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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 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:

• The article on [[wp:Tarjan's_strongly_connected_components_algorithm|Wikipedia]].

## C#

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);
}
}

## 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
}
}
}

{{out}}

[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().

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))")

{{out}}

Results in the zero-base scheme: Array{Int64,1}[[2, 1, 0], [6, 5], [4, 3], [7]]

## Kotlin

// 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"))
}

{{out}}

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

## Perl

{{trans|Perl 6}}

use feature 'state';
use List::Util qw(min);

sub tarjan {
our(%k) = @_;
our @connected = ();

sub strong_connect {
my(\$vertex) = @_;
state \$index = 0;
\$index{\$vertex}   = \$index;
push @stack, \$vertex;
\$onstack{\$vertex} = 1;
for my \$connection (@{\$k{\$vertex}}) {
if (not \$index{\$connection}) {
strong_connect(\$connection);
} 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(\$_) 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);

{{out}}

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

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

## Perl 6

{{works with|Rakudo|2018.09}}

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
}

# TESTING

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

# hash of vertex, edge list pairs
(((1),(2),(0),(1,2,4),(3,5),(2,6),(5),(4,6,7)).pairs.hash),

# 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>
)

{{out}}

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

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

## Phix

{{trans|Go}} Same data as other examples, but with 1-based indexes.

constant g = {{2}, {3}, {1}, {2,3,5}, {6,4}, {3,7}, {6}, {5,8,7}}

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"))

{{out}}

{1,2,3}
{6,7}
{4,5}
{8}

## Racket

### Manual implementation

{{trans|Kotlin}}

#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])))

{{out}}

'((7) (3 4) (5 6) (2 1 0))

### With the graph library

#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)

{{out}}

'((7) (3 4) (5 6) (1 0 2))

## Sidef

{{trans|Perl 6}}

func tarjan (k) {

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}) {
}
}

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)
}

{{out}}

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

## 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);

// 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
}
}
}
// 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);

{{out}}

0,1,2
5,6
3,4
7