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Advanced Topic 36 of 51 Algorithms

Floyd-Warshall Algorithm

All-pairs shortest paths via dynamic programming.

How it works

Floyd-Warshall computes shortest paths between all pairs of vertices with three nested loops, allowing each vertex k in turn as an intermediate point and updating dist[i][j] to min(dist[i][j], dist[i][k] + dist[k][j]). It runs in O(V cubed) time and O(V squared) space, handles negative weights and detects negative cycles via negative diagonal entries.

Where it is used

All-pairs routing tables, transitive closure of relations, finding the graph diameter, computing reachability matrices and small dense graphs in network analysis use Floyd-Warshall.

Common pitfall

The loop order matters: the intermediate vertex k must be the outermost loop. Infinity plus a value can overflow integers, so guard it. It is too slow for large sparse graphs, where running Dijkstra from every vertex is better. Reconstructing paths needs a next-hop matrix.

Where it fits in the course

Floyd-Warshall Algorithm is topic 36 of 51 in Algorithms, rated Advanced. It follows Bellman-Ford Algorithm. The next topic is A* Search.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Algorithms quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Floyd-Warshall Algorithm?
All-pairs shortest paths via dynamic programming. It belongs to Algorithms and is an advanced-level topic.
Where do people use Floyd-Warshall Algorithm?
All-pairs routing tables, transitive closure of relations, finding the graph diameter, computing reachability matrices and small dense graphs in network analysis use Floyd-Warshall.
What is the most common mistake with Floyd-Warshall Algorithm?
The loop order matters: the intermediate vertex k must be the outermost loop.
What should I learn before this topic?
Start with Bellman-Ford Algorithm, the topic just before it in Algorithms.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Algorithms quiz. You can also explore Graph Traversal in the Visual Lab.
Is Floyd-Warshall Algorithm hard to learn?
It is an Advanced topic. Expect to read the explanation more than once and to connect it with earlier topics. Working through a small example by hand helps most.
How long does it take to learn Floyd-Warshall Algorithm?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Floyd-Warshall Algorithm?
Bellman-Ford Algorithm, A* Search, Dijkstra's Algorithm are closely connected. The full list is in the related topics section of this page.
What should I study after Floyd-Warshall Algorithm?
Continue with A* Search, the next topic in Algorithms.

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