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

Recall & Review

beginner

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

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

beginner

In Union Find, what does the 'find' function return?

The 'find' function returns the root or representative of the set that a particular element belongs to.

intermediate

How does path compression affect the time complexity of Union Find operations?

Path compression reduces the time complexity of 'find' and 'union' operations to nearly O(1) amortized, making the structure very efficient for many queries.

intermediate

Explain the key step in the path compression technique during the 'find' operation.

During 'find', after recursively finding the root, each node on the path is updated to point directly to the root, shortening the path for future finds.

beginner

Why is path compression important in real-life applications of Union Find?

Because it makes repeated queries on connected components very fast, which is useful in network connectivity, image processing, and clustering problems.

What does path compression do in the Union Find data structure?

A Merges two sets by size

B Splits a set into two

C Deletes nodes from the set

D Makes every node on the find path point directly to the root

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

Which operation benefits directly from path compression?

A Find

B Union

C Insert

D Delete

The 'find' operation is optimized by path compression, making it faster by flattening the tree.

What is the typical time complexity of Union Find with path compression for m operations on n elements?

A O(m α(n)) where α is the inverse Ackermann function

B O(m n)

C O(m log n)

D O(m^2)

With path compression, the amortized time complexity is almost constant, O(m α(n)), where α(n) grows very slowly.

In path compression, when is the parent pointer updated?

A Before recursive find call

B Never updated

C After recursive find call returns the root

D Only during union operation

Parent pointers are updated after the recursive find call returns the root, pointing nodes directly to the root.

Which of these is NOT a benefit of path compression?

A Faster repeated find operations

B Increased memory usage

C Flatter tree structure

D Improved amortized time complexity

Path compression does not increase memory usage; it optimizes the structure without extra memory.

Describe how path compression works during the find operation in Union Find.

Think about what happens to nodes on the path after finding the root.

Explain why path compression improves the efficiency of Union Find operations.

Consider how the structure changes after multiple find operations.

Practice

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

easy

A. To merge two sets by size

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

C. To initialize the parent array

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 B

4. Quick Check:

   Path compression = flatten tree for faster find [OK]

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

Common Mistakes:

• Confusing path compression with union by size/rank
• Thinking it merges sets instead of optimizing find
• Assuming it initializes data structures

2. Which of the following is the correct Java method signature for the find function implementing path compression in Union Find?

easy

A. public int find(int x)

B. public void find(int x)

C. public boolean find(int x)

D. public int find()"

Solution

1. Step 1: Understand the return type of find

   The find method returns the representative (root) of the set containing element x, so it returns an integer.

2. Step 2: Check method parameters

   The method must accept the element x as input to find its root.

3. Final Answer:

   public int find(int x) -> Option A

4. Quick Check:

   find returns root index as int [OK]

Hint: find returns int root and takes int parameter [OK]

Common Mistakes:

• Using void return type for find
• Missing parameter for element to find
• Returning boolean instead of int

3. Consider the following Java code snippet implementing path compression in Union Find:

int[] parent = {0, 0, 1, 2, 3};

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

System.out.println(find(4));
System.out.println(Arrays.toString(parent));

What is the output?

medium

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

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

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

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

Solution

1. Step 1: Trace find(4) call with path compression

   parent[4]=3 -> find(3); parent[3]=2 -> find(2); parent[2]=1 -> find(1); parent[1]=0 -> find(0)=0 (base); parent[1]=0; parent[2]=0; parent[3]=0; parent[4]=0. Returns 0.

2. Step 2: Update parent array after path compression

   Entire path flattens: parent = [0, 0, 0, 0, 0].

3. Final Answer:

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

4. Quick Check:

   find(4) = 0, all path nodes point to root 0 [OK]

Hint: Path compression flattens entire path to root [OK]

Common Mistakes:

• Thinking only leaf updates (partial compression)
• Expecting no change in parent array
• Misreading recursion depth or base case

4. The following find method is intended to implement path compression but has a bug:

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

What is the bug and how to fix it?

medium

A. Missing path compression assignment; fix by adding parent[x] = find(parent[x]);

B. Incorrect base case; fix by changing parent[x] == x to parent[x] != x

C. Wrong return type; fix by changing return type to void

D. No bug; code is correct

Solution

1. Step 1: Identify missing path compression step

   The method recursively finds the root but does not update parent[x] to root, so no path compression happens.

2. Step 2: Fix by assigning compressed parent

   Adding parent[x] = find(parent[x]); before returning compresses the path.

3. Final Answer:

   Missing path compression assignment; fix by adding parent[x] = find(parent[x]); -> Option A

4. Quick Check:

   Assign parent[x] = find(parent[x]) for compression [OK]

Hint: Always assign parent[x] = find(parent[x]) to compress path [OK]

Common Mistakes:

• Forgetting to update parent[x]
• Changing base case incorrectly
• Returning void instead of int

5. Given a Union Find structure with parent = {0, 0, 1, 2, 4, 5} and rank = {0, 1, 2, 3, 4, 5}, after performing union(4, 5) and then find(5) with path compression, what will be the updated parent array?

hard

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

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

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

D. [0, 0, 1, 2, 5, 5]

Solution

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

   Ranks: rank[4]=4, rank[5]=5. Since rank[5] > rank[4], parent[4] = 5. But ranks given are increasing, so actually rank[5] is highest, so parent[4] = 5.

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

   find(5) returns 5 (root). Path compression updates parent of 5 to itself (no change). But parent[4] points to 5, so parent array is [0,0,1,2,5,5].

3. Step 3: Check options

   [0, 0, 1, 2, 4, 4] shows parent[5] = 4, which contradicts union by rank. [0, 0, 1, 2, 3, 4] no change. [0, 0, 1, 2, 3, 3] parent[5]=3 incorrect. [0, 0, 1, 2, 5, 5] parent[4]=5 and parent[5]=5 matches expected.

4. Final Answer:

   [0, 0, 1, 2, 5, 5] -> Option D

5. Quick Check:

   Union by rank sets parent of lower rank to higher [OK]

Hint: Union by rank attaches lower rank root to higher rank root [OK]

Common Mistakes:

• Confusing which parent to update in union
• Ignoring rank comparison
• Assuming parent array unchanged after union