# Tarjan

Tarjan
You are encouraged to solve this task according to the task description, using any language you may know.
 This page uses content from Wikipedia. The original article was at Graph. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

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 for alternative methods including Kosaraju's algorithm and the path-based strong component algorithm.

Tarjan's Algorithm is named for its discoverer, Robert Tarjan.

References

## 11l

Translation of: Python: As class

<lang 11l>T Graph

```  String name
[Char = [Char]] graph
Int _order
[Char = Int] disc
[Char = Int] low
[Char] stack
Char scc
```
```  F (name, connections)
.name = name
```
```     DefaultDict[Char, [Char]] g
L(n) connections
V (n1, n2) = (n[0], n[1])
I n1 != n2
g[n1].append(n2)
E
g[n1]
g[n2]
```
```     .graph = Dict(move(g))
.tarjan_algo()
```
```  F _visitor(this) -> N
‘
Recursive function that finds SCC's
using DFS traversal of vertices.
```
```       Arguments:
this        --> Vertex to be visited in this call.
disc{}      --> Discovery order of visited vertices.
low{}       --> Connected vertex of earliest discovery order
stack       --> Ancestor node stack during DFS.
’
.disc[this] = .low[this] = ._order
._order++
.stack.append(this)
```
```     L(neighbr) .graph[this]
I neighbr !C .disc
._visitor(neighbr)
.low[this] = min(.low[this], .low[neighbr])
```
```        E I neighbr C .stack
.low[this] = min(.low[this], .disc[neighbr])
```
```     I .low[this] == .disc[this]
[Char] new
L
V top = .stack.pop()
new.append(top)
I top == this
L.break
.scc.append(new)
```
```  F tarjan_algo()
‘
Recursive function that finds strongly connected components
using the Tarjan Algorithm and function _visitor() to visit nodes.
’
._order = 0
```
```     L(vertex) sorted(.graph.keys())
I vertex !C .disc
._visitor(vertex)
```

L(n, m) [(‘Tx1’, ‘10 02 21 03 34’.split(‘ ’)),

```        (‘Tx2’, ‘01 12 23’.split(‘ ’)),
(‘Tx3’, ‘01 12 20 13 14 16 35 45’.split(‘ ’)),
(‘Tx4’, ‘01 03 12 14 20 26 32 45 46 56 57 58 59 64 79 89 98 AA’.split(‘ ’)),
(‘Tx5’, ‘01 12 23 24 30 42’.split(‘ ’))
]
print("\n\nGraph('#.', #.):\n".format(n, m))
V g = Graph(n, m)
print(‘               :  ’sorted(g.disc.keys()).map(v -> String(v)).join(‘  ’))
print(‘    DISC of ’(g.name‘:’)‘ ’sorted(g.disc.items()).map((_, v) -> v))
print(‘     LOW of ’(g.name‘:’)‘ ’sorted(g.low.items()).map((_, v) -> v))
V scc = (I !g.scc.empty {String(g.scc).replace(‘'’, ‘’).replace(‘,’, ‘’)[1 .< (len)-1]} E ‘’)
print("\n   SCC's of "(g.name‘:’)‘ ’scc)</lang>
```
Output:
```

Graph('Tx1', [10, 02, 21, 03, 34]):

:  0  1  2  3  4
DISC of Tx1: [0, 2, 1, 3, 4]
LOW of Tx1: [0, 0, 0, 3, 4]

SCC's of Tx1: [4] [3] [1 2 0]

Graph('Tx2', [01, 12, 23]):

:  0  1  2  3
DISC of Tx2: [0, 1, 2, 3]
LOW of Tx2: [0, 1, 2, 3]

SCC's of Tx2: [3] [2] [1] [0]

Graph('Tx3', [01, 12, 20, 13, 14, 16, 35, 45]):

:  0  1  2  3  4  5  6
DISC of Tx3: [0, 1, 2, 3, 5, 4, 6]
LOW of Tx3: [0, 0, 0, 3, 5, 4, 6]

SCC's of Tx3: [5] [3] [4] [6] [2 1 0]

Graph('Tx4', [01, 03, 12, 14, 20, 26, 32, 45, 46, 56, 57, 58, 59, 64, 79, 89, 98, AA]):

:  0  1  2  3  4  5  6  7  8  9  A
DISC of Tx4: [0, 1, 2, 9, 4, 5, 3, 6, 8, 7, 10]
LOW of Tx4: [0, 0, 0, 2, 3, 3, 3, 6, 7, 7, 10]

SCC's of Tx4: [8 9] [7] [5 4 6] [3 2 1 0] [A]

Graph('Tx5', [01, 12, 23, 24, 30, 42]):

:  0  1  2  3  4
DISC of Tx5: [0, 1, 2, 3, 4]
LOW of Tx5: [0, 0, 0, 0, 2]

SCC's of Tx5: [4 3 2 1 0]
```

## C

<lang c>#include <stddef.h>

1. include <stdlib.h>
2. include <stdbool.h>
1. ifndef min
2. define min(x, y) ((x)<(y) ? (x) : (y))
3. endif

struct edge { void *from; void *to; };

struct components { int nnodes; void **nodes; struct components *next; };

struct node { int index; int lowlink; bool onStack; void *data; };

struct tjstate { int index; int sp; int nedges; struct edge *edges; struct node **stack; struct components *head; struct components *tail; };

static int nodecmp(const void *l, const void *r) { return (ptrdiff_t)l -(ptrdiff_t)((struct node *)r)->data; }

static int strongconnect(struct node *v, struct tjstate *tj) { struct node *w;

/* Set the depth index for v to the smallest unused index */ v->index = tj->index; v->lowlink = tj->index; tj->index++; tj->stack[tj->sp] = v; tj->sp++; v->onStack = true;

for (int i = 0; i<tj->nedges; i++) { /* Only consider nodes reachable from v */ if (tj->edges[i].from != v) { continue; } w = tj->edges[i].to; /* Successor w has not yet been visited; recurse on it */ if (w->index == -1) { int r = strongconnect(w, tj); if (r != 0) return r; v->lowlink = min(v->lowlink, w->lowlink); /* Successor w is in stack S and hence in the current SCC */ } else if (w->onStack) { v->lowlink = min(v->lowlink, w->index); } }

