Disjoint Set
Disjoint Set / Union-Find tracks a collection of non-overlapping groups and supports two key operations: find which group an item belongs to, and union two groups.
Start with this mental model
Disjoint Set / Union-Find tracks a collection of non-overlapping groups and supports two key operations: find which group an item belongs to, and union two groups.
How it works, without unnecessary jargon
Each set has a representative root.
Path compression and union by rank/size make repeated operations almost constant amortized time.
Read the algorithm as a sequence of decisions
Identify the invariant that must always remain true.
Follow the read/update path through the structure.
After a mutation, restore the invariant before returning.
Read a tiny input by hand and write down the state after every important operation. For Disjoint Set, the goal is to make the invariant visible rather than to memorize code.
amortized near O(1)
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
Good fit
- Kruskal MST
- Connectivity queries
- Cycle detection in undirected graphs
What usually goes wrong
- Memorizing complexity without understanding representation.
- Ignoring worst-case shape or resizing/rebalancing costs.
A professional answer shape
Name both optimizations—path compression and union by rank/size—and explain that the amortized complexity is inverse-Ackermann, effectively constant for practical sizes.
What problem does Disjoint Set solve?
Disjoint Set / Union-Find tracks a collection of non-overlapping groups and supports two key operations: find which group an item belongs to, and union two groups.
What should I mention about complexity?
Its typical cost here is amortized near O(1), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
What is the most common mistake?
Memorizing complexity without understanding representation.