Kosaraju
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Kosaraju's algorithm (also known as the Kosaraju–Sharir algorithm) is a linear time algorithm to find the strongly connected components of a directed graph. Aho, Hopcroft and Ullman credit it to an unpublished paper from 1978 by S. Rao Kosaraju. The same algorithm was independently discovered by Micha Sharir and published by him in 1981. It makes use of the fact that the transpose graph (the same graph with the direction of every edge reversed) has exactly the same strongly connected components as the original graph.
- References
- The article on Wikipedia.
C#
<lang csharp>using System; using System.Collections.Generic;
class Node { public enum Colors { Black, White, Gray }
public Colors color { get; set; } public int N { get; }
public Node(int n) { N = n; color = Colors.White; } }
class Graph { public HashSet<Node> V { get; } public Dictionary<Node, HashSet<Node>> Adj { get; }
/// <summary> /// Kosaraju's strongly connected components algorithm /// </summary> public void Kosaraju() { var L = new HashSet<Node>();
Action<Node> Visit = null; Visit = (u) => { if (u.color == Node.Colors.White) { u.color = Node.Colors.Gray;
foreach (var v in Adj[u]) Visit(v);
L.Add(u); } };
Action<Node, Node> Assign = null; Assign = (u, root) => { if (u.color != Node.Colors.Black) { if (u == root) Console.Write("SCC: ");
Console.Write(u.N + " "); u.color = Node.Colors.Black;
foreach (var v in Adj[u]) Assign(v, root);
if (u == root) Console.WriteLine(); } };
foreach (var u in V) Visit(u);
foreach (var u in L) Assign(u, u); } }</lang>
Go
<lang go>package main
import "fmt"
var g = [][]int{
0: {1},
1: {2},
2: {0},
3: {1, 2, 4},
4: {3, 5},
5: {2, 6},
6: {5},
7: {4, 6, 7},
}
func main() {
fmt.Println(kosaraju(g))
}
func kosaraju(g [][]int) []int {
// 1. For each vertex u of the graph, mark u as unvisited. Let L be empty.
vis := make([]bool, len(g))
L := make([]int, len(g))
x := len(L) // index for filling L in reverse order
t := make([][]int, len(g)) // transpose graph
// 2. recursive subroutine:
var Visit func(int)
Visit = func(u int) {
if !vis[u] {
vis[u] = true
for _, v := range g[u] {
Visit(v)
t[v] = append(t[v], u) // construct transpose
}
x--
L[x] = u
}
}
// 2. For each vertex u of the graph do Visit(u)
for u := range g {
Visit(u)
}
c := make([]int, len(g)) // result, the component assignment
// 3: recursive subroutine:
var Assign func(int, int)
Assign = func(u, root int) {
if vis[u] { // repurpose vis to mean "unassigned"
vis[u] = false
c[u] = root
for _, v := range t[u] {
Assign(v, root)
}
}
}
// 3: For each element u of L in order, do Assign(u,u)
for _, u := range L {
Assign(u, u)
}
return c
}</lang>
- Output:
[0 0 0 3 3 5 5 7]
Kotlin
<lang scala>// version 1.1.3
/* the list index is the first vertex in the edge(s) */ val g = listOf(
intArrayOf(1), // 0 intArrayOf(2), // 1 intArrayOf(0), // 2 intArrayOf(1, 2, 4), // 3 intArrayOf(3, 5), // 4 intArrayOf(2, 6), // 5 intArrayOf(5), // 6 intArrayOf(4, 6, 7) // 7
)
fun kosaraju(g: List<IntArray>): List<List<Int>> {
// 1. For each vertex u of the graph, mark u as unvisited. Let l be empty.