/* If v is a root node, pop the stack and generate an SCC */ if (v->lowlink == v->index) { struct components *ng = malloc(sizeof(struct components)); if (ng == NULL) { return 2; } if (tj->tail == NULL) { tj->head = ng; } else { tj->tail->next = ng; } tj->tail = ng; ng->next = NULL; ng->nnodes = 0; do { tj->sp--; w = tj->stack[tj->sp]; w->onStack = false; ng->nnodes++; } while (w != v); ng->nodes = malloc(ng->nnodes*sizeof(void *)); if (ng == NULL) { return 2; } for (int i = 0; i<ng->nnodes; i++) { ng->nodes[i] = tj->stack[tj->sp+i]->data; } } return 0; }

static int ptrcmp(const void *l, const void *r) { return (ptrdiff_t)((struct node *)l)->data - (ptrdiff_t)((struct node *)r)->data; }

/**

```* Calculate the strongly connected components using Tarjan's algorithm:
* en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm
*
* Returns NULL when there are invalid edges and sets the error to:
* 1 if there was a malformed edge
* 2 if malloc failed
*
* @param number of nodes
* @param data of the nodes (assumed to be unique)
* @param number of edges
* @param data of edges
* @param pointer to error code
*/
```

struct components *tarjans( int nnodes, void *nodedata[], int nedges, struct edge *edgedata[], int *error) { struct node nodes[nnodes]; struct edge edges[nedges]; struct node *stack[nnodes]; struct node *from, *to; struct tjstate tj = {0, 0, nedges, edges, stack, NULL, .tail=NULL};

// Populate the nodes for (int i = 0; i<nnodes; i++) { nodes[i] = (struct node){-1, -1, false, nodedata[i]}; } qsort(nodes, nnodes, sizeof(struct node), ptrcmp);

// Populate the edges for (int i = 0; i<nedges; i++) { from = bsearch(edgedata[i]->from, nodes, nnodes, sizeof(struct node), nodecmp); if (from == NULL) { *error = 1; return NULL; } to = bsearch(edgedata[i]->to, nodes, nnodes, sizeof(struct node), nodecmp); if (to == NULL) { *error = 1; return NULL; } edges[i] = (struct edge){.from=from, .to=to}; }

//Tarjan's for (int i = 0; i < nnodes; i++) { if (nodes[i].index == -1) { *error = strongconnect(&nodes[i], &tj); if (*error != 0) return NULL; } } return tj.head; } </lang>

## C#

<lang csharp>using System; using System.Collections.Generic;

class Node {

```   public int LowLink { get; set; }
public int Index { get; set; }
public int N { get; }
```
```   public Node(int n)
{
N = n;
Index = -1;
}
```

}

class Graph {

```   public HashSet<Node> V { get; }
public Dictionary<Node, HashSet<Node>> Adj { get; }
```
```   /// <summary>
/// Tarjan's strongly connected components algorithm
/// </summary>
public void Tarjan()
{
var index = 0; // number of nodes
var S = new Stack<Node>();
```
```       Action<Node> StrongConnect = null;
StrongConnect = (v) =>
{
// Set the depth index for v to the smallest unused index
v.Index = index;
```
```           index++;
S.Push(v);
```
```           // Consider successors of v
if (w.Index < 0)
{
// Successor w has not yet been visited; recurse on it
StrongConnect(w);
}
else if (S.Contains(w))
// Successor w is in stack S and hence in the current SCC
```
```           // If v is a root node, pop the stack and generate an SCC
{
Console.Write("SCC: ");
```
```               Node w;
do
{
w = S.Pop();
Console.Write(w.N + " ");
} while (w != v);
```
```               Console.WriteLine();
}
};
```
```       foreach (var v in V)
if (v.Index < 0)
StrongConnect(v);
}
```

}</lang>

## C++

<lang cpp>// // C++ implementation of Tarjan's strongly connected components algorithm // See https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm //

1. include <algorithm>
2. include <iostream>
3. include <list>
4. include <string>
5. include <vector>

struct noncopyable {

```   noncopyable() {}
noncopyable(const noncopyable&) = delete;
noncopyable& operator=(const noncopyable&) = delete;
```

};

template <typename T> class tarjan;

template <typename T> class vertex : private noncopyable { public:

```   explicit vertex(const T& t) : data_(t) {}
neighbours_.push_back(v);
}
neighbours_.insert(neighbours_.end(), vs);
}
const T& get_data() {
return data_;
}
```

private:

```   friend tarjan<T>;
T data_;
int index_ = -1;
bool on_stack_ = false;
std::vector<vertex*> neighbours_;
```

};

template <typename T> class graph : private noncopyable { public:

```   vertex<T>* add_vertex(const T& t) {
vertexes_.emplace_back(t);
return &vertexes_.back();
}
```

private:

```   friend tarjan<T>;
std::list<vertex<T>> vertexes_;
```

};

template <typename T> class tarjan  : private noncopyable { public:

```   using component = std::vector<vertex<T>*>;
std::list<component> run(graph<T>& graph) {
index_ = 0;
stack_.clear();
strongly_connected_.clear();
for (auto& v : graph.vertexes_) {
if (v.index_ == -1)
strongconnect(&v);
}
return strongly_connected_;
}
```

private:

```   void strongconnect(vertex<T>* v) {
v->index_ = index_;
++index_;
stack_.push_back(v);
v->on_stack_ = true;
for (auto w : v->neighbours_) {
if (w->index_ == -1) {
strongconnect(w);
}
else if (w->on_stack_) {
}
}
strongly_connected_.push_back(component());
component& c = strongly_connected_.back();
for (;;) {
auto w = stack_.back();
stack_.pop_back();
w->on_stack_ = false;
c.push_back(w);
if (w == v)
break;
}
}
}
int index_ = 0;
std::list<vertex<T>*> stack_;
std::list<component> strongly_connected_;
```

