Floyd–Warshall Algorithm
Floyd–Warshall computes shortest paths between every pair of vertices with dynamic programming over allowed intermediate vertices.
Start with this mental model
Floyd–Warshall computes shortest paths between every pair of vertices with dynamic programming over allowed intermediate vertices.
How it works, without unnecessary jargon
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.
Read the algorithm as a sequence of decisions
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Do not quote this complexity mechanically.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.
Good fit
- Use the algorithm only when its graph assumptions match the problem.
What usually goes wrong
- Forgetting disconnected components.
- Mixing weighted and unweighted shortest-path algorithms.
- Not protecting against revisits/cycles.
A professional answer shape
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.
What should I mention about complexity?
Its typical cost here is O(V³), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
What is the most common mistake?
Forgetting disconnected components.