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

Topological Sort Using Depth-First Search (DFS)

Problem Statement

Topological Sort is a linear ordering of vertices in a directed graph such that for every directed edge u → v, vertex u comes before v in the ordering.

It is only applicable to Directed Acyclic Graphs (DAGs) and is used in scenarios like task scheduling, build systems, and dependency resolution.

Examples

Graph (Edges) Topological Sort Description
0 → 1, 0 → 2, 1 → 3, 2 → 3 0 → 2 → 1 → 3 Multiple valid orderings possible, DFS pushes deeper nodes first
5 → 0, 5 → 2, 4 → 0, 4 → 1, 2 → 3, 3 → 1 5 → 4 → 2 → 3 → 1 → 0 Topological sort based on dependencies
1 → 2, 2 → 3 1 → 2 → 3 Straightforward chain
None Any order Empty graph or no edges
0 → 1, 1 → 2, 2 → 0 Not possible Cycle detected, not a DAG

Solution

Understanding the Problem

Topological sorting is a way of arranging the nodes of a directed graph in a linear order such that for every directed edge u → v, node u comes before node v in the ordering.

This type of sorting is only possible if the graph is a Directed Acyclic Graph (DAG), meaning it has no cycles. It's commonly used in scenarios like task scheduling, where one task must be completed before another, or in resolving build dependencies.

Step-by-Step Solution with Example

step 1: Represent the Graph

We begin by representing the graph using an adjacency list. This structure allows us to keep track of all nodes and their direct dependencies.


Graph:
A → B
B → C
A → C
Adjacency List:
{
  A: [B, C],
  B: [C],
  C: []
}

step 2: Initialize Required Data Structures

We use a visited set to track visited nodes and a stack to store the topological order as we finish visiting each node’s dependencies.

step 3: Perform DFS from Each Unvisited Node

We perform a Depth-First Search (DFS) starting from each unvisited node. In DFS, we go deeper into the graph until we reach nodes with no outgoing edges, then we add those to our stack as we backtrack.

step 4: Push Nodes to Stack After Exploring All Neighbors

Once we finish visiting all neighbors of a node, we push it to the stack. This ensures that nodes with no dependencies are added first, and their dependents come later in the final order.

step 5: Reverse the Stack to Get the Topological Order

Since the first node we complete is the one with no outgoing edges, we reverse the stack to get the correct topological order from start to finish.

step 6: Apply the Steps to Our Example

Let’s apply this to our earlier example:

  • Start DFS from A → visit B → visit C
  • Push C to stack (C has no neighbors)
  • Backtrack to B → all neighbors visited → push B
  • Backtrack to A → all neighbors visited → push A

Stack (before reversing): [C, B, A]

Final Topological Order (after reversing): A, B, C

Edge Cases

Disconnected Graph Components

If the graph has disconnected components (not all nodes are reachable from a single starting point), we must initiate DFS from every unvisited node to ensure all components are included in the result.

Already Sorted Graph

If the graph already has nodes arranged in a valid topological order, the algorithm still works correctly and preserves the order after processing.

Cycle in Graph

If the graph contains a cycle (e.g., A → B → C → A), then topological sorting is not possible. A DFS-based approach needs an extra mechanism, like tracking nodes in the recursion stack, to detect cycles and prevent incorrect results.

Finally

Topological sort using DFS is a powerful technique to handle task sequencing problems in graphs. By understanding how DFS backtracks and uses a stack, we can ensure correct ordering of tasks based on dependencies. However, always verify that the input graph is acyclic, as cycles make topological sorting invalid.

This step-by-step approach builds intuition for beginners and prepares you to handle both regular and edge-case scenarios effectively.

Algorithm Steps

  1. Initialize a visited set and a stack (for order).
  2. Define a recursive dfs(node):
    • Mark node as visited.
    • For each neighbor of node:
      • If not visited, call dfs(neighbor).
    • After visiting all neighbors, push node to stack.
  3. Call dfs on all unvisited nodes.
  4. Reverse the stack to get the topological order.

Code

JavaScript
function topologicalSort(graph) {
  const visited = new Set();
  const stack = [];

  function dfs(node) {
    visited.add(node);
    for (const neighbor of graph[node] || []) {
      if (!visited.has(neighbor)) {
        dfs(neighbor);
      }
    }
    stack.push(node);
  }

  for (const node in graph) {
    if (!visited.has(Number(node))) {
      dfs(Number(node));
    }
  }

  return stack.reverse();
}

const graph = {
  5: [0, 2],
  4: [0, 1],
  2: [3],
  3: [1],
  0: [],
  1: []
};

console.log("Topological Sort:", topologicalSort(graph));

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(V + E)Even in the best case, each vertex and its edges must be visited to ensure proper order.
Average CaseO(V + E)Every node is visited once and all its outgoing edges are checked once during DFS.
Worst CaseO(V + E)In the worst case, the graph is dense, and all vertices and edges must be processed.

Space Complexity

O(V + E)

Explanation: We use a visited set (O(V)), a stack (O(V)), and an adjacency list (O(V + E)) to represent the graph.


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