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.
先从这个心智模型开始
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.
它如何工作，不堆砌不必要的术语
Each set has a representative root.
Path compression and union by rank/size make repeated operations almost constant amortized time.
把算法读成一连串决策
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Do not quote this complexity mechanically.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.
适用场景
- Kruskal MST
- Connectivity queries
- Cycle detection in undirected graphs
通常哪里会出错
- Memorizing complexity without understanding representation.
- Ignoring worst-case shape or resizing/rebalancing costs.
专业回答结构
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.
关于复杂度应该提到什么？
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.
最常见的错误是什么？
Memorizing complexity without understanding representation.