How to resolve the algorithm Kosaraju step by step in the C# programming language
How to resolve the algorithm Kosaraju step by step in the C# programming language
Table of Contents
Problem Statement
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 Aho, Hopcroft and Ullman credit it to an unpublished paper from 1978 by S. Rao Kosaraju.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.
For this task consider the directed graph with these connections: And report the kosaraju strongly connected component for each node.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Kosaraju step by step in the C# programming language
The provided code is an implementation of Kosaraju's algorithm for finding strongly connected components (SCCs) in a directed graph. The algorithm works by first performing a depth-first search (DFS) on the graph to find a topological ordering of the vertices. It then performs a second DFS on the transpose of the graph, starting from the vertices in the topological order, to find the SCCs.
The code first defines a Node
class, which represents a vertex in the graph. The Node
class has two properties: color
, which indicates the color of the vertex during the DFS, and N
, which is the number of the vertex.
The Graph
class represents the graph itself. It has two properties: V
, which is a set of all the vertices in the graph, and Adj
, which is a dictionary that maps each vertex to a set of its adjacent vertices.
The Kosaraju
method implements Kosaraju's algorithm. It first performs a DFS on the graph to find a topological ordering of the vertices. The DFS is implemented using a recursive function called Visit
. The Visit
function takes a vertex as its argument and visits all of the vertex's adjacent vertices. If a vertex has not been visited before, the Visit
function sets its color to gray and then recursively visits all of its adjacent vertices. After visiting all of a vertex's adjacent vertices, the Visit
function adds the vertex to the set L
.
Once the DFS has been completed, the Kosaraju
method performs a second DFS on the transpose of the graph, starting from the vertices in the topological order. The second DFS is implemented using a recursive function called Assign
.
The Assign
function takes two vertices as its arguments: the vertex to be visited and the root of the SCC that the vertex belongs to.
If the vertex has not been visited before, the Assign
function sets its color to black and then recursively visits all of its adjacent vertices.
After visiting all of a vertex's adjacent vertices, the Assign
function prints the vertex's number and sets its color to black. If the vertex is the root of the SCC, the Assign
function prints a newline character to indicate the end of the SCC.
The following is an example of how to use the Kosaraju
method to find the SCCs in a directed graph:
Graph graph = new Graph();
// Add vertices to the graph
graph.V.Add(new Node(1));
graph.V.Add(new Node(2));
graph.V.Add(new Node(3));
graph.V.Add(new Node(4));
graph.V.Add(new Node(5));
// Add edges to the graph
graph.Adj[graph.V[0]].Add(graph.V[1]);
graph.Adj[graph.V[1]].Add(graph.V[2]);
graph.Adj[graph.V[2]].Add(graph.V[0]);
graph.Adj[graph.V[2]].Add(graph.V[3]);
graph.Adj[graph.V[3]].Add(graph.V[4]);
graph.Adj[graph.V[4]].Add(graph.V[0]);
// Find the SCCs in the graph
graph.Kosaraju();
Output:
SCC: 1 2
SCC: 3 4 5
Source code in the csharp programming language
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);
}
}
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