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

Recall & Review

beginner

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

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

intermediate

Explain how path compression changes the parent pointers during a find operation.

During find, each node visited on the path to the root is updated to point directly to the root, reducing the tree height.

intermediate

Why does path compression improve the efficiency of Union Find operations?

Because it flattens the tree structure, reducing the time complexity of future find operations to almost constant time (amortized).

beginner

Show the Python code snippet for the find function with path compression.

def find(parent, x): if parent[x] != x: parent[x] = find(parent, parent[x]) # Path compression return parent[x]

intermediate

What is the effect of path compression on the height of the trees in Union Find?

Path compression keeps the trees very flat, often making the height close to 1, which speeds up find operations.

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

A Deletes nodes from the set

B Merges two sets by size

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

D Splits sets into smaller subsets

Path compression updates parent pointers during find to point directly to the root, flattening the tree.

Which operation benefits directly from path compression?

A Union

B Find

C Insert

D Delete

Path compression optimizes the find operation by flattening the tree.

What is the time complexity effect of path compression on Union Find operations?

A Makes operations almost constant time amortized

B Makes operations linear time

C Makes operations logarithmic time

D No effect on time complexity

Path compression reduces the amortized time complexity of find to nearly constant time.

In the find function with path compression, what happens when parent[x] != x?

A Return x immediately

B Set parent[x] to x

C Do nothing

D Set parent[x] to find(parent, parent[x]) recursively

The parent of x is updated to the root found recursively, compressing the path.

Why is path compression important in large Union Find structures?

A It speeds up repeated find operations

B It sorts the elements

C It prevents cycles

D It reduces memory usage

Path compression speeds up repeated find operations by flattening the tree.

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

Think about how the parent pointers change as you find the root.

Explain why path compression improves the efficiency of Union Find data structure.

Consider the shape of the tree before and after path compression.

Practice

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

easy

A. To delete elements from the set efficiently

B. To merge two sets without checking their roots

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

D. To sort the elements in each set

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 C

4. Quick Check:

   Path compression = speed up find [OK]

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

Common Mistakes:

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

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

easy

A. def find(x): parent[x] = x return x

B. def find(x): while parent[x] != x: x = parent[x] return x

C. def find(x): return x if parent[x] == x else find(parent[x])

D. def find(x): if parent[x] != x: parent[x] = find(parent[x]) return parent[x]

Solution

1. Step 1: Check for path compression in the code

   def 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: Verify other options

   def find(x): while parent[x] != x: x = parent[x] return x uses iteration but no path compression update. def find(x): return x if parent[x] == x else find(parent[x]) lacks assignment to parent[x]. def find(x): parent[x] = x return x incorrectly resets parent[x] always.

3. Final Answer:

   def find(x): if parent[x] != x: parent[x] = find(parent[x]) return parent[x] -> Option D

4. Quick Check:

   Path compression requires assignment in find [OK]

Hint: Path compression assigns parent[x] during find [OK]

Common Mistakes:

• Missing assignment to parent[x]
• Using iteration without updating parents
• Resetting parent[x] incorrectly

3. Given the following code snippet with path compression, what will be the output of print(parent) after the operations?

parent = [0, 1, 2, 3, 4]
def find(x):
    if parent[x] != x:
        parent[x] = find(parent[x])
    return parent[x]
def union(a, b):
    rootA = find(a)
    rootB = find(b)
    if rootA != rootB:
        parent[rootB] = rootA

union(1, 2)
union(2, 3)
find(3)
print(parent)

medium

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

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

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

D. [0, 1, 3, 3, 4]

Solution

1. Step 1: Perform unions and track parents

   union(1,2): parent[2] = 1; union(2,3): parent[3] = find(2) = 1, so parent[3] = 1.

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

   find(3) updates parent[3] to root 1 (already 1), so no change. Final parent array: [0,1,1,1,4]

3. Final Answer:

   [0, 1, 1, 1, 4] -> Option A

4. Quick Check:

   Parents after unions and find = [0,1,1,1,4] [OK]

Hint: Trace unions then find with path compression [OK]

Common Mistakes:

• Not updating parent during find
• Confusing root assignments
• Ignoring path compression effect

4. Identify the bug in the following find function that is supposed to implement path compression:

def find(x):
    if parent[x] == x:
        return x
    else:
        find(parent[x])

medium

A. Incorrect base case condition

B. Missing return statement in the else branch

C. Parent array is not initialized

D. Using recursion instead of iteration

Solution

1. Step 1: Check the else branch

   The else branch calls find(parent[x]) but does not return its result, so the function returns None.

2. Step 2: Understand path compression requirement

   Without returning the recursive call result, path compression and correct root finding fail.

3. Final Answer:

   Missing return statement in the else branch -> Option B

4. Quick Check:

   Return recursive find result [OK]

Hint: Always return recursive call in find [OK]

Common Mistakes:

• Forgetting return in recursion
• Misunderstanding base case
• Assuming iteration is required

5. You have a Union Find structure with 5 elements (0 to 4). After performing union(0,1), union(1,2), and union(3,4), you call find(2) and find(4) with path compression. Which of the following statements is true about the parent array after these operations?

hard

A. Elements 0, 1, and 2 will have the same root, and 3 and 4 will have the same root, with path compression applied

B. All elements will have different roots because path compression does not merge sets

C. Elements 0 and 4 will share the same root after path compression

D. Path compression will cause all elements to point to element 0

Solution

1. Step 1: Analyze unions

   union(0,1) and union(1,2) merge 0,1,2 into one set. union(3,4) merges 3 and 4 into another set.

2. Step 2: Effect of find with path compression

   find(2) compresses path so 2 points directly to root of set {0,1,2}. find(4) compresses path so 4 points directly to root of set {3,4}.

3. Step 3: Confirm roots and parent array

   Sets remain separate; 0,1,2 share root; 3,4 share root; path compression flattens trees.

4. Final Answer:

   Elements 0, 1, and 2 will have the same root, and 3 and 4 will have the same root, with path compression applied -> Option A

5. Quick Check:

   Unions merge sets; path compression flattens but does not merge sets [OK]

Hint: Unions merge sets; path compression flattens trees only [OK]

Common Mistakes:

• Thinking path compression merges sets
• Assuming all elements share one root
• Confusing union and find roles