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Add Tarjan's strongly connected components algorithm (#561)
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‎Algorithms.Tests/Graph/TarjanStronglyConnectedComponentsTests.cs‎

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using Algorithms.Graph;
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using NUnit.Framework;
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using FluentAssertions;
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using System.Linq;
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namespace Algorithms.Tests.Graph;
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public class TarjanStronglyConnectedComponentsTests
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{
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[Test]
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public void FindSCCs_SimpleGraph_ReturnsCorrectSCCs()
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{
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var tarjan = new TarjanStronglyConnectedComponents(3);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 0);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(1);
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sccs[0].Should().BeEquivalentTo(new[] { 0, 1, 2 });
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}
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[Test]
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public void FindSCCs_TwoSeparateSCCs_ReturnsBothSCCs()
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{
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var tarjan = new TarjanStronglyConnectedComponents(4);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 0);
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tarjan.AddEdge(2, 3);
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tarjan.AddEdge(3, 2);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(2);
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}
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[Test]
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public void FindSCCs_DisconnectedVertices_EachVertexIsSCC()
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{
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var tarjan = new TarjanStronglyConnectedComponents(3);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(3);
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sccs.Should().OnlyContain(scc => scc.Count == 1);
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}
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[Test]
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public void FindSCCs_ComplexGraph_ReturnsCorrectSCCs()
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{
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var tarjan = new TarjanStronglyConnectedComponents(8);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 0);
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tarjan.AddEdge(2, 3);
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tarjan.AddEdge(3, 4);
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tarjan.AddEdge(4, 5);
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tarjan.AddEdge(5, 3);
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tarjan.AddEdge(5, 6);
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tarjan.AddEdge(6, 7);
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tarjan.AddEdge(7, 6);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(3);
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}
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[Test]
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public void GetSccCount_AfterFindingSCCs_ReturnsCorrectCount()
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{
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var tarjan = new TarjanStronglyConnectedComponents(5);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 0);
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tarjan.AddEdge(2, 3);
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tarjan.AddEdge(3, 4);
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tarjan.AddEdge(4, 2);
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tarjan.FindSCCs();
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var count = tarjan.GetSccCount();
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count.Should().Be(2);
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}
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[Test]
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public void InSameScc_VerticesInSameScc_ReturnsTrue()
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{
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var tarjan = new TarjanStronglyConnectedComponents(3);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 0);
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var result = tarjan.InSameScc(0, 2);
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result.Should().BeTrue();
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}
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[Test]
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public void InSameScc_VerticesInDifferentSccs_ReturnsFalse()
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{
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var tarjan = new TarjanStronglyConnectedComponents(4);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 0);
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tarjan.AddEdge(2, 3);
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var result = tarjan.InSameScc(0, 2);
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result.Should().BeFalse();
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}
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[Test]
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public void GetScc_ValidVertex_ReturnsSccContainingVertex()
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{
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var tarjan = new TarjanStronglyConnectedComponents(3);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 0);
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var scc = tarjan.GetScc(1);
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scc.Should().NotBeNull();
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scc.Should().Contain(1);
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scc.Should().HaveCount(3);
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}
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[Test]
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public void BuildCondensationGraph_ComplexGraph_ReturnsDAG()
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{
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var tarjan = new TarjanStronglyConnectedComponents(6);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 0);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 3);
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tarjan.AddEdge(3, 4);
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tarjan.AddEdge(4, 5);
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tarjan.AddEdge(5, 3);
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var condensation = tarjan.BuildCondensationGraph();
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condensation.Should().NotBeNull();
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condensation.Length.Should().Be(3); // {0,1}, {2}, {3,4,5}
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}
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[Test]
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public void AddEdge_InvalidVertex_ThrowsException()
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{
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var tarjan = new TarjanStronglyConnectedComponents(3);
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var act = () => tarjan.AddEdge(0, 5);
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act.Should().Throw<ArgumentOutOfRangeException>();
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}
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[Test]
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public void FindSCCs_SingleVertex_ReturnsSingleSCC()
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{
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var tarjan = new TarjanStronglyConnectedComponents(1);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(1);
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sccs[0].Should().BeEquivalentTo(new[] { 0 });
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}
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[Test]
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public void FindSCCs_LinearChain_EachVertexIsSCC()
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{
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var tarjan = new TarjanStronglyConnectedComponents(4);
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tarjan.AddEdge(0, 1);
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tarjan.AddEdge(1, 2);
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tarjan.AddEdge(2, 3);
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var sccs = tarjan.FindSCCs();
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sccs.Should().HaveCount(4);
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}
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[Test]
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public void FindSCCs_SelfLoop_VertexIsSCC()
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{
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var tarjan = new TarjanStronglyConnectedComponents(2);
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tarjan.AddEdge(0, 0);
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tarjan.AddEdge(0, 1);
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var sccs = tarjan.FindSCCs();
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sccs.Should().Contain(scc => scc.Contains(0) && scc.Count == 1);
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}
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}

