Path Compression in Union Find in DSA Typescript - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of find operation with path compression

What is the output of the following TypeScript code that uses path compression in a Union Find structure?

DSA Typescript

class UnionFind {
  parent: number[];
  constructor(size: number) {
    this.parent = Array.from({ length: size }, (_, i) => i);
  }
  find(x: number): number {
    if (this.parent[x] !== x) {
      this.parent[x] = this.find(this.parent[x]);
    }
    return this.parent[x];
  }
  union(a: number, b: number): void {
    const rootA = this.find(a);
    const rootB = this.find(b);
    if (rootA !== rootB) {
      this.parent[rootB] = rootA;
    }
  }
}

const uf = new UnionFind(5);
uf.union(0, 1);
uf.union(1, 2);
uf.union(3, 4);
console.log(uf.find(2));
console.log(uf.parent);

A

2
[0, 1, 2, 3, 4]

B

0
[0, 0, 0, 3, 3]

C

1
[0, 0, 1, 3, 3]

D

3
[0, 1, 2, 3, 4]

💡 Hint

Remember that path compression updates the parent of each node visited during find to the root.

The find operation with path compression sets the parent of nodes 1 and 2 directly to 0, the root of their set. So uf.find(2) returns 0, and the parent array updates to [0, 0, 0, 3, 3].

🧠 Conceptual

intermediate

Effect of path compression on parent array

After performing union(0,1), union(1,2), and then calling find(2) with path compression, what will be the state of the parent array for a Union Find of size 4 initialized as [0,1,2,3]?

A [0, 0, 0, 3]

B [0, 1, 0, 3]

C [0, 1, 2, 3]

D [1, 1, 1, 3]

💡 Hint

Path compression sets all nodes on the find path directly to the root.

Calling find(2) compresses the path so that nodes 1 and 2 point directly to 0, the root. So parent array becomes [0, 0, 0, 3].

🔧 Debug

advanced

Identify the error in path compression implementation

What error will the following TypeScript code produce when calling uf.find(2)?

DSA Typescript

class UnionFind {
  parent: number[];
  constructor(size: number) {
    this.parent = Array.from({ length: size }, (_, i) => i);
  }
  find(x: number): number {
    if (this.parent[x] !== x) {
      this.find(this.parent[x]);
    }
    return this.parent[x];
  }
}

const uf = new UnionFind(3);
uf.parent = [0, 2, 1];
console.log(uf.find(2));

A Returns 0 without error

B TypeError: Cannot read property of undefined

C Stack overflow due to infinite recursion

D Returns 2 without error

💡 Hint

Check if the recursive call updates the parent array.

The find method calls itself recursively but does not update parent[x] with the result, causing infinite recursion when parent[x] != x.

❓ Predict Output

advanced

Parent array after multiple unions and finds with path compression

What is the parent array after executing the following code?

DSA Typescript

class UnionFind {
  parent: number[];
  constructor(size: number) {
    this.parent = Array.from({ length: size }, (_, i) => i);
  }
  find(x: number): number {
    if (this.parent[x] !== x) {
      this.parent[x] = this.find(this.parent[x]);
    }
    return this.parent[x];
  }
  union(a: number, b: number): void {
    const rootA = this.find(a);
    const rootB = this.find(b);
    if (rootA !== rootB) {
      this.parent[rootB] = rootA;
    }
  }
}

const uf = new UnionFind(6);
uf.union(0, 1);
uf.union(1, 2);
uf.union(3, 4);
uf.union(4, 5);
uf.union(2, 5);
uf.find(5);
uf.find(4);
console.log(uf.parent);

A [0, 0, 0, 0, 0, 0]

B [0, 0, 0, 0, 3, 3]

C [0, 1, 2, 3, 4, 5]

D [0, 0, 0, 3, 3, 0]

💡 Hint

The union operations connect all nodes into one set with root 0. Path compression updates parents.

All nodes 0-5 become connected with root 0. The find(5) compresses paths so all parents point to 0.

🧠 Conceptual

expert

Time complexity impact of path compression in Union Find

Which statement best describes the effect of path compression on the time complexity of Union Find operations?

A Path compression changes the space complexity of Union Find from linear to logarithmic.

B Path compression increases the worst-case time complexity of union operations to linear time.

C Path compression has no effect on the time complexity of find or union operations.

D Path compression reduces the amortized time complexity of find operations to nearly constant, specifically inverse Ackermann function time.

💡 Hint

Consider how path compression flattens the tree structure over time.

