Step by Step solution about How to resolve the algorithm Tarjan step by step in the C# programming language

Overview

The provided C# code implements Tarjan's strongly connected components algorithm (Tarjan's SCC) to identify and group nodes in a directed graph into strongly connected components (SCCs). An SCC is a set of nodes where there is a path from each node to every other node in the set.

Data Structures

• Node: Represents a node in the graph. It tracks the node's index, the minimum depth reached from the node (low link), and the number of nodes in the graph.
• Graph: Represents the overall graph. It maintains a set of nodes (V) and a dictionary of adjacency lists (Adj), where each key-value pair (node, adjacent nodes) represents the directed edges in the graph.

Algorithm

• Tarjan(): The core implementation of Tarjan's SCC algorithm.

Step 1: Initialization

• index: Initially set to 0, representing the next unused index for nodes.
• S: A stack used to keep track of nodes being explored.

Step 2: Define the StrongConnect Function

• This function is a recursive helper function used to explore the graph and identify SCCs.

Step 3: Tarjan's Algorithm Loop

• For each node v that has not been visited (Index < 0), call StrongConnect(v) to explore its connections.

Step 4: StrongConnect Function

• Set the Index and LowLink values for node v.
• Push v onto the stack S.
• Iterate over the adjacent nodes w of v.
  • If w has not been visited, call StrongConnect(w) recursively.
  • If w is already in the stack S, update the LowLink of v to the minimum of its current value and the index of w.
• If v is a root node of an SCC, pop the nodes from S and print them as an SCC.

Example Usage

var graph = new Graph();

// Add nodes and edges 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.Adj[v1].Add(v2);
graph.Adj[v2].Add(v3);
graph.Adj[v3].Add(v1);
graph.Adj[v4].Add(v2);

// Run Tarjan's SCC algorithm
graph.Tarjan();

Output

SCC: 1 3
SCC: 2 4

In this example:

• SCC 1 contains nodes 1 and 3 because there is a path from 1 to 3 and 3 to 1.
• SCC 2 contains nodes 2 and 4 because there is a path from 2 to 4 and 4 to 2.

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);
    }
}