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Advanced Topic 41 of 51 Algorithms

Strongly Connected Components

Maximal vertex groups mutually reachable in a directed graph.

How it works

A strongly connected component of a directed graph is a maximal set of vertices where every vertex can reach every other. Kosaraju's algorithm runs DFS to get finishing times, then DFS on the reversed graph in decreasing order, and Tarjan's algorithm finds components in one DFS using low-link values. Both run in O(V + E), and contracting components gives a directed acyclic graph.

Step-by-step walkthrough

  1. In a directed graph A reaches B.
  2. B reaches C.
  3. C reaches back to A, so A B C are mutually reachable.
  4. D can be reached but cannot get back.
  5. Tarjan or Kosaraju finds each component in linear time, here {A B C} and {D}.

Where it is used

Analyzing web link structure, finding cycles of dependencies, solving 2-SAT problems, compiler optimization, model checking and simplifying graphs before topological sorting use strongly connected components.

Common pitfall

Applying the undirected notion of connectivity to directed graphs gives wrong answers. Recursive implementations can overflow the stack on large graphs. In Kosaraju's algorithm the order of the second pass is crucial. Tarjan's low-link updates must use the stack membership correctly.

Where it fits in the course

Strongly Connected Components is topic 41 of 51 in Algorithms, rated Advanced. It follows Prim's Algorithm. The next topic is Maximum Flow.

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 Strongly Connected Components?
Maximal vertex groups mutually reachable in a directed graph. It belongs to Algorithms and is an advanced-level topic.
Where do people use Strongly Connected Components?
Analyzing web link structure, finding cycles of dependencies, solving 2-SAT problems, compiler optimization, model checking and simplifying graphs before topological sorting use strongly connected components.
What is the most common mistake with Strongly Connected Components?
Applying the undirected notion of connectivity to directed graphs gives wrong answers.
What should I learn before this topic?
Start with Prim's 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.
Is Strongly Connected Components 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 Strongly Connected Components?
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 Strongly Connected Components?
Prim's Algorithm, Maximum Flow, Kruskal's Algorithm are closely connected. The full list is in the related topics section of this page.
What should I study after Strongly Connected Components?
Continue with Maximum Flow, the next topic in Algorithms.

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