val size = g.size
val vis = BooleanArray(size) // all false by default
val l = IntArray(size) // all zero by default
var x = size // index for filling l in reverse order
val t = List(size) { mutableListOf<Int>() } // transpose graph
// Recursive subroutine 'visit':
fun visit(u: Int) {
if (!vis[u]) {
vis[u] = true
for (v in g[u]) {
visit(v)
t[v].add(u) // construct transpose
}
l[--x] = u
}
}
// 2. For each vertex u of the graph do visit(u) for (u in g.indices) visit(u) val c = IntArray(size) // used for component assignment
// Recursive subroutine 'assign':
fun assign(u: Int, root: Int) {
if (vis[u]) { // repurpose vis to mean 'unassigned'
vis[u] = false
c[u] = root
for (v in t[u]) assign(v, root)
}
}
// 3: For each element u of l in order, do assign(u, u) for (u in l) assign(u, u)
// Obtain list of SCC's from 'c' and return it
return c.withIndex()
.groupBy { it.value }.values
.map { ivl -> ivl.map { it.index } }
}
fun main(args: Array<String>) {
println(kosaraju(g).joinToString("\n"))
}</lang>
- Output:
[0, 1, 2] [3, 4] [5, 6] [7]
Racket
<lang racket>#lang racket
(require racket/dict)
- G is a dictionary of vertex -> (list vertex)
(define (Kosuraju G)
(letrec
((vertices (remove-duplicates (append (dict-keys G) (append* (dict-values G)))))
(visited?-dict (make-hash)) ; or any mutable dict type
(assigned-dict (make-hash)) ; or any mutable dict type
(neighbours:in (λ (u) (for/list (([v outs] (in-dict G)) #:when (member u outs)) v)))
(visit! (λ (u L)
(cond [(dict-ref visited?-dict u #f) L]
[else (dict-set! visited?-dict u #t)
(cons u (for/fold ((L L)) ((v (in-list (dict-ref G u)))) (visit! v L)))])))
(assign! (λ (u root)
(unless (dict-ref assigned-dict u #f)
(dict-set! assigned-dict u root)
(for ((v (in-list (neighbours:in u)))) (assign! v root)))))
(L (for/fold ((l null)) ((u (in-dict-keys G))) (visit! u l))))
(for ((u (in-list L))) (assign! u u))
(map (curry map car) (group-by cdr (dict->list assigned-dict) =))))
(module+ test
(Kosuraju '((0 1)
(2 0)
(5 2 6)
(6 5)
(1 2)
(3 1 2 4) ; equvalent to (3 . (1 2 4))
(4 5 3)
(7 4 7 6))))</lang>
- Output:
'((7) (6 5) (4 3) (2 1 0))
zkl
<lang zkl>const VISITED=0,ASSIGNED=1;
fcn visit(u,G,L){ // u is ((visited,assigned), (id,edges))
u0:=u[0];
if(u0[VISITED]) return();
u0[VISITED]=True;
foreach idx in (u[1][1,*]){ visit(G[idx],G,L) } // vist out-neighbours
L.insert(0,u); // prepend u to L
} fcn assign(u,root,G){ // u as above, root is a list of strong components
u0:=u[0];
if(u0[ASSIGNED]) return();
root.append(u[1][0]);
u0[ASSIGNED]=True;
uid:=u[1][0];
foreach v in (G){ // traverse graph to find in-neighbours, fugly
n,ins := v[1][0],v[1][1,*];
if(ins.holds(uid)) assign(G[n],root,G); // assign in-neighbour
}
} fcn kosaraju(graph){ // Use Tarjan's algorithm instead of this one
// input: graph G = (V, Es) // output: set of strongly connected components (sets of vertices)
// convert graph to ( (index,lowlink,onStack),(id,links)), ...)
// sorted by id
G:=List.createLong(graph.len(),0);
foreach v in (graph){ G[v[0]]=T( List(False,False),v) }
L:=List();
foreach u in (G){ visit(u,G,L) }
components:=List.createLong(graph.len(),List.copy,True);
foreach u in (L){ assign(u,components[u[1][0]],G) }
components=components.filter();
println("List of strongly connected components:");
foreach c in (components){ println(c.reverse().concat(",")) }
return(components);
}</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) );
kosaraju(graph);</lang>
- Output:
List of strongly connected components: 1,2,0 4,3 6,5 7