Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree
GraphsGraphs46
  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

Strongly Connected Components Using Kosaraju's Algorithm

Problem Statement

Given a directed graph with V vertices (numbered from 0 to V - 1) and E directed edges, your task is to find the number of Strongly Connected Components (SCCs).

A strongly connected component is a maximal group of vertices such that every vertex is reachable from every other vertex in the group via directed paths.

Examples

Graph (Adjacency List) Vertices Edges Output Description
{ 0: [1], 1: [2], 2: [0], 3: [4] } 5 [(0→1), (1→2), (2→0), (3→4)] 2 Two SCCs: {0,1,2} and {3,4}
{ 0: [1], 1: [2], 2: [3], 3: [] } 4 [(0→1), (1→2), (2→3)] 4 No cycles, each node is its own SCC
{} 0 [] 0 Empty graph has no components
{ 0: [0] } 1 [(0→0)] 1 Self loop is a valid SCC
{ 0: [1], 1: [2], 2: [0], 3: [2,4], 4: [3] } 5 [(0→1), (1→2), (2→0), (3→2), (3→4), (4→3)] 2 {0,1,2} and {3,4} are two SCCs

Solution

Understanding the Problem

In a directed graph, a Strongly Connected Component (SCC) is a group of vertices where every vertex is reachable from every other vertex in that group. In simpler terms, if you can start at any node in the group and reach all the others through the directed edges—and also get back to the starting node—then those nodes form an SCC.

Our goal is to identify all such strongly connected components in a given directed graph. We'll solve this using Kosaraju's Algorithm, which is efficient and beginner-friendly once understood step-by-step.

Step-by-Step Solution with Example

Step 1: Do a DFS on the original graph and record finish times

Start by doing a standard DFS traversal of the graph. As we finish exploring each node (i.e., all its neighbors are visited), we push it onto a stack. This stack captures the nodes in the order of their finishing times—the last finished node ends up on top.

Why? This helps us process nodes in the right order during the second DFS on the transposed graph.

Step 2: Transpose the graph (reverse all edges)

Next, we reverse all the directed edges in the graph. That is, if there was an edge from A → B, in the transposed graph it becomes B → A.

This reversal helps us discover which nodes can be reached in reverse direction, a key aspect in identifying SCCs.

Step 3: Do DFS on the transposed graph using the stack order

Now, we pop nodes one by one from the stack and perform DFS on the transposed graph. Each time we start a new DFS from an unvisited node, we find one SCC—all the nodes reached in this DFS form a strongly connected component.

Since the stack ensures we process nodes in decreasing order of finish times, we are guaranteed that whenever we start a new DFS, we start from a root node of some SCC.

Example: Step-by-step on a sample graph


Graph:
Vertices = {0, 1, 2, 3, 4}
Edges = {
  0 → 2,
  2 → 1,
  1 → 0,
  0 → 3,
  3 → 4
}

Step 1: DFS Finish Order (pushed to stack): [4, 3, 0, 1, 2]
Step 2: Transpose Graph:
  2 → 0
  1 → 2
  0 → 1
  3 → 0
  4 → 3

Step 3: DFS on Transposed Graph in Stack Order:
- Start DFS from 2 → reaches 2, 1, 0 → SCC1 = [2,1,0]
- Next node in stack is 1 → already visited
- Next is 0 → already visited
- Next is 3 → reaches 3 → SCC2 = [3]
- Next is 4 → reaches 4 → SCC3 = [4]

Final SCCs: [2,1,0], [3], [4]

Edge Cases

  • Empty Graph: No vertices means no SCCs to find.
  • Single Node: A single node with or without a self-loop is an SCC by itself.
  • Disconnected Nodes: Nodes with no outgoing or incoming edges form their own SCCs.
  • Multiple Cycles: Kosaraju's algorithm naturally groups cycles correctly into separate SCCs.

Finally

Kosaraju’s Algorithm is a powerful and intuitive method to identify strongly connected components. Its beauty lies in the simplicity of using two DFS traversals—first to record finish order, and second to explore reversed connectivity.

It runs in linear time O(V + E), making it suitable even for large graphs. As you work through the steps, always remember that SCCs are about mutual reachability, and DFS with finish order is the key to unlocking that structure.

Algorithm Steps

  1. Initialize a visited array and an empty stack.
  2. For each unvisited vertex, perform DFS and push the vertex onto the stack after visiting all its descendants.
  3. Reverse all edges of the graph (create a transpose).
  4. Reset the visited array.
  5. While the stack is not empty:
    1. Pop a vertex from the stack.
    2. If it hasn't been visited, perform DFS from that node in the transposed graph.
    3. Each such DFS traversal gives one strongly connected component.

Code

JavaScript
function kosarajuSCC(V, adj) {
  const visited = new Array(V).fill(false);
  const stack = [];

  function dfs(v) {
    visited[v] = true;
    for (let nei of adj[v] || []) {
      if (!visited[nei]) dfs(nei);
    }
    stack.push(v);
  }

  for (let i = 0; i < V; i++) {
    if (!visited[i]) dfs(i);
  }

  const transpose = Array.from({ length: V }, () => []);
  for (let u = 0; u < V; u++) {
    for (let v of adj[u] || []) {
      transpose[v].push(u);
    }
  }

  visited.fill(false);
  let count = 0;

  function reverseDFS(v) {
    visited[v] = true;
    for (let nei of transpose[v]) {
      if (!visited[nei]) reverseDFS(nei);
    }
  }

  while (stack.length > 0) {
    const node = stack.pop();
    if (!visited[node]) {
      reverseDFS(node);
      count++;
    }
  }

  return count;
}

// Example usage
const graph = {
  0: [1],
  1: [2],
  2: [0],
  3: [4]
};

console.log("Number of SCCs:", kosarajuSCC(5, graph)); // Output: 2

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