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

Shortest Path in a DAG Using Topological Sort

Problem Statement

Given a Directed Acyclic Graph (DAG) with n vertices and weighted edges, find the shortest path from a given source node to all other nodes.

For this problem, we assume that the source node is always node 0. The graph is represented using an edge list where each edge has a weight. Your task is to compute the shortest distances from the source to all other nodes using topological sorting.

Examples

Nodes Edges Output Description
6 [[0,1,2],[0,4,1],[1,2,3],[4,2,2],[2,3,6],[4,5,4],[5,3,1]] [0,2,3,6,1,5] Shortest paths from node 0 using topological order and edge relaxation
3 [[0,1,5],[1,2,3]] [0,5,8] Path 0 → 1 → 2 accumulates weight
4 [[0,1,1],[0,2,4],[1,3,2],[2,3,1]] [0,1,4,3] Shortest path from 0 to 3 is via node 2
1 [] [0] Single node, no edges
3 [[1,2,1]] [0, ∞, ∞] No outgoing edges from source 0

Solution

Understanding the Problem

We are given a Directed Acyclic Graph (DAG) and a source node. Our goal is to find the shortest path from this source to all other nodes in the graph.

Unlike general graphs where we might use Dijkstra’s algorithm, here we can take advantage of the acyclic nature of the graph. Since there are no cycles, a topological order of nodes exists. In this order, for every directed edge u → v, node u comes before v. This means we can process each node and update distances of its neighbors in a single pass, efficiently solving the problem.

Step-by-Step Solution with Example

Step 1: Represent the Graph

Suppose we are given the following edge list for a graph with 6 nodes (0 to 5):


edges = [
  (0, 1, 2),
  (0, 4, 1),
  (1, 2, 3),
  (4, 2, 2),
  (2, 3, 6),
  (4, 5, 4),
  (5, 3, 1)
]

Each tuple (u, v, w) represents a directed edge from u to v with weight w. We first convert this into an adjacency list for easier processing.

Step 2: Topological Sort

We perform a topological sort on the DAG. This gives us an order in which we can safely process the nodes, ensuring that all prerequisites for a node are processed before it.

For the above graph, one possible topological order is: [0, 1, 4, 5, 2, 3].

Step 3: Initialize Distances

We set up an array dist to keep track of the shortest distance from the source node to every other node. Initially, all distances are set to infinity except for the source (let's say source is node 0), which is set to 0:


dist = [0, ∞, ∞, ∞, ∞, ∞]

Step 4: Process Nodes in Topological Order

Now we traverse the nodes in topological order. For each node, we look at its outgoing edges and try to "relax" them. Relaxing means: if the current known distance to the neighbor is greater than the distance to the current node plus the edge weight, we update it.

Let's walk through it:

  • From node 0 → 1 (weight 2): dist[1] = min(∞, 0+2) = 2
  • From node 0 → 4 (weight 1): dist[4] = min(∞, 0+1) = 1
  • From node 1 → 2 (weight 3): dist[2] = min(∞, 2+3) = 5
  • From node 4 → 2 (weight 2): dist[2] = min(5, 1+2) = 3
  • From node 4 → 5 (weight 4): dist[5] = min(∞, 1+4) = 5
  • From node 5 → 3 (weight 1): dist[3] = min(∞, 5+1) = 6
  • From node 2 → 3 (weight 6): dist[3] = min(6, 3+6) = 6 (no change)

Final dist array: [0, 2, 3, 6, 1, 5]

Edge Cases

  • Disconnected Nodes: If a node is not reachable from the source, its distance will remain ∞. Always check for such cases before interpreting the results.
  • Multiple Valid Topological Orders: The topological sort is not unique. As long as it's a valid topological order, the final distances will be the same.
  • Zero-weight Edges: Edges with weight 0 are still valid and can affect shortest paths. Our relaxation logic should include them.
  • Negative Weights: Since we’re in a DAG, negative edge weights are safe and won’t create cycles. The algorithm works fine with them.

Finally

Using topological sort for shortest path in DAGs is optimal and elegant. It avoids unnecessary overhead of priority queues or repeated visits. Understanding how topological order guarantees that each node is processed only after all of its dependencies is the key to this approach.

Just remember: this method only works for DAGs. If your graph has cycles, you must use other algorithms like Dijkstra or Bellman-Ford depending on the context.

Algorithm Steps

  1. Construct an adjacency list from the given edges.
  2. Perform a topological sort of the DAG using DFS.
  3. Initialize a distance array with Infinity for all nodes, except the source (node 0) which is set to 0.
  4. Iterate through each node in topological order:
    1. For each neighbor v of the current node u, update dist[v] = min(dist[v], dist[u] + weight)
  5. Return the final distance array. Nodes unreachable from source will retain Infinity.

Code

JavaScript
function shortestPathDAG(n, edges) {
  const adj = Array.from({ length: n }, () => []);
  for (const [u, v, w] of edges) {
    adj[u].push([v, w]);
  }

  const visited = new Array(n).fill(false);
  const stack = [];

  function topoSort(node) {
    visited[node] = true;
    for (const [neighbor] of adj[node]) {
      if (!visited[neighbor]) {
        topoSort(neighbor);
      }
    }
    stack.push(node);
  }

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

  const dist = new Array(n).fill(Infinity);
  dist[0] = 0;

  while (stack.length) {
    const node = stack.pop();
    if (dist[node] !== Infinity) {
      for (const [neighbor, weight] of adj[node]) {
        if (dist[node] + weight < dist[neighbor]) {
          dist[neighbor] = dist[node] + weight;
        }
      }
    }
  }

  return dist;
}

// Example usage:
console.log("Shortest Path:", shortestPathDAG(6, [[0,1,2],[0,4,1],[1,2,3],[4,2,2],[2,3,6],[4,5,4],[5,3,1]])); // Output: [0,2,3,6,1,5]

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