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Tarjan's algorithm is an algorithm in graph theory for finding the strongly connected components of a graph. It runs in linear time, matching the time bound of alternative methods such as Kosaraju's algorithm and the path-based strong component algorithm. Tarjan's Algorithm is named for its discoverer, Robert Tarjan.

Overview
The algorithm takes a directed graph as input, and produces a partition of its nodes into strongly connected components. Each node is assigned to exactly one strongly connected component. Any node not on a directed cycle forms a strongly connected component by itself: a node whose in-degree or out-degree is 0, or any node of an acyclic graph is so.

Each pass of the algorithm collects the strongly-connected components downstream of an arbitrary start node and removes their nodes from the graph. At the start of the algorithm, we have the whole graph and an empty set of strongly connected components. At the end, we have an empty graph and a complete set of strongly connected components. At any stage, we have a set of strongly connected components and a subgraph restricted to (formally, induced by) the nodes outside of them.

The pass conducts a depth-first search from the start node, visiting in turn all nodes downstream of it. As usual, it visits every node exactly once, declining to visit any node already visited. The strongly connected components are recovered as certain paths of this tree. The roots of the sub-trees are called the roots of the strongly connected components. Any node can be a root, which is just the first node of the strongly connected component that the search discovers.

Stack invariant
Nodes are pushed onto a stack as they are visited. When the depth-first search visits a node and recursively visits its descendants, the node is not necessarily popped from the stack when the visit ends. The crucial invariant property is that a node remains on the stack after it has been visited if and only if there is a path from it to some node lower on the stack.

At the end of the call that visits v and its descendants, we know whether v itself has a path to any node lower on the stack. If so, the call returns, leaving v on the stack, preserving the invariant. If not, then v is the root of its strongly connected component, which consists of v together with all the nodes above it on the stack (such nodes all have paths back to v but not to any lower node, because if any of them had a path to a lower node, then v would too, which is false). The connected component rooted at v is then popped from the stack and collected, again preserving the invariant.

Bookkeeping
Each node v Therefore <tt>v</tt> must be left on the stack at the end of the visit if <tt>v.lowlink < v.index</tt>, but removed as the root of a strongly connected component if <tt>v.lowlink = v.index</tt>. The value <tt>v.lowlink</tt> is computed during the depth-first search from <tt>v</tt>, as this finds the nodes that are reachable from <tt>v</tt>.
 * is assigned a unique integer <tt>v.index</tt>, which numbers the nodes consecutively as they are discovered; and
 * maintains a number <tt>v.lowlink</tt> that — throughout the visit — points to the lowest node on the stack found to be reachable from <tt>v</tt>, including <tt>v</tt> itself.

The algorithm in pseudocode
algorithm tarjan is input: graph G = (V, E) output: set of strongly connected components (sets of vertices) index := 0 S    := [] // empty array SCCS := {} // empty set for each v in V do if (v.index is undefined) then strongconnect(v) end if end for function strongconnect(v) v.index  := index v.lowlink := index index    := index + 1

S.push(v) v.onStack := true for each (v, w) in E do // v is bound; w is bound to each successor of v in turn if (w.index is undefined) then strongconnect(w) v.lowlink := min(v.lowlink, w.lowlink) else if (w.onStack) then // w is in stack S, hence not in a previously completed SCC // Note: The next line may look odd - but is correct. // It says w.index not w.lowlink; that is deliberate and from the original paper v.lowlink := min(v.lowlink, w.index) end if end for // If v is a root node, pop the stack and generate an SCC if (v.lowlink = v.index) then SCC := {} repeat w := S.pop w.onStack := false SCC.add(w) add w to current strongly connected component until (w = v) output the current strongly connected component end if end function

The <tt>index</tt> variable is the depth-first search node number counter. <tt>S</tt> is the node stack, which starts out empty and stores the history of nodes explored but not yet committed to a strongly connected component. Note that <tt>S</tt> is not the normal depth-first search stack, as nodes are not popped as the search returns up the tree; they are only popped when an entire strongly connected component has been found.

The outermost loop searches each node that has not yet been visited, ensuring that nodes which are not reachable from the first node are still eventually traversed. The function <tt>strongconnect</tt> performs a single depth-first search of the graph, finding all successors from the node <tt>v</tt>, and reporting all strongly connected components of that subgraph.

When each node finishes recursing, if its lowlink is still set to its index, then it is the root node of a strongly connected component, formed by all of the nodes above it on the stack. The algorithm pops the stack up to and including the current node, and presents all of these nodes as a strongly connected component.

Complexity
Complexity: The Tarjan procedure is called once for each node; the forall statement considers each edge at most once. The algorithm's running time is therefore linear in the number of edges and nodes in G, i.e. $$O(|V|+|E|)$$.

In order to achieve this complexity, the test for whether <tt>w</tt> is on the stack should be done in constant time. This may be done, for example, by storing a flag on each node that indicates whether it is on the stack, and performing this test by examining the flag.

Additional remarks
While there is nothing special about the order of the nodes within each strongly connected component, one useful property of the algorithm is that no strongly connected component will be identified before any of its successors. Therefore, the order in which the strongly connected components are identified constitutes a reverse topological sort of the DAG formed by the strongly connected components.

Donald Knuth described Tarjan's algorithm as one of Knuth's favorite implementations in his book The Stanford GraphBase. He also wrote: "The data structures that he devised for this problem fit together in an amazingly beautiful way, so that the quantities you need to look at while exploring a directed graph are always magically at your fingertips. And his algorithm also does topological sorting as a byproduct."

