Tarjan's algorithm is a graph theory algorithm used to find the strongly connected components (SCCs) of a directed graph. A strongly connected component is a maximal subgraph where every vertex is reachable from every other vertex within the subgraph. This algorithm is named after its inventor, Robert Tarjan.

Key Principles

The algorithm uses a depth-first search (DFS) to traverse the graph and identify SCCs. It maintains two arrays, disc and low, to keep track of the discovery times of nodes and the lowest points reachable from each node, respectively. Additionally, a stack is used to store the nodes of the current DFS tree.

Algorithm Steps

1. Initialization: Initialize the disc and low arrays to -1, and create an empty stack to store nodes.

2. DFS Traversal: Perform a DFS traversal starting from an arbitrary node. For each node u: Set disc[u] and low[u] to the current time. Push u onto the stack and mark it as part of the current SCC. For each neighbor v of u: If v is not visited, recursively perform DFS on v, and update low[u] based on low[v]. If v is already on the stack, update low[u] based on disc[v]. After processing all neighbors, if disc[u] equals low[u], u is the root of an SCC. Pop all nodes from the stack up to u to form the SCC.