};

template <typename T> void print_vector(const std::vector<vertex<T>*>& vec) {

```   if (!vec.empty()) {
auto i = vec.begin();
std::cout << (*i)->get_data();
for (++i; i != vec.end(); ++i)
std::cout << ' ' << (*i)->get_data();
}
std::cout << '\n';
```

}

int main() {

```   graph<std::string> g;
```
```   andy->add_neighbour(bart);
```
```   tarjan<std::string> t;
for (auto&& s : t.run(g))
print_vector(s);
return 0;
```

}</lang>

Output:
```Carl Bart Andy
Gary Fred
Earl Dave
Hank
```

## Go

<lang go>package main

import (

```   "fmt"
"math/big"
```

)

// (same data as zkl example) var g = [][]int{

```   0: {1},
2: {0},
5: {2, 6},
6: {5},
1: {2},
3: {1, 2, 4},
4: {5, 3},
7: {4, 7, 6},
```

}

func main() {

```   tarjan(g, func(c []int) { fmt.Println(c) })
```

}

// the function calls the emit argument for each component identified. // each component is a list of nodes. func tarjan(g [][]int, emit func([]int)) {

```   var indexed, stacked big.Int
index := make([]int, len(g))
x := 0
var S []int
var sc func(int) bool
sc = func(n int) bool {
index[n] = x
indexed.SetBit(&indexed, n, 1)
x++
S = append(S, n)
stacked.SetBit(&stacked, n, 1)
for _, nb := range g[n] {
if indexed.Bit(nb) == 0 {
if !sc(nb) {
return false
}
}
} else if stacked.Bit(nb) == 1 {
}
}
}
var c []int
for {
last := len(S) - 1
w := S[last]
S = S[:last]
stacked.SetBit(&stacked, w, 0)
c = append(c, w)
if w == n {
emit(c)
break
}
}
}
return true
}
for n := range g {
if indexed.Bit(n) == 0 && !sc(n) {
return
}
}
```

}</lang>

Output:
```[2 1 0]
[6 5]
[4 3]
[7]
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq

In this adaptation of the #Kotlin/#Wren implementations:

• a Node is represented by JSON object with .n being its id;
• a DirectedGraph is represented by a JSON object {vs, es} where vs is an array of Nodes and es is an array of integer ids;
• a Stack is reprsented by an array, with .[-1] as the active point;
• the output of `tarjan` is an array each item of which is an array of Node ids.

<lang jq># Input: an integer def Node:

``` { n: .,
index: -1,    # -1 signifies undefined
onStack: false
} ;
```
1. Input: a DirectedGraph
2. Output: a stream of Node ids

def successors(\$vn): .es[\$vn][];

1. Input: a DirectedGraph
2. Output: an array of integer arrays

def tarjan:

``` . + { sccs: [],    # strongly connected components
index: 0,
s: []        # Stack
}
# input: {es, vs, sccs, index, s}
| def strongConnect(\$vn):
# Set the depth index for v to the smallest unused index
.vs[\$vn].index = .index
| .index += 1
| .s += [ \$vn ]
| .vs[\$vn].onStack = true

# consider successors of v
| reduce successors(\$vn) as \$wn (.;
if .vs[\$wn].index < 0
```

then

```               # Successor w has not yet been visited; recurse on it
strongConnect(\$wn)
elif .vs[\$wn].onStack
```

then

```               # Successor w is in stack s and hence in the current SCC
```

else .

```           end
)
# If v is a root node, pop the stack and generate an SCC
| if .vs[\$vn] | (.lowLink == .index)
then .scc = []
```

| .stop = false

```           | until(.stop;
.s[-1] as \$wn
```

| .s |= .[:-1] # pop

```               | .vs[\$wn].onStack = false
| .scc += [\$wn]
| if \$wn == \$vn then .stop = true else . end )
| .sccs += [.scc]
else .scc = []
```

end

```   ;

reduce .vs[].n as \$vn (.;
if .vs[\$vn].index < 0
then strongConnect(\$vn)
else . end
)
| .sccs
```
1. Vertices

def vs: [range(0;8) | Node ];

1. Edges

def es:

``` [ [1],
[2],
[0],
[1, 2, 4],
[5, 3],
[2, 6],
[5],
[4, 7, 6]
]
```

{ vs: vs, es: es } | tarjan</lang>

Output:
```[[2,1,0],[6,5],[4,3],[7]]
```

## Julia

LightGraphs uses Tarjan's algorithm by default. The package can also use Kosaraju's algorithm with the function strongly_connected_components_kosaraju(). <lang julia>using LightGraphs

edge_list=[(1,2),(3,1),(6,3),(6,7),(7,6),(2,3),(4,2),(4,3),(4,5),(5,6),(5,4),(8,5),(8,8),(8,7)]

grph = SimpleDiGraph(Edge.(edge_list))

tarj = strongly_connected_components(grph)

inzerobase(arrarr) = map(x -> sort(x .- 1, rev=true), arrarr)

println("Results in the zero-base scheme: \$(inzerobase(tarj))")

</lang>
Output:
```Results in the zero-base scheme: Array{Int64,1}[[2, 1, 0], [6, 5], [4, 3], [7]]
```

## Kotlin

<lang scala>// version 1.1.3

import java.util.Stack

typealias Nodes = List<Node>

class Node(val n: Int) {

```   var index   = -1  // -1 signifies undefined
var onStack = false
```
```   override fun toString()  = n.toString()
```

}

class DirectedGraph(val vs: Nodes, val es: Map<Node, Nodes>)

fun tarjan(g: DirectedGraph): List<Nodes> {