‎Algorithms/Graph/TarjanStronglyConnectedComponents.cs‎

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using System;
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using System.Collections.Generic;
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using System.Linq;
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namespace Algorithms.Graph;
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/// <summary>
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/// Tarjan's algorithm for finding strongly connected components in a directed graph.
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/// Uses depth-first search with a stack to identify SCCs in O(V + E) time.
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/// </summary>
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public class TarjanStronglyConnectedComponents
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{
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private readonly List<int>[] graph;
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private readonly int[] ids;
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private readonly int[] low;
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private readonly bool[] onStack;
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private readonly Stack<int> stack;
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private readonly List<List<int>> sccs;
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private int id;
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public TarjanStronglyConnectedComponents(int vertices)
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{
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graph = new List<int>[vertices];
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ids = new int[vertices];
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low = new int[vertices];
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onStack = new bool[vertices];
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stack = new Stack<int>();
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sccs = new List<List<int>>();
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for (int i = 0; i < vertices; i++)
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{
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graph[i] = new List<int>();
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ids[i] = -1;
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}
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}
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/// <summary>
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/// Adds a directed edge from u to v.
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/// </summary>
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public void AddEdge(int u, int v)
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{
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if (u < 0 || u >= graph.Length || v < 0 || v >= graph.Length)
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{
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throw new ArgumentOutOfRangeException(nameof(u), "Vertex indices must be within valid range.");
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}
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graph[u].Add(v);
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}
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/// <summary>
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/// Finds all strongly connected components.
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/// </summary>
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/// <returns>List of SCCs, where each SCC is a list of vertex indices.</returns>
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public List<List<int>> FindSCCs()
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{
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for (int i = 0; i < graph.Length; i++)
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{
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if (ids[i] == -1)
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{
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Dfs(i);
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}
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}
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return sccs;
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}
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/// <summary>
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/// Gets the number of strongly connected components.
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/// </summary>
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public int GetSccCount() => sccs.Count;
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/// <summary>
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/// Checks if two vertices are in the same SCC.
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/// </summary>
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public bool InSameScc(int u, int v)
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{
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if (sccs.Count == 0)
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{
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FindSCCs();
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}
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foreach (var scc in sccs)
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{
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if (scc.Contains(u) && scc.Contains(v))
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{
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return true;
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}
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}
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return false;
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}
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/// <summary>
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/// Gets the SCC containing the given vertex.
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/// </summary>
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public List<int>? GetScc(int vertex)
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{
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if (sccs.Count == 0)
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{
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FindSCCs();
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}
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return sccs.FirstOrDefault(scc => scc.Contains(vertex));
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}
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/// <summary>
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/// Builds the condensation graph (DAG of SCCs).
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/// </summary>
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public List<int>[] BuildCondensationGraph()
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{
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if (sccs.Count == 0)
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{
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FindSCCs();
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}
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var sccIndex = new int[graph.Length];
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for (int i = 0; i < sccs.Count; i++)
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{
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foreach (var vertex in sccs[i])
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{
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sccIndex[vertex] = i;
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}
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}
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var condensation = new List<int>[sccs.Count];
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for (int i = 0; i < sccs.Count; i++)
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{
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condensation[i] = new List<int>();
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}
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var edges = new HashSet<(int, int)>();
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for (int u = 0; u < graph.Length; u++)
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{
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foreach (var v in graph[u])
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{
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int sccU = sccIndex[u];
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int sccV = sccIndex[v];
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if (sccU != sccV && !edges.Contains((sccU, sccV)))
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{
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condensation[sccU].Add(sccV);
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edges.Add((sccU, sccV));
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}
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}
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}
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return condensation;
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}
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private void Dfs(int at)
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{
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stack.Push(at);
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onStack[at] = true;
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ids[at] = low[at] = id++;
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foreach (var to in graph[at])
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{
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if (ids[to] == -1)
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{
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Dfs(to);
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}
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if (onStack[to])
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{
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low[at] = Math.Min(low[at], low[to]);
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}
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}
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if (ids[at] == low[at])
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{
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var scc = new List<int>();
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while (true)
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{
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int node = stack.Pop();
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onStack[node] = false;
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scc.Add(node);
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if (node == at)
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{
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break;
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}
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}
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sccs.Add(scc);
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}
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}
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}

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