Tarjan
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.
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) |
- References
- The article on Wikipedia.
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; 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); } }</lang>
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))
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
}
}
}</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: $(inzerobase(tarj))")
</lang>
- Output:
Results in the zero-base scheme: Array{Int64,1}[[2, 1, 0], [6, 5], [4, 3], [7]]
Kotlin
<lang scala>// version 1.1.3
import java.util.Stack
typealias Nodes = List<Node>
class Node(val n: Int) {
var index = -1 // -1 signifies undefined var lowLink = -1 var onStack = false
override fun toString() = n.toString()
}
class DirectedGraph(val vs: Nodes, val es: Map<Node, Nodes>)
fun tarjan(g: DirectedGraph): List<Nodes> {
val sccs = mutableListOf<Nodes>()
var index = 0
val s = Stack<Node>()
fun strongConnect(v: Node) {
// Set the depth index for v to the smallest unused index
v.index = index
v.lowLink = index
index++
s.push(v)
v.onStack = true
// consider successors of v
for (w in g.es[v]!!) {
if (w.index < 0) {
// Successor w has not yet been visited; recurse on it
strongConnect(w)
v.lowLink = minOf(v.lowLink, w.lowLink)
}
else if (w.onStack) {
// Successor w is in stack s and hence in the current SCC
v.lowLink = minOf(v.lowLink, w.index)
}
}
// If v is a root node, pop the stack and generate an SCC
if (v.lowLink == v.index) {
val scc = mutableListOf<Node>()
do {
val w = s.pop()
w.onStack = false
scc.add(w)
}
while (w != v)
sccs.add(scc)
}
}
for (v in g.vs) if (v.index < 0) strongConnect(v) return sccs
}
fun main(args: Array<String>) {
val vs = (0..7).map { Node(it) }
val es = mapOf(
vs[0] to listOf(vs[1]),
vs[2] to listOf(vs[0]),
vs[5] to listOf(vs[2], vs[6]),
vs[6] to listOf(vs[5]),
vs[1] to listOf(vs[2]),
vs[3] to listOf(vs[1], vs[2], vs[4]),
vs[4] to listOf(vs[5], vs[3]),
vs[7] to listOf(vs[4], vs[7], vs[6])
)
val g = DirectedGraph(vs, es)
val sccs = tarjan(g)
println(sccs.joinToString("\n"))
}</lang>
- Output:
[2, 1, 0] [6, 5] [4, 3] [7]
Perl
<lang perl>use feature 'state'; use List::Util qw(min);
sub tarjan {
our(%k) = @_; our(%onstack, %index, %lowlink, @stack); our @connected = ();
sub strong_connect {
my($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);</lang>
- 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
<lang perl6>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>
)</lang>
- Output:
Strongly connected components: (0 1 2)(3 4)(5 6)(7) Strongly connected components: (Andy Bart Carl)(Dave Earl)(Fred Gary)(Hank)
Phix
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
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 true
end 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 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}
Racket
Manual implementation
<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-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 h)
(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 ([(u vs) (in-hash h)]) (values (make-node u) (map make-node vs))))
(tarjan (make-graph #hash([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)
(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)</lang>
- Output:
'((7) (3 4) (5 6) (1 0 2))
Sidef
<lang ruby>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)
}</lang>
- Output:
Strongly connected components: [["0", "1", "2"], ["3", "4"], ["5", "6"], ["7"]] Strongly connected components: [["Andy", "Bart", "Carl"], ["Dave", "Earl"], ["Fred", "Gary"], ["Hank"]]
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
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
} }
}</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