```   val sccs = mutableListOf<Nodes>()
var index = 0
val s = Stack<Node>()
```
```   fun strongConnect(v: Node) {
// Set the depth index for v to the smallest unused index
v.index = index
index++
s.push(v)
v.onStack = true
```
```       // consider successors of v
for (w in g.es[v]!!) {
if (w.index < 0) {
// Successor w has not yet been visited; recurse on it
strongConnect(w)
}
else if (w.onStack) {
// Successor w is in stack s and hence in the current SCC
}
}
```
```       // If v is a root node, pop the stack and generate an SCC
val scc = mutableListOf<Node>()
do {
val w = s.pop()
w.onStack = false
}
while (w != v)
}
}
```
```   for (v in g.vs) if (v.index < 0) strongConnect(v)
return sccs
```

}

fun main(args: Array<String>) {

```   val vs = (0..7).map { Node(it) }
val es = mapOf(
vs[0] to listOf(vs[1]),
vs[2] to listOf(vs[0]),
vs[5] to listOf(vs[2], vs[6]),
vs[6] to listOf(vs[5]),
vs[1] to listOf(vs[2]),
vs[3] to listOf(vs[1], vs[2], vs[4]),
vs[4] to listOf(vs[5], vs[3]),
vs[7] to listOf(vs[4], vs[7], vs[6])
)
val g = DirectedGraph(vs, es)
val sccs = tarjan(g)
println(sccs.joinToString("\n"))
```

}</lang>

Output:
```[2, 1, 0]
[6, 5]
[4, 3]
[7]
```

## Nim

Translation of: Kotlin

<lang Nim>import sequtils, strutils, tables

type

``` Node = ref object
val: int
index: int
onStack: bool
```
``` Nodes = seq[Node]
```
``` DirectedGraph = object
nodes: seq[Node]
edges: Table[int, Nodes]
```

func initNode(n: int): Node =

``` Node(val: n, index: -1, lowLink: -1, onStack: false)
```

func `\$`(node: Node): string = \$node.val

func tarjan(g: DirectedGraph): seq[Nodes] =

``` var index = 0
var s: seq[Node]
var sccs: seq[Nodes]
```

``` func strongConnect(v: Node) =
```
```   # Set the depth index for "v" to the smallest unused index.
v.index = index
inc index
v.onStack = true
```
```   # Consider successors of "v".
for w in g.edges[v.val]:
if w.index < 0:
# Successor "w" has not yet been visited; recurse on it.
w.strongConnect()
elif w.onStack:
# Successor "w" is in stack "s" and hence in the current SCC.
```
```   # If "v" is a root node, pop the stack and generate an SCC.
var scc: Nodes
while true:
let w = s.pop()
w.onStack = false
if w == v: break
```

``` for v in g.nodes:
if v.index < 0:
v.strongConnect()
result = move(sccs)
```

when isMainModule:

``` let vs = toSeq(0..7).map(initNode)
let es = {0: @[vs[1]],
1: @[vs[2]],
2: @[vs[0]],
3: @[vs[1], vs[2], vs[4]],
4: @[vs[5], vs[3]],
5: @[vs[2], vs[6]],
6: @[vs[5]],
7: @[vs[4], vs[7], vs[6]]}.toTable
var g = DirectedGraph(nodes: vs, edges: es)
let sccs = g.tarjan()
echo sccs.join("\n")</lang>
```
Output:
```@[2, 1, 0]
@[6, 5]
@[4, 3]
@[7]```

## Perl

Translation of: Raku

<lang perl>use strict; use warnings; use feature <say state current_sub>; use List::Util qw(min);

sub tarjan {

```   my (%k) = @_;
my (%onstack, %index, %lowlink, @stack, @connected);
```
```   my sub strong_connect {
my (\$vertex, \$i) = @_;
\$index{\$vertex}   = \$i;
\$onstack{\$vertex} = 1;
push @stack, \$vertex;
for my \$connection (@{\$k{\$vertex}}) {
if (not defined \$index{\$connection}) {
__SUB__->(\$connection, \$i + 1);
}
elsif (\$onstack{\$connection}) {
}
}
my @node;
do {
push @node, pop @stack;
\$onstack{\$node[-1]} = 0;
} while \$node[-1] ne \$vertex;
push @connected, [@node];
}
}
```
```   for (sort keys %k) {
strong_connect(\$_, 0) unless \$index{\$_};
}
@connected;
```

}

my %test1 = (

```            0 => [1],
1 => [2],
2 => [0],
3 => [1, 2, 4],
4 => [3, 5],
5 => [2, 6],
6 => [5],
7 => [4, 6, 7]
);
```

my %test2 = (

```            'Andy' => ['Bart'],
'Bart' => ['Carl'],
'Carl' => ['Andy'],
'Dave' => [qw<Bart Carl Earl>],
'Earl' => [qw<Dave Fred>],
'Fred' => [qw<Carl Gary>],
'Gary' => ['Fred'],
'Hank' => [qw<Earl Gary Hank>]
);
```

print "Strongly connected components:\n"; print join(', ', sort @\$_) . "\n" for tarjan(%test1); print "\nStrongly connected components:\n"; print join(', ', sort @\$_) . "\n" for tarjan(%test2);</lang>

Output:
```Strongly connected components:
0, 1, 2
5, 6
3, 4
7

Strongly connected components:
Andy, Bart, Carl
Fred, Gary
Dave, Earl
Hank```

## Phix

Translation of: Go

Same data as other examples, but with 1-based indexes. <lang Phix>constant g = {{2}, {3}, {1}, {2,3,5}, {6,4}, {3,7}, {6}, {5,8,7}}

sequence index, lowlink, stacked, stack integer x

function strong_connect(integer n, r_emit)

```   index[n] = x
stacked[n] = 1
stack &= n
x += 1
for b=1 to length(g[n]) do
integer nb = g[n][b]
if index[nb] == 0 then
if not strong_connect(nb,r_emit) then
return false
end if
end if
elsif stacked[nb] == 1 then
end if
end if
end for
sequence c = {}
while true do
integer w := stack[\$]
stack = stack[1..\$-1]
stacked[w] = 0
c = prepend(c, w)
if w == n then
call_proc(r_emit,{c})
exit
end if
end while
end if
return true
```

end function

procedure tarjan(sequence g, integer r_emit)

```   index   = repeat(0,length(g))
stacked = repeat(0,length(g))
stack = {}
x := 1
for n=1 to length(g) do
if index[n] == 0
and not strong_connect(n,r_emit) then
return
end if
end for
```

end procedure

procedure emit(object c) -- called for each component identified. -- each component is a list of nodes.

