# Tarjan

Tarjan
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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

## 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;		LowLink = 0;	}} 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;			v.LowLink = index; 			index++;			S.Push(v); 			// Consider successors of v			foreach (var w in Adj[v])				if (w.Index < 0)				{					// Successor w has not yet been visited; recurse on it					StrongConnect(w);					v.LowLink = Math.Min(v.LowLink, w.LowLink);				}				else if (S.Contains(w))					// Successor w is in stack S and hence in the current SCC					v.LowLink = Math.Min(v.LowLink, w.Index); 			// If v is a root node, pop the stack and generate an SCC			if (v.LowLink == v.Index)			{				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))    lowlink := 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)        lowlink[n] = x        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                }                if lowlink[nb] < lowlink[n] {                    lowlink[n] = lowlink[nb]                }            } else if stacked.Bit(nb) == 1 {                if index[nb] < lowlink[n] {                    lowlink[n] = index[nb]                }            }        }        if lowlink[n] == index[n] {            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        }    }}`
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().

`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))") `
Output:
```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 lowLink = -1    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        v.lowLink = 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)                v.lowLink = minOf(v.lowLink, w.lowLink)            }            else if (w.onStack) {                // Successor w is in stack s and hence in the current SCC                v.lowLink = minOf(v.lowLink, w.index)            }        }         // If v is a root node, pop the stack and generate an SCC        if (v.lowLink == v.index) {            val scc = mutableListOf<Node>()            do {                val w = s.pop()                w.onStack = false                scc.add(w)            }             while (w != v)            sccs.add(scc)        }    }     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"))   }`
Output:
```[2, 1, 0]
[6, 5]
[4, 3]
[7]
```

## Perl

Translation of: Perl 6
`use feature 'state';use List::Util qw(min); sub tarjan {    our(%k) = @_;    our(%onstack, %index, %lowlink, @stack);    our @connected = ();     sub strong_connect {        my(\$vertex) = @_;         state \$index = 0;         \$index{\$vertex}   = \$index;         \$lowlink{\$vertex} = \$index++;         push @stack, \$vertex;         \$onstack{\$vertex} = 1;         for my \$connection (@{\$k{\$vertex}}) {             if (not \$index{\$connection}) {                 strong_connect(\$connection);                 \$lowlink{\$vertex} = min(\$lowlink{\$connection},\$lowlink{\$vertex});             } elsif (\$onstack{\$connection}) {                 \$lowlink{\$vertex} = min(\$lowlink{\$connection},\$lowlink{\$vertex});             }        }        if (\$lowlink{\$vertex} eq \$index{\$vertex}) {            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);`
Output:
```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 version 2018.09
`sub tarjan (%k) {    my %onstack;    my %index;    my %lowlink;    my @stack;    my @connected;     sub strong-connect (\$vertex) {         state \$index      = 0;         %index{\$vertex}   = \$index;         %lowlink{\$vertex} = \$index++;         %onstack{\$vertex} = True;         @stack.push: \$vertex;         for |%k{\$vertex} -> \$connection {             if not %index{\$connection}.defined {                 strong-connect(\$connection);                 %lowlink{\$vertex} min= %lowlink{\$connection};             }             elsif %onstack{\$connection} {                 %lowlink{\$vertex} min= %index{\$connection};             }        }        if %lowlink{\$vertex} eq %index{\$vertex} {            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>)`
Output:
```Strongly connected components: (0 1 2)(3 4)(5 6)(7)

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

## Phix

Translation of: 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}} sequence index, lowlink, stacked, stackinteger x function strong_connect(integer n, r_emit)    index[n] = x    lowlink[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            if lowlink[nb] < lowlink[n] then                lowlink[n] = lowlink[nb]            end if        elsif stacked[nb] == 1 then            if index[nb] < lowlink[n] then                lowlink[n] = index[nb]            end if        end if    end for    if lowlink[n] == index[n] then        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 trueend function procedure tarjan(sequence g, integer r_emit)    index   = repeat(0,length(g))    lowlink = 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 forend procedure procedure emit(object c)-- called for each component identified.-- each component is a list of nodes.    ?cend procedure tarjan(g,routine_id("emit"))`
Output:
```{1,2,3}
{6,7}
{4,5}
{8}
```

## Racket

### Manual implementation

Translation of: 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-low-link! v index)    (set-node-on?! v #t)    (change! s (cons v _))    (change! index add1)     (for ([w (in-list (hash-ref g v '()))])      (match-define (node _ index low-link on?) w)      (cond        [(not index) (dfs w)                     (change! node-low-link v (min (node-low-link 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])))`
Output:
```'((7) (3 4) (5 6) (2 1 0))
```

### With the graph library

`#lang racket (require graph) (define g (unweighted-graph/adj '([0 1]                                  [2 0]                                  [5 2 6]                                  [6 5]                                  [1 2]                                  [3 1 2 4]                                  [4 5 3]                                  [7 4 7 6]))) (scc g)`
Output:
```'((7) (3 4) (5 6) (1 0 2))
```

## Sidef

Translation of: Perl 6
`func tarjan (k) {     var(:onstack, :index, :lowlink, *stack, *connected)     func strong_connect (vertex, i=0) {          index{vertex}   = i         lowlink{vertex} = i+1         onstack{vertex} = true         stack << vertex          for connection in (k{vertex}) {             if (index{connection} == nil) {                 strong_connect(connection, i+1)                 lowlink{vertex} `min!` lowlink{connection}             }             elsif (onstack{connection}) {                 lowlink{vertex} `min!` index{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)}`
Output:
```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   const INDEX=0, LOW_LINK=1, ON_STACK=2;   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   }   fcn strongConnect(v){  // v is ( (index,lowlink,onStack), (id,links) )      // Set the depth index for v to the smallest unused index      v0:=v[0]; v0[INDEX]=v0[LOW_LINK]=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	    v0[LOW_LINK]=v0[LOW_LINK].min(w0[LOW_LINK]);	 }	 else if(w0[ON_STACK])	    // Successor w is in stack S and hence in the current SCC	    v0[LOW_LINK]=v0[LOW_LINK].min(w0[INDEX]);      }      // If v is a root node, pop the stack and generate an SCC      if(v0[LOW_LINK]==v0[INDEX]){         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 unsortedgraph:=	  // ( (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);`
Output:
```0,1,2
5,6
3,4
7
```