Path compression flattens the tree, making future find operations very fast. The amortized time per operation becomes almost constant, bounded by the inverse Ackermann function.

Practice

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

easy

A. To sort elements inside each set

B. To speed up the find operation by flattening the tree

C. To delete elements from the set efficiently

D. To merge two sets without checking their roots

Solution

1. Step 1: Understand the find operation

   The find operation locates the root of an element's set in Union Find.

2. Step 2: Role of path compression

   Path compression flattens the tree by making nodes point directly to the root, speeding up future find calls.

3. Final Answer:

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

4. Quick Check:

   Path compression = faster find [OK]

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

Common Mistakes:

• Confusing path compression with union operation
• Thinking it deletes elements
• Assuming it sorts elements

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

easy

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

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

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

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

Solution

1. Step 1: Identify path compression in code

   Path compression updates parent[x] to root during recursion.

2. Step 2: Check each option

   function find(x: number): number { if (parent[x] !== x) parent[x] = find(parent[x]); return parent[x]; } updates parent[x] recursively, implementing path compression correctly.

3. Final Answer:

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

4. Quick Check:

   Path compression updates parent[x] recursively [OK]

Hint: Look for recursive parent update in find [OK]

Common Mistakes:

• Missing parent update for path compression
• Using loop without updating parent
• Returning without compression

3. Given the following TypeScript 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: number): number {
  if (parent[x] !== x) parent[x] = find(parent[x]);
  return parent[x];
}
find(3);
console.log(parent);

medium

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

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

C. [0, 0, 2, 2, 4]

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

Solution

1. Step 1: Trace find(3) calls

   find(3) checks parent[3] = 2, calls find(2). find(2) checks parent[2] = 1, calls find(1). find(1) checks parent[1] = 0, calls find(0). find(0) returns 0 since parent[0] = 0.

2. Step 2: Apply path compression updates

   Unwinding: parent[1] = 0 (already), returns 0; parent[2] = 0, returns 0; parent[3] = 0, returns 0. Final parent = [0, 0, 0, 0, 4].

3. Final Answer:

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

4. Quick Check:

   Path compression sets all path nodes to root [OK]

Hint: Follow recursive calls and update parents on return [OK]

Common Mistakes:

• Not updating parent array after recursion
• Confusing indices and values
• Assuming all parents become root immediately

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

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

medium

A. Missing assignment to parent[x] for path compression

B. Incorrect base case condition

C. Function should return void

D. Using recursion without a return statement

Solution

1. Step 1: Analyze recursion and return

   The function calls find(parent[x]) but does not assign its result back to parent[x].

2. Step 2: Identify missing path compression update

   Without parent[x] = find(parent[x]), path compression does not happen, so the tree is not flattened.

3. Final Answer:

   Missing assignment to parent[x] for path compression -> Option A

4. Quick Check:

   Path compression requires updating parent[x] [OK]

Hint: Ensure recursive find result is assigned to parent[x] [OK]

Common Mistakes:

• Forgetting to assign recursive call result
• Returning without recursion
• Wrong base case logic

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

hard

A. [0, 0, 1, 2, 3, 4]

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

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

D. [0, 0, 1, 2, 3, 0]

Solution

1. Step 1: Perform union(4, 5)

   Initial parent = [0, 0, 1, 2, 3, 5]. Simple union links by setting parent[5] = 4. Now parent = [0, 0, 1, 2, 3, 4].

2. Step 2: Call find(5) with path compression

   find(5): parent[5]=4 -> find(4): parent[4]=3 -> find(3)=2 -> find(2)=1 -> find(1)=0 -> find(0)=0.

3. Step 3: Update parent array after compression

   Unwinding sets parent[4]=0, parent[3]=0, parent[2]=0, parent[1]=0, but parent[5] remains 4 because find(5) returns 0 but parent[5] was set to 4 and not updated in this code snippet. So after path compression, parent array is [0, 0, 1, 2, 0, 4] if path compression is applied only on the path from 5 to root. However, since find(5) calls find(4), which calls find(3), etc., and path compression updates parent[x] = find(parent[x]) recursively, parent[5] should be updated to 0 as well. So final parent array is [0, 0, 0, 0, 0, 0].

4. Final Answer:

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

5. Quick Check:

   Union sets and compress path update parents [OK]

Hint: Union sets then compress path to update parents [OK]

Common Mistakes:

• Not updating parent after union
• Ignoring path compression updates
• Assuming parent stays same after find