```   ?c
```

end procedure

tarjan(g,routine_id("emit"))</lang>

Output:
```{1,2,3}
{6,7}
{4,5}
{8}
```

## Python

### Python: As function

<lang python>from collections import defaultdict

def from_edges(edges):

```   translate list of edges to list of nodes
```
```   class Node:
def __init__(self):
# root is one of:
#   None: not yet visited
#   non-negative integer: what Wikipedia pseudo code calls 'lowlink'
self.root = None
self.succ = []
```
```   nodes = defaultdict(Node)
for v,w in edges:
nodes[v].succ.append(nodes[w])
```
```   for i,v in nodes.items(): # name the nodes for final output
v.id = i
```
```   return nodes.values()
```

def trajan(V):

```   def strongconnect(v, S):
v.root = pos = len(S)
S.append(v)
```
```       for w in v.succ:
if w.root is None:  # not yet visited
yield from strongconnect(w, S)
```
```           if w.root >= 0:  # still on stack
v.root = min(v.root, w.root)
```
```       if v.root == pos:  # v is the root, return everything above
res, S[pos:] = S[pos:], []
for w in res:
w.root = -1
yield [r.id for r in res]
```
```   for v in V:
if v.root is None:
yield from strongconnect(v, [])
```

tables = [ # table 1

```           [(1,2), (3,1), (3,6), (6,7), (7,6), (2,3), (4,2),
(4,3), (4,5), (5,6), (5,4), (8,5), (8,7), (8,6)],
```
```           # table 2
[('A', 'B'), ('B', 'C'), ('C', 'A'), ('A', 'Other')]]
```

for table in (tables):

```   for g in trajan(from_edges(table)):
print(g)
print()</lang>
```
Output:
```[6, 7]
[1, 2, 3]
[4, 5]
[8]

['Other']
['A', 'B', 'C']
```

### Python: As class

This takes inspiration from the Geeks4Geeks explanation and uses its five examples.

Tx1
```+---+     +---+     +---+     +---+
| 1 | --> | 0 | --> | 3 | --> | 4 |
+---+     +---+     +---+     +---+
^         |
|         |
|         v
|       +---+
+------ | 2 |
+---+
```
Tx2
```+---+     +---+     +---+     +---+
| 0 | --> | 1 | --> | 2 | --> | 3 |
+---+     +---+     +---+     +---+
```
Tx3
```
+----------------------------------+
v                                  |
+---+     +---+     +---+     +---+  |
| 0 | --> |   | --> | 3 | --> | 5 |  |
+---+     |   |     +---+     +---+  |
|   |                 ^    |
| 1 |                 |    |
|   |                 |    |
+---+     |   |     +---+       |    |
| 6 | <-- |   | --> | 2 | ------+----+
+---+     +---+     +---+       |
|                   |
|                   |
v                   |
+---+                 |
| 4 | ----------------+
+---+
```
Tx4
```
+-----------------------------+
|                             |
|       +---+                 |
|       | A |                 |         +-------------------+
|       +---+                 |         |                   |
|                             |         |                   |
|                   +---------+---------+---------+         |                   +---------+
|                   |         v         v         v         |                   v         |
+---+     +---+     +---+     +---+     +---+     +---+     +---+     +---+     +---+     +---+
| 3 | <-- | 0 | --> | 1 | --> | 2 | --> | 6 | --> | 4 | --> |   | --> | 7 | --> | 9 | --> | 8 |
+---+     +---+     +---+     +---+     +---+     +---+     |   |     +---+     +---+     +---+
^                   |         ^         |       |   |                 ^         ^
+-------------------+         +---------+       | 5 | ----------------+         |
|   |                           |
|   |                           |
|   | --------------------------+
+---+
```
Tx5
```
+--------------+
|              |
|              |
+-------------------+---------+    |
v                   v         |    |
+---+     +---+     +---+     +---+  |
| 0 | --> | 1 | --> | 2 | --> | 3 |  |
+---+     +---+     +---+     +---+  |
|              |
|              |
v              |
+---+            |
| 4 | -----------+
+---+
```
Code

<lang python>from collections import defaultdict

class Graph:

```   "Directed Graph Tarjan's strongly connected components algorithm"
```
```   def __init__(self, name, connections):
self.name = name
self.connections = connections
g = defaultdict(list)  # map node vertex to direct connections
for n1, n2 in connections:
if n1 != n2:
g[n1].append(n2)
else:
g[n1]
for _, n2 in connections:
g[n2]   # For leaf nodes having no edges from themselves
self.graph = dict(g)
self.tarjan_algo()
```
```   def _visitor(self, this, low, disc, stack):

Recursive function that finds SCC's
using DFS traversal of vertices.
```
```       Arguments:
this        --> Vertex to be visited in this call.
disc{}      --> Discovery order of visited vertices.
low{}       --> Connected vertex of earliest discovery order
stack       --> Ancestor node stack during DFS.

```
```       disc[this] = low[this] = self._order
self._order += 1
stack.append(this)
```
```       for neighbr in self.graph[this]:
if neighbr not in disc:
# neighbour not visited so do DFS recurrence.
self._visitor(neighbr, low, disc, stack)
low[this] = min(low[this], low[neighbr])  # Prior connection?
```
```           elif neighbr in stack:
# Update low value of this only if neighbr in stack
low[this] = min(low[this], disc[neighbr])
```
```       if low[this] == disc[this]:
# Head node found of SCC
top, new = None, []
while top != this:
top = stack.pop()
new.append(top)
self.scc.append(new)
```
```   def tarjan_algo(self):

