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
- 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>
C++
<lang cpp>// // C++ implementation of Tarjan's strongly connected components algorithm // See https://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm //
- include <algorithm>
- include <iostream>
- include <list>
- include <string>
- 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) {}
void add_neighbour(vertex* v) {
neighbours_.push_back(v);
}
void add_neighbours(const std::initializer_list<vertex*>& vs) {
neighbours_.insert(neighbours_.end(), vs);
}
const T& get_data() {
return data_;
}
private:
friend tarjan<T>; T data_; int index_ = -1; int lowlink_ = -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_;
v->lowlink_ = index_;
++index_;
stack_.push_back(v);
v->on_stack_ = true;
for (auto w : v->neighbours_) {
if (w->index_ == -1) {
strongconnect(w);
v->lowlink_ = std::min(v->lowlink_, w->lowlink_);
}
else if (w->on_stack_) {
v->lowlink_ = std::min(v->lowlink_, w->index_);
}
}
if (v->lowlink_ == v->index_) {
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;
auto andy = g.add_vertex("Andy");
auto bart = g.add_vertex("Bart");
auto carl = g.add_vertex("Carl");
auto dave = g.add_vertex("Dave");
auto earl = g.add_vertex("Earl");
auto fred = g.add_vertex("Fred");
auto gary = g.add_vertex("Gary");
auto hank = g.add_vertex("Hank");
andy->add_neighbour(bart);
bart->add_neighbour(carl);
carl->add_neighbour(andy);
dave->add_neighbours({bart, carl, earl});
earl->add_neighbours({dave, fred});
fred->add_neighbours({carl, gary});
gary->add_neighbour(fred);
hank->add_neighbours({earl, gary, hank});
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))
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: $(inzerobase(tarj))")
</lang>
- Output:
Results in the zero-base scheme: Array{Int64,1}[[2, 1, 0], [6, 5], [4, 3], [7]]
Kotlin
<lang scala>// version 1.1.3
import java.util.Stack
typealias Nodes = List<Node>
class Node(val n: Int) {
var index = -1 // -1 signifies undefined var lowLink = -1 var onStack = false
override fun toString() = n.toString()
}
class DirectedGraph(val vs: Nodes, val es: Map<Node, Nodes>)
fun tarjan(g: DirectedGraph): List<Nodes> {
val sccs = mutableListOf<Nodes>() var index = 0 val s = Stack<Node>()
fun strongConnect(v: Node) {
// Set the depth index for v to the smallest unused index
v.index = index
v.lowLink = index
index++
s.push(v)
v.onStack = true
// consider successors of v
for (w in g.es[v]!!) {
if (w.index < 0) {
// Successor w has not yet been visited; recurse on it
strongConnect(w)
v.lowLink = minOf(v.lowLink, w.lowLink)
}
else if (w.onStack) {
// Successor w is in stack s and hence in the current SCC
v.lowLink = minOf(v.lowLink, w.index)
}
}
// If v is a root node, pop the stack and generate an SCC
if (v.lowLink == v.index) {
val scc = mutableListOf<Node>()
do {
val w = s.pop()
w.onStack = false
scc.add(w)
}
while (w != v)
sccs.add(scc)
}
}
for (v in g.vs) if (v.index < 0) strongConnect(v) return sccs
}
fun main(args: Array<String>) {
val vs = (0..7).map { Node(it) }
val es = mapOf(
vs[0] to listOf(vs[1]),
vs[2] to listOf(vs[0]),
vs[5] to listOf(vs[2], vs[6]),
vs[6] to listOf(vs[5]),
vs[1] to listOf(vs[2]),
vs[3] to listOf(vs[1], vs[2], vs[4]),
vs[4] to listOf(vs[5], vs[3]),
vs[7] to listOf(vs[4], vs[7], vs[6])
)
val g = DirectedGraph(vs, es)
val sccs = tarjan(g)
println(sccs.joinToString("\n"))
}</lang>
- Output:
[2, 1, 0] [6, 5] [4, 3] [7]
Perl
<lang perl>use strict; use warnings; use feature <say state current_sub>; use List::Util qw(min);
sub tarjan {
my (%k) = @_; my (%onstack, %index, %lowlink, @stack, @connected);
my sub strong_connect {
my ($vertex, $i) = @_;
$index{$vertex} = $i;
$lowlink{$vertex} = $i + 1;
$onstack{$vertex} = 1;
push @stack, $vertex;
for my $connection (@{$k{$vertex}}) {
if (not defined $index{$connection}) {
__SUB__->($connection, $i + 1);
$lowlink{$vertex} = min($lowlink{$connection}, $lowlink{$vertex});
}
elsif ($onstack{$connection}) {
$lowlink{$vertex} = min($index{$connection}, $lowlink{$vertex});
}
}
if ($lowlink{$vertex} eq $index{$vertex}) {
my @node;
do {
push @node, pop @stack;
$onstack{$node[-1]} = 0;
} while $node[-1] ne $vertex;
push @connected, [@node];
}
}
for (sort keys %k) {
strong_connect($_, 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
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}
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
# -1: already processed
# 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
self.gv = self._to_gv()
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
<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 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)
(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))
Raku
(formerly 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)
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
If lowlink.nb < lowlink.n Then
lowlink.n = lowlink.nb
End
Else Do
If stacked.nb = 1 Then
If index.nb < lowlink.n Then
lowlink.n = index.nb
end
end
if lowlink.n=index.n then Do
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 low=low lowlink.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
- [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()
}
}
- [derive(Clone)]
struct VertexState {
index: usize, low_link: 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,
low_link: 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;
v_ref.low_link = 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 w_low_link = self.state[w].low_link;
let v_ref = &mut self.state[v];
v_ref.low_link = v_ref.low_link.min(w_low_link);
} else if w_ref.on_stack {
let w_index = self.state[w].index;
let v_ref = &mut self.state[v];
v_ref.low_link = v_ref.low_link.min(w_index);
}
}
let v_ref = &self.state[v];
if v_ref.low_link == v_ref.index {
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.add_edge(0, 1);
g.add_edge(2, 0);
g.add_edges(5, vec![2, 6]);
g.add_edge(6, 5);
g.add_edge(1, 2);
g.add_edges(3, vec![1, 2, 4]);
g.add_edges(4, vec![5, 3]);
g.add_edges(7, vec![4, 7, 6]);
g
};
for component in Tarjan::walk(&graph) {
println!("{:?}", component);
}
}</lang>
- Output:
{0, 1, 2}
{5, 6}
{3, 4}
{7}
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"]]
Wren
<lang ecmascript>import "/seq" for Stack import "/dynamic" for Tuple import "/math" for Math
class Node {
construct new(n) {
_n = n
_index = -1 // -1 signifies undefined
_lowLink = -1
_onStack = false
}
n { _n }
index { _index }
index=(v) { _index = v }
lowLink { _lowLink }
lowLink=(v) { _lowLink = 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
v.lowLink = 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)
v.lowLink = Math.min(v.lowLink, w.lowLink)
} else if (w.onStack) {
// 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) {
var scc = []
while (true) {
var w = s.pop()
w.onStack = false
scc.add(w)
if (w == v) break
}
sccs.add(scc)
}
}
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
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