Path Compression in Union Find in DSA Typescript

Mental Model

Path compression makes finding the group leader faster by flattening the tree structure during searches.

Analogy: Imagine a family tree where each person points to their parent. Path compression is like updating everyone on the path to point directly to the oldest ancestor, so next time you find the ancestor faster.

Before path compression:
1 -> 2 -> 3 -> 4 -> null
After path compression:
1 -> 4
2 -> 4
3 -> 4
4 -> null

Dry Run Walkthrough

Input: Union-Find with elements 1 to 4, unions: union(1,2), union(2,3), union(3,4), then find(1)

Goal: Show how path compression flattens the tree when finding the leader of element 1

Step 1: union(1,2) links 1's leader to 2

1 -> 2 -> null
2 -> null
3 -> null
4 -> null

Why: We connect 1 and 2 so they share the same leader

Step 2: union(2,3) links 2's leader to 3

1 -> 2 -> 3 -> null
2 -> 3 -> null
3 -> null
4 -> null

Why: We connect 2 and 3, extending the chain

Step 3: union(3,4) links 3's leader to 4

1 -> 2 -> 3 -> 4 -> null
2 -> 3 -> 4 -> null
3 -> 4 -> null
4 -> null

Why: We connect 3 and 4, making 4 the leader

Step 4: find(1) traverses from 1 to 4, compressing path

1 -> 4
2 -> 4
3 -> 4
4 -> null

Why: Path compression updates all nodes on path to point directly to leader 4

Result:

1 -> 4
2 -> 4
3 -> 4
4 -> null
Leader of 1 is 4

Annotated Code

DSA Typescript

class UnionFind {
  parent: number[];

  constructor(size: number) {
    this.parent = Array(size + 1).fill(0).map((_, i) => i);
  }

  find(x: number): number {
    if (this.parent[x] !== x) {
      this.parent[x] = this.find(this.parent[x]); // path compression
    }
    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[rootA] = rootB; // link rootA to rootB
    }
  }
}

// Driver code
const uf = new UnionFind(4);
uf.union(1, 2);
uf.union(2, 3);
uf.union(3, 4);

console.log('Parent array after unions:', uf.parent.slice(1).join(' '));

const leader = uf.find(1);
console.log('Leader of 1 is', leader);
console.log('Parent array after path compression:', uf.parent.slice(1).join(' '));

if (this.parent[x] !== x) {

check if current node is not leader

this.parent[x] = this.find(this.parent[x]); // path compression

recursively find leader and update current node to point directly to leader

return this.parent[x];

return leader of current node

const rootA = this.find(a);

find leader of element a

const rootB = this.find(b);

find leader of element b

if (rootA !== rootB) {

only union if leaders differ

this.parent[rootA] = rootB; // link rootA to rootB

attach rootA's tree under rootB to merge sets

Output

Parent array after unions: 2 3 4 4

Leader of 1 is 4

Parent array after path compression: 4 4 4 4

Complexity Analysis

Time: O(α(n)) per find/union operation because path compression flattens trees making future finds almost constant time

Space: O(n) for storing parent array for n elements

vs Alternative: Without path compression, find can be O(n) in worst case due to tall trees; path compression reduces this to nearly O(1)

Edge Cases

Finding leader of an element that is its own leader

Returns the element itself immediately without recursion

DSA Typescript

if (this.parent[x] !== x) {

Union of two elements already in the same set

No change to parent array as leaders are equal

DSA Typescript

if (rootA !== rootB) {

Union and find on single element set

Find returns element itself; union links sets correctly

DSA Typescript

if (this.parent[x] !== x) {

When to Use This Pattern

When you see problems about grouping elements and quickly checking if two elements belong to the same group, use Union Find with path compression to speed up repeated queries.

Common Mistakes

Mistake: Not updating the parent during find, missing path compression
Fix: Assign parent[x] = find(parent[x]) to flatten the tree

Mistake: Linking parents incorrectly in union causing cycles
Fix: Always link root of one set to root of another after finding leaders

Summary

Path compression speeds up finding the leader by making nodes point directly to the leader.

Use it when you need fast repeated union and find operations on disjoint sets.

The key insight is flattening the tree during find to make future finds faster.