Recursive function that finds strongly connected components
using the Tarjan Algorithm and function _visitor() to visit nodes.

```
```       self._order = 0         # Visitation order counter
disc, low = {}, {}
stack = []
```
```       self.scc = []           # SCC result accumulator
for vertex in sorted(self.graph):
if vertex not in disc:
self._visitor(vertex, low, disc, stack)
self._disc, self._low = disc, low
```

if __name__ == '__main__':

```   for n, m in [('Tx1', '10 02 21 03 34'.split()),
('Tx2', '01 12 23'.split()),
('Tx3', '01 12 20 13 14 16 35 45'.split()),
('Tx4', '01 03 12 14 20 26 32 45 46 56 57 58 59 64 79 89 98 AA'.split()),
('Tx5', '01 12 23 24 30 42'.split()),
]:
print(f"\n\nGraph({repr(n)}, {m}):\n")
g = Graph(n, m)
print("               : ", '  '.join(str(v) for v in sorted(g._disc)))
print("    DISC of", g.name + ':', [v for _, v in sorted(g._disc.items())])
print("     LOW of", g.name + ':', [v for _, v in sorted(g._low.items())])
scc = repr(g.scc if g.scc else ).replace("'", ).replace(',', )[1:-1]
print("\n   SCC's of", g.name + ':', scc)</lang>
```
Output:
```Graph('Tx1', ['10', '02', '21', '03', '34']):

:  0  1  2  3  4
DISC of Tx1: [0, 2, 1, 3, 4]
LOW of Tx1: [0, 0, 0, 3, 4]

SCC's of Tx1: [4] [3] [1 2 0]

Graph('Tx2', ['01', '12', '23']):

:  0  1  2  3
DISC of Tx2: [0, 1, 2, 3]
LOW of Tx2: [0, 1, 2, 3]

SCC's of Tx2: [3] [2] [1] [0]

Graph('Tx3', ['01', '12', '20', '13', '14', '16', '35', '45']):

:  0  1  2  3  4  5  6
DISC of Tx3: [0, 1, 2, 3, 5, 4, 6]
LOW of Tx3: [0, 0, 0, 3, 5, 4, 6]

SCC's of Tx3: [5] [3] [4] [6] [2 1 0]

Graph('Tx4', ['01', '03', '12', '14', '20', '26', '32', '45', '46', '56', '57', '58', '59', '64', '79', '89', '98', 'AA']):

:  0  1  2  3  4  5  6  7  8  9  A
DISC of Tx4: [0, 1, 2, 9, 4, 5, 3, 6, 8, 7, 10]
LOW of Tx4: [0, 0, 0, 2, 3, 3, 3, 6, 7, 7, 10]

SCC's of Tx4: [8 9] [7] [5 4 6] [3 2 1 0] [A]

Graph('Tx5', ['01', '12', '23', '24', '30', '42']):

:  0  1  2  3  4
DISC of Tx5: [0, 1, 2, 3, 4]
LOW of Tx5: [0, 0, 0, 0, 2]

SCC's of Tx5: [4 3 2 1 0]```

## Racket

### Manual implementation

Translation of: Kotlin

<lang racket>#lang racket

(require syntax/parse/define

```        fancy-app
(for-syntax racket/syntax))
```

(struct node (name index low-link on?) #:transparent #:mutable

``` #:methods gen:custom-write
[(define (write-proc v port mode) (fprintf port "~a" (node-name v)))])
```

(define-syntax-parser change!

``` [(_ x:id f) #'(set! x (f x))]
[(_ accessor:id v f)
#:with mutator! (format-id this-syntax "set-~a!" #'accessor)
#'(mutator! v (f (accessor v)))])
```

(define (tarjan g)

``` (define sccs '())
(define index 0)
(define s '())
```
``` (define (dfs v)
(set-node-index! v index)
(set-node-on?! v #t)
(change! s (cons v _))
```
```   (for ([w (in-list (hash-ref g v '()))])
(match-define (node _ index low-link on?) w)
(cond
[(not index) (dfs w)
[on? (change! node-low-link v (min index _))]))
```
```   (when (= (node-low-link v) (node-index v))
(define-values (scc* s*) (splitf-at s (λ (w) (not (eq? w v)))))
(set! s (rest s*))
(define scc (cons (first s*) scc*))
(for ([w (in-list scc)]) (set-node-on?! w #f))
(change! sccs (cons scc _))))
```
``` (for* ([(u _) (in-hash g)] #:when (not (node-index u))) (dfs u))
sccs)
```

(define (make-graph xs)

``` (define store (make-hash))
(define (make-node v) (hash-ref! store v (thunk (node v #f #f #f))))
```
``` ;; it's important that we use hasheq instead of hash so that we compare
;; reference instead of actual value. Had we use the actual value,
;; the key would be a mutable value, which causes undefined behavior
(for/hasheq ([vs (in-list xs)]) (values (make-node (first vs)) (map make-node (rest vs)))))
```

(tarjan (make-graph '([0 1]

```                     [2 0]
[5 2 6]
[6 5]
[1 2]
[3 1 2 4]
[4 5 3]
[7 4 7 6])))</lang>
```
Output:
```'((7) (3 4) (5 6) (2 1 0))
```

### With the graph library

<lang racket>#lang racket

(require graph)

```                                 [2 0]
[5 2 6]
[6 5]
[1 2]
[3 1 2 4]
[4 5 3]
[7 4 7 6])))
```

(scc g)</lang>

Output:
```'((7) (3 4) (5 6) (1 0 2))
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2018.09

<lang perl6>sub tarjan (%k) {

```   my %onstack;
my %index;
my @stack;
my @connected;
```
```   sub strong-connect (\$vertex) {
state \$index      = 0;
%index{\$vertex}   = \$index;
%onstack{\$vertex} = True;
@stack.push: \$vertex;
for |%k{\$vertex} -> \$connection {
if not %index{\$connection}.defined {
strong-connect(\$connection);
}
elsif %onstack{\$connection} {
}
}
my @node;
repeat {
@node.push: @stack.pop;
%onstack{@node.tail} = False;
} while @node.tail ne \$vertex;
@connected.push: @node;
}
}
```
```   .&strong-connect unless %index{\$_} for %k.keys;
```
```   @connected
```

}

1. TESTING

-> \$test { say "\nStrongly connected components: ", |tarjan(\$test).sort».sort } for

1. hash of vertex, edge list pairs

(((1),(2),(0),(1,2,4),(3,5),(2,6),(5),(4,6,7)).pairs.hash),

1. Same layout test data with named vertices instead of numbered.

%(:Andy<Bart>,