Overview
The algorithm takes a directed graph as input, and produces a partition of the graph's vertices into the graph's strongly connected components. Each vertex of the graph appears in exactly one of the strongly connected components. Any vertex that is not on a directed cycle forms a strongly connected component all by itself: for example, a vertex whose in-degree or out-degree is 0, or any vertex of an acyclic graph.

The basic idea of the algorithm is this: a depth-first search begins from an arbitrary start node (and subsequent depth-first searches are conducted on any nodes that have not yet been found). As usual with depth-first search, the search visits every node of the graph exactly once, declining to revisit any node that has already been visited. Thus, the collection of search trees is a spanning forest of the graph. The strongly connected components will be recovered as certain subtrees of this forest. The roots of these subtrees are called the "roots" of the strongly connected components. Any node of a strongly connected component might serve as the root, if it happens to be the first node of the component that is discovered by the search.

Stack invariant
Nodes are placed on a stack in the order in which they are visited. When the depth-first search recursively visits a node <tt>v</tt> and its descendants, those nodes are not all necessarily popped from the stack when this recursive call returns. The crucial invariant property is that a node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node lower on the stack.

At the end of the call that visits <tt>v</tt> and its descendants, we know whether <tt>v</tt> itself has a path to any node lower on the stack. If so, the call returns, leaving <tt>v</tt> on the stack, preserving the invariant. If not, then <tt>v</tt> must be the root of its strongly connected component, which consists of <tt>v</tt> together with any nodes higher on the stack than <tt>v</tt> (all such nodes have paths back to <tt>v</tt> but not to any lower node, because if they had paths to lower nodes then <tt>v</tt> would also have paths to lower nodes, which is false). The connected component rooted at <tt>v</tt> is then popped from the stack and returned, again preserving the invariant.

Bookkeeping
Each node v is assigned a unique integer <tt>v.index</tt>, which numbers the nodes consecutively in the order in which they are discovered. It also maintains a value <tt>v.lowlink</tt> that represents (roughly speaking) the smallest index of any node known to be reachable from <tt>v</tt>, including <tt>v</tt> itself. Therefore <tt>v</tt> must be left on the stack if <tt>v.lowlink < v.index</tt>, whereas v must be removed as the root of a strongly connected component if <tt>v.lowlink == v.index</tt>. The value <tt>v.lowlink</tt> is computed during the depth-first search from <tt>v</tt>, as this finds the nodes that are reachable from <tt>v</tt>.

The algorithm in pseudocode
algorithm tarjan is input: graph G = (V, E) output: set of strongly connected components (sets of vertices) index := 0 S := empty array for each v in V do if (v.index is undefined) then strongconnect(v) end if end for function strongconnect(v) // Set the depth index for v to the smallest unused index v.index := index v.lowlink := index index := index + 1 S.push(v) v.onStack := true // Consider successors of v for each (v, w) in E do if (w.index is undefined) then // Successor w has not yet been visited; recurse on it strongconnect(w) v.lowlink := min(v.lowlink, w.lowlink) else if (w.onStack) then // Successor w is in stack S and hence in the current SCC // Note: The next line may look odd - but is correct. // It says w.index not w.lowlink; that is deliberate and from the original paper v.lowlink := min(v.lowlink, w.index) end if end for // If v is a root node, pop the stack and generate an SCC if (v.lowlink = v.index) then start a new strongly connected component repeat w := S.pop w.onStack := false add w to current strongly connected component while (w != v) output the current strongly connected component end if end function

The <tt>index</tt> variable is the depth-first search node number counter. <tt>S</tt> is the node stack, which starts out empty and stores the history of nodes explored but not yet committed to a strongly connected component. Note that this is not the normal depth-first search stack, as nodes are not popped as the search returns up the tree; they are only popped when an entire strongly connected component has been found.

The outermost loop searches each node that has not yet been visited, ensuring that nodes which are not reachable from the first node are still eventually traversed. The function <tt>strongconnect</tt> performs a single depth-first search of the graph, finding all successors from the node <tt>v</tt>, and reporting all strongly connected components of that subgraph.

When each node finishes recursing, if its lowlink is still set to its index, then it is the root node of a strongly connected component, formed by all of the nodes above it on the stack. The algorithm pops the stack up to and including the current node, and presents all of these nodes as a strongly connected component.

Complexity
Complexity: The Tarjan procedure is called once for each node; the forall statement considers each edge at most once. The algorithm's running time is therefore linear in the number of edges and nodes in G, i.e. $$O(|V|+|E|)$$.

In order to achieve this complexity, the test for whether <tt>w</tt> is on the stack should be done in constant time. This may be done, for example, by storing a flag on each node that indicates whether it is on the stack, and performing this test by examining the flag.

Additional remarks
While there is nothing special about the order of the nodes within each strongly connected component, one useful property of the algorithm is that no strongly connected component will be identified before any of its successors. Therefore, the order in which the strongly connected components are identified constitutes a reverse topological sort of the DAG formed by the strongly connected components.

Donald Knuth described Tarjan's algorithm as one of Knuth's favorite implementations in his book The Stanford GraphBase. He also wrote: "The data structures that he devised for this problem fit together in an amazingly beautiful way, so that the quantities you need to look at while exploring a directed graph are always magically at your fingertips. And his algorithm also does topological sorting as a byproduct."