Floyd–Warshall Algorithm
Floyd–Warshall computes shortest paths between every pair of vertices with dynamic programming over allowed intermediate vertices.
先从这个心智模型开始
Floyd–Warshall computes shortest paths between every pair of vertices with dynamic programming over allowed intermediate vertices.
它如何工作，不堆砌不必要的术语
For each intermediate k, try improving dist[i][j] via i→k→j.
The triple loop gives O(V³) time and the distance matrix uses O(V²) space.
把算法读成一连串决策
Define directed/undirected and weighted/unweighted assumptions.
Choose the graph representation.
Track algorithm state and handle disconnected/cyclic cases explicitly.
Read a tiny input by hand and write down the state after every important operation. For Floyd–Warshall Algorithm, the goal is to make the invariant visible rather than to memorize code.
O(V³)
Do not quote this complexity mechanically.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
- Use the algorithm only when its graph assumptions match the problem.
通常哪里会出错
- Forgetting disconnected components.
- Mixing weighted and unweighted shortest-path algorithms.
- Not protecting against revisits/cycles.
专业回答结构
Use it when all-pairs shortest paths are needed and V is moderate. It is simple but not ideal for huge sparse graphs.
What problem does Floyd–Warshall Algorithm solve?
Floyd–Warshall computes shortest paths between every pair of vertices with dynamic programming over allowed intermediate vertices.
关于复杂度应该提到什么？
Its typical cost here is O(V³), 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？
Forgetting disconnected components.