``` :Bart<Carl>,
:Carl<Andy>,
:Dave<Bart Carl Earl>,
:Earl<Dave Fred>,
:Fred<Carl Gary>,
:Gary<Fred>,
:Hank<Earl Gary Hank>
```

)</lang>

Output:
```Strongly connected components: (0 1 2)(3 4)(5 6)(7)

Strongly connected components: (Andy Bart Carl)(Dave Earl)(Fred Gary)(Hank)```

## REXX

<lang rexx>/* REXX - Tarjan's Algorithm */ /* Vertices are numbered 1 to 8 (instead of 0 to 7) */ g='[2] [3] [1] [2 3 5] [4 6] [3 7] [6] [5 7 8]' gg=g Do i=1 By 1 While gg>

``` Parse Var gg '[' g.i ']' gg
name.i=i-1
End
```

g.0=i-1 index.=0 lowlink.=0 stacked.=0 stack.=0 x=1 Do n=1 To g.0

``` If index.n=0 Then
If strong_connect(n)=0 Then
Return
End
```

Exit

strong_connect: Procedure Expose x g. index. lowlink. stacked. stack. name. Parse Arg n index.n = x lowlink.n = x stacked.n = 1 Call stack n x=x+1 Do b=1 To words(g.n)

``` Call show_all
nb=word(g.n,b)
If index.nb=0 Then Do
If strong_connect(nb)=0 Then
Return 0
End
Else Do
If stacked.nb = 1 Then
end
end
```

``` c=
Do z=stack.0 By -1
w=stack.z
stacked.w=0
stack.0=stack.0-1
c=name.w c
If w=n Then Do
Say '['space(c)']'
Return 1
End
End
End
```

Return 1

stack: Procedure Expose stack. Parse Arg m z=stack.0+1 stack.z=m stack.0=z Return

/* The following were used for debugging (and understanding) */

show_all: Return ind='Index ' low='Lowlink' sta='Stacked' Do z=1 To g.0

``` ind=ind index.z
sta=sta stacked.z
End
```

Say ind Say low Say sta Return

show_stack: ol='Stack' Do z=1 To stack.0

``` ol=ol stack.z
End
```

Say ol Return</lang>

Output:
```[0 1 2]
[5 6]
[3 4]
[7]```

## Rust

<lang Rust>use std::collections::{BTreeMap, BTreeSet};

// Using a naked BTreeMap would not be very nice, so let's make a simple graph representation

1. [derive(Clone, Debug)]

pub struct Graph {

```   neighbors: BTreeMap<usize, BTreeSet<usize>>,
```

}

impl Graph {

```   pub fn new(size: usize) -> Self {
Self {
neighbors: (0..size).fold(BTreeMap::new(), |mut acc, x| {
acc.insert(x, BTreeSet::new());
acc
}),
}
}
```
```   pub fn edges<'a>(&'a self, vertex: usize) -> impl Iterator<Item = usize> + 'a {
self.neighbors[&vertex].iter().cloned()
}
```
```   pub fn add_edge(&mut self, from: usize, to: usize) {
assert!(to < self.len());
self.neighbors.get_mut(&from).unwrap().insert(to);
}
```
```   pub fn add_edges(&mut self, from: usize, to: impl IntoIterator<Item = usize>) {
let limit = self.len();
```
```       self.neighbors
.get_mut(&from)
.unwrap()
.extend(to.into_iter().filter(|x| {
assert!(*x < limit);
true
}));
}
```
```   pub fn is_empty(&self) -> bool {
self.neighbors.is_empty()
}
```
```   pub fn len(&self) -> usize {
self.neighbors.len()
}
```

}

1. [derive(Clone)]

struct VertexState {

```   index: usize,
on_stack: bool,
```

}

// The easy way not to fight with Rust's borrow checker is moving the state in // a structure and simply invoke methods on that structure.

pub struct Tarjan<'a> {

```   graph: &'a Graph,
index: usize,
stack: Vec<usize>,
state: Vec<VertexState>,
components: Vec<BTreeSet<usize>>,
```

}

impl<'a> Tarjan<'a> {

```   // Having index: Option<usize> would look nicer, but requires just
// some unwraps and Vec has actual len limit isize::MAX anyway, so
// we can reserve this large value as the invalid one.
const INVALID_INDEX: usize = usize::MAX;
```
```   pub fn walk(graph: &'a Graph) -> Vec<BTreeSet<usize>> {
Self {
graph,
index: 0,
stack: Vec::new(),
state: vec![
VertexState {
index: Self::INVALID_INDEX,
on_stack: false
};
graph.len()
],
components: Vec::new(),
}
.visit_all()
}
```
```   fn visit_all(mut self) -> Vec<BTreeSet<usize>> {
for vertex in 0..self.graph.len() {
if self.state[vertex].index == Self::INVALID_INDEX {
self.visit(vertex);
}
}
```
```       self.components
}
```
```   fn visit(&mut self, v: usize) {
let v_ref = &mut self.state[v];
v_ref.index = self.index;
self.index += 1;
self.stack.push(v);
v_ref.on_stack = true;
```
```       for w in self.graph.edges(v) {
let w_ref = &self.state[w];
if w_ref.index == Self::INVALID_INDEX {
self.visit(w);
let v_ref = &mut self.state[v];
} else if w_ref.on_stack {
let w_index = self.state[w].index;
let v_ref = &mut self.state[v];
}
}
```
```       let v_ref = &self.state[v];
let mut component = BTreeSet::new();
```
```           loop {
let w = self.stack.pop().unwrap();
self.state[w].on_stack = false;
component.insert(w);
if w == v {
break;
}
}
```
```           self.components.push(component);
}
}
```

}

fn main() {

