A disjoint-set (union-find) tracks elements partitioned into non-overlapping groups and answers "are these two in the same group?" and "" in . With და , both operations run in — effectively O(1) (α is the inverse Ackermann function).
A disjoint-set (union-find) tracks elements partitioned into non-overlapping groups and answers "are these two in the same group?" and "" in . With და , both operations run in — effectively O(1) (α is the inverse Ackermann function).
Each set is a tree; the root is the set's representative. find walks to the root; union links one root under another.
find(x): follow parents to the root
union(a,b): attach the shorter tree under the taller (by rank)
path compression: after find, point nodes DIRECTLY at the root
before: a->b->c->root after: a->root, b->root, c->root
class DSU:
def __init__(self, n):
self.parent = list(range(n))
self.rank = [0]*n
def find(self, x): # path compression
while self.parent[x] != x:
self.parent[x] = self.parent[self.parent[x]] # halve path
x = self.parent[x]
return x
def union(self, a, b): # union by rank
ra, rb = self.find(a), self.find(b)
if ra == rb: return False
if self.rank[ra] < self.rank[rb]: ra, rb = rb, ra
self.parent[rb] = ra
if self.rank[ra] == self.rank[rb]: self.rank[ra] += 1
return True
| Operation | Time (with both optimizations) |
|---|---|
| find | O(α(n)) ≈ O(1) |
| union | O(α(n)) ≈ O(1) |
Union-find solves grouping and connectivity problems that would otherwise need repeated O(n) graph traversals, collapsing them to almost O(1) per query.
It is a go-to tool whenever you must incrementally merge sets and test connectivity — a frequent senior-level interview topic.