find(2) checks parent[2] = 1, not 2, so calls find(1). find(1) checks parent[1] = 0, not 1, so calls find(0). find(0) returns 0 since parent[0] = 0.

2. Step 2: Apply path compression updates

   parent[1] = 0, parent[2] = 0, parent[3] = 0 after recursion. But initial array was [0,0,1,2,4]. After updates, parent becomes [0,0,0,0,4]. However, the code only updates parent[x] during recursion. The first call updates parent[3] = find(2) = 0, parent[2] = find(1) = 0, parent[1] = find(0) = 0. So final parent array is [0,0,0,0,4].

3. Final Answer:

   [0, 0, 0, 0, 4] -> Option A

4. Quick Check:

   Path compression sets parents to root = [0,0,0,0,4] [OK]

Hint: Follow recursion and update parents to root [OK]

Common Mistakes:

• Not updating all parents during recursion
• Confusing parent indices
• Assuming no change after find

4. Identify the bug in the following find function implementing path compression:

function find(x) {
  if (parent[x] === x) return x;
  else find(parent[x]);
}

medium

A. Function should use iteration instead of recursion

B. Incorrect base condition for recursion

C. Parent array is not updated for path compression

D. Missing return statement in the else branch

Solution

1. Step 1: Check the recursive calls and returns

   The else branch calls find(parent[x]) but does not return its result, so the function returns undefined for non-root nodes.

2. Step 2: Confirm missing return causes bug

   Without returning the recursive call, the function fails to return the root, breaking the find operation.

3. Final Answer:

   Missing return statement in the else branch -> Option D

4. Quick Check:

   Recursive calls must return values [OK]

Hint: Always return recursive find calls [OK]

Common Mistakes:

• Forgetting to return recursive call result
• Misunderstanding base case
• Thinking iteration is required

5. You have a Union Find structure with parent = [0, 0, 2, 2, 4] and rank = [0, 1, 0, 0, 0]. After performing union(3, 4) and then find(3), what will be the updated parent array assuming path compression is applied in find?

hard

A. [0, 0, 2, 2, 4]

B. [0, 0, 4, 2, 4]

C. [0, 0, 4, 4, 4]

D. [0, 0, 2, 4, 4]

Solution

1. Step 1: Perform union(3, 4)

   Find roots: find(3) = 2, find(4) = 4. Since rank[2] = 0 and rank[4] = 0, attach 3's root (2) to 4. So parent[2] = 4.

2. Step 2: Perform find(3) with path compression

   find(3) calls find(2), which calls find(4). find(4) returns 4. Path compression sets parent[2] = 4 and parent[3] = parent[2] = 4.

3. Final Answer:

   [0, 0, 4, 4, 4] -> Option C

4. Quick Check:

   Union attaches root, find compresses path [OK]

Hint: Union attaches root; find compresses path to root [OK]

Common Mistakes:

• Not updating parent after union
• Ignoring path compression updates
• Confusing rank usage