```   let graph = {
let mut g = Graph::new(8);
g
};
```
```   for component in Tarjan::walk(&graph) {
println!("{:?}", component);
}
```

}</lang>

Output:
```{0, 1, 2}
{5, 6}
{3, 4}
{7}
```

## Sidef

Translation of: Raku

<lang ruby>func tarjan (k) {

```   var(:onstack, :index, :lowlink, *stack, *connected)
```
```   func strong_connect (vertex, i=0) {
```
```        index{vertex}   = i
onstack{vertex} = true
stack << vertex
```
```        for connection in (k{vertex}) {
if (index{connection} == nil) {
strong_connect(connection, i+1)
}
elsif (onstack{connection}) {
}
}
```
```       if (lowlink{vertex} == index{vertex}) {
var *node
do {
node << stack.pop
onstack{node.tail} = false
} while (node.tail != vertex)
connected << node
}
}
```
```   { strong_connect(_) if !index{_} } << k.keys
```
```   return connected
```

}

var tests = [

```   Hash(
0 => <1>,
1 => <2>,
2 => <0>,
3 => <1 2 4>,
4 => <3 5>,
5 => <2 6>,
6 => <5>,
7 => <4 6 7>,
),
Hash(
:Andy => <Bart>,
:Bart => <Carl>,
:Carl => <Andy>,
:Dave => <Bart Carl Earl>,
:Earl => <Dave Fred>,
:Fred => <Carl Gary>,
:Gary => <Fred>,
:Hank => <Earl Gary Hank>,
)
```

]

tests.each {|t|

```   say ("Strongly connected components: ", tarjan(t).map{.sort}.sort)
```

}</lang>

Output:
```Strongly connected components: [["0", "1", "2"], ["3", "4"], ["5", "6"], ["7"]]
Strongly connected components: [["Andy", "Bart", "Carl"], ["Dave", "Earl"], ["Fred", "Gary"], ["Hank"]]
```

## Wren

Translation of: Kotlin
Library: Wren-seq
Library: Wren-dynamic

<lang ecmascript>import "/seq" for Stack import "/dynamic" for Tuple

class Node {

```   construct new(n) {
_n = n
_index = -1   // -1 signifies undefined
_onStack = false
}
```
```   n           { _n }
index       { _index }
index=(v)   { _index = v }
onStack     { _onStack }
onStack=(v) { _onStack = v }
```
```   toString    { _n.toString }
```

}

var DirectedGraph = Tuple.create("DirectedGraph", ["vs", "es"])

var tarjan = Fn.new { |g|

```   var sccs = []
var index = 0
var s = Stack.new()
```
```   var strongConnect // recursive closure
strongConnect = Fn.new { |v|
// Set the depth index for v to the smallest unused index
v.index = index
index = index + 1
s.push(v)
v.onStack = true
```
```       // consider successors of v
for (w in g.es[v.n]) {
if (w.index < 0) {
// Successor w has not yet been visited; recurse on it
strongConnect.call(w)
} else if (w.onStack) {
// Successor w is in stack s and hence in the current SCC
}
}
```
```       // If v is a root node, pop the stack and generate an SCC
var scc = []
while (true) {
var w = s.pop()
w.onStack = false
if (w == v) break
}
}
}
```
```   for (v in g.vs) if (v.index < 0) strongConnect.call(v)
return sccs
```

}

var vs = (0..7).map { |i| Node.new(i) }.toList var es = {

```   0: [vs[1]],
2: [vs[0]],
5: [vs[2], vs[6]],
6: [vs[5]],
1: [vs[2]],
3: [vs[1], vs[2], vs[4]],
4: [vs[5], vs[3]],
7: [vs[4], vs[7], vs[6]]
```

} var g = DirectedGraph.new(vs, es) var sccs = tarjan.call(g) System.print(sccs.join("\n"))</lang>

Output:
```[2, 1, 0]
[6, 5]
[4, 3]
[7]
```

## zkl

<lang zkl>class Tarjan{

```  // input: graph G = (V, Es)
// output: set of strongly connected components (sets of vertices)
// Ick: class holds global state for strongConnect(), otherwise inert
fcn init(graph){
var index=0, stack=List(), components=List(),
G=List.createLong(graph.len(),0);
```
```     // convert graph to ( (index,lowlink,onStack),(id,links)), ...)
// sorted by id
foreach v in (graph){ G[v[0]]=T( L(Void,Void,False),v) }
```
```     foreach v in (G){ if(v[0][INDEX]==Void) strongConnect(v) }
```
```     println("List of strongly connected components:");
foreach c in (components){ println(c.reverse().concat(",")) }
```
```     returnClass(components);  // over-ride return of class instance
}
// Set the depth index for v to the smallest unused index
index+=1;
v0[ON_STACK]=True;
stack.push(v);
```
```      // Consider successors of v
foreach idx in (v[1][1,*]){  // links of v to other vs
w,w0 := G[idx],w[0];   // well, that is pretty vile
if(w[0][INDEX]==Void){
strongConnect(w); // Successor w not yet visited; recurse on it
}
else if(w0[ON_STACK])
// Successor w is in stack S and hence in the current SCC
}
// If v is a root node, pop the stack and generate an SCC
strong:=List();  // start a new strongly connected component
do{
w,w0 := stack.pop(), w[0];
w0[ON_STACK]=False;
strong.append(w[1][0]); // add w to strongly connected component
}while(w.id!=v.id);
components.append(strong); // output strongly connected component
}
}
```

}</lang> <lang zkl> // graph from https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm

```  // with vertices id zero based (vs 1 based in article)
// ids start at zero and are consecutive (no holes), graph is unsorted
```

graph:= // ( (id, links/Edges), ...)

```  T( T(0,1), T(2,0),     T(5,2,6), T(6,5),
T(1,2), T(3,1,2,4), T(4,5,3), T(7,4,7,6) );
```

Tarjan(graph);</lang>

Output:
```0,1,2
5,6
3,4
7
```