If you are unfamiliar with graphs, check out some of my earlier posts on them.

## Resources:

Takeaways:

- A
**strongly connected component**in a directed graph is a partition or sub-graph where each vertex of the component is reachable from every other vertex in the component. - Strongly connected components are always the
*maximal*sub-graph, meaning none of their vertices are part of another strongly connected component. - Finding strongly connected components is very similar to finding articulation points & bridges. Many of the solutions for finding them involve
**depth-first search**(DFS). - One way to find strongly connected components is using
**Tarjan's Algorithm**. - Tarjan's Algorithm uses DFS and a stack to find strongly connected components.
- The algorithm overview is:
- For every unvisited vertex
`v`

in the graph, perform DFS. - At the start of each DFS routine, mark the current vertex
`v`

as visited, push`v`

onto the stack, and assign`v`

an ID and low-link value. - Initially, like with articulation points & bridges, the ID and low-link values of
`v`

will be the same. - Increment the integer value we are using as the ID/low-link seed. So the next vertex
`w`

to enter our DFS routine will get a higher value for both (compared to`v`

). - For every adjacent vertex
`u`

of`v`

, if`u`

is unvisited, perform DFS (recursive call). - On exit of the recursive DFS call, or if
`u`

had already been visited, check if`u`

is on the stack. - If
`u`

is on the stack, update the low-link value of`v`

to be smallest between`low-link[v]`

and`low-link[u]`

- low-link represents the
*earliest*ancestor of a vertex (a lower value means the vertex entered DFS earlier). In other words, low-link is the ID of the earliest vertex`y`

vertex`v`

is connected to.

- low-link represents the
- After exploring all adjacent vertices of
`v`

check if the ID of`v`

is the same as it's low-link value - If they are the same, then
`v`

is the root of a strongly connected component. - If they are the same, pop all vertices off the stack up to and including
`v`

. All of these vertices represent a single strongly connected component - Complete the above steps for the entire graph, and all strongly connected components will be discovered.

- For every unvisited vertex
- Time complexity of Tarjan's Algorithm is
`O(v + e)`

- where`v`

is the number of vertices, and`e`

the number of edges, in a graph. - Space is
`O(v)`

Below are implementations for finding strongly connected components in undirected adjacency list & adjacency matrix representations of graphs:

As always, if you found any errors in this post please let me know!

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