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

Dijkstra’s Algorithm Using Priority Queue

Problem Statement

You are given a weighted, undirected, and connected graph with V vertices represented using an adjacency list. Each entry adj[i] contains a list of lists, where each sublist contains two integers j and weight, representing an edge from vertex i to vertex j with a given weight.

Given a source vertex S, your task is to find the shortest distances from S to all other vertices using Dijkstra’s Algorithm and return them as a list.

Examples

Vertices Adjacency List Source Shortest Distances
3 [ [ [1, 1], [2, 4] ], [ [0, 1], [2, 2] ], [ [0, 4], [1, 2] ] ] 0 [0, 1, 3]
2 [ [ [1, 5] ], [ [0, 5] ] ] 1 [5, 0]
1 [ [] ] 0 [0]
4 [ [ [1, 2] ], [ [0, 2], [2, 1] ], [ [1, 1], [3, 5] ], [ [2, 5] ] ] 0 [0, 2, 3, 8]

Solution

Understanding the Problem

We are given a weighted graph and a starting node (source). Our goal is to find the shortest distance from this source to all other nodes in the graph.

In other words, for every node in the graph, we want to calculate the minimum possible sum of edge weights required to reach that node from the source.

This problem is a classic case of finding single-source shortest paths in graphs with non-negative weights, and Dijkstra’s Algorithm is an efficient way to solve it using a priority queue.

Step-by-Step Solution with Example

Step 1: Choose an Example

Let's consider the following weighted graph with 5 nodes (0 to 4), and the edges:


0 --(2)--> 1
0 --(4)--> 2
1 --(1)--> 2
1 --(7)--> 3
2 --(3)--> 4
3 --(1)--> 4

We want to find the shortest distance from node 0 to all other nodes.

Step 2: Initialize the Distance Array

Create a distance[] array and fill it with Infinity to represent that we don’t yet know the shortest path. Set the distance of the source node to 0.


distance = [0, ∞, ∞, ∞, ∞]

Step 3: Use a Min-Heap (Priority Queue)

Use a priority queue (min-heap) to always pick the next node with the smallest known distance. Initially, insert the source node with distance 0:


priorityQueue = [(0, 0)]  // (distance, node)

Step 4: Start the Main Loop

While the priority queue is not empty:

  1. Extract the node with the smallest distance.
  2. Explore all its neighbors.
  3. If a shorter path to a neighbor is found, update the distance and insert it into the queue.

Step 5: Trace the Execution for Our Example

  • Start with node 0, distance 0.
  • Check edges: (0 → 1, weight 2), (0 → 2, weight 4)
  • Update: distance[1] = 2, distance[2] = 4
  • Queue: [(2,1), (4,2)]
  • Next node: 1, distance 2
  • Check edges: (1 → 2, weight 1) → 2 + 1 = 3 is better than current 4, update distance[2] = 3
  • Also (1 → 3, weight 7) → distance[3] = 9
  • Queue: [(3,2), (4,2), (9,3)]
  • Next node: 2, distance 3
  • Check edge (2 → 4, weight 3) → distance[4] = 6
  • Queue: [(4,2), (9,3), (6,4)]
  • Next node: 4, distance 6 → Already the best.
  • Check edge (4 → 3, weight 1) → 6 + 1 = 7 is better than 9 → update distance[3] = 7

Step 6: Final Distance Array


distance = [0, 2, 3, 7, 6]

This gives the shortest path from node 0 to every other node in the graph.

Edge Cases

  • Disconnected Graph: If a node is not reachable from the source, its distance will remain Infinity.
  • Zero Weight Edges: Dijkstra’s algorithm handles them correctly as long as weights are non-negative.
  • Multiple Edges Between Nodes: Always consider the one with the minimum weight during edge relaxation.
  • Self-loops: They can be ignored unless required for the specific application.

Finally

Dijkstra’s Algorithm is a powerful tool for finding the shortest paths in a graph with non-negative weights. Using a priority queue helps efficiently expand the closest node at every step. Always remember to initialize distances properly and carefully update them when a shorter path is found. This algorithm is widely used in routing, pathfinding, and network optimization problems.

Algorithm Steps

  1. Initialize dist[] with Infinity, set dist[S] = 0.
  2. Create a priority queue (min-heap) and insert {0, S}.
  3. While the queue is not empty:
    1. Pop the vertex with the minimum distance from the queue.
    2. For each adjacent vertex v of the current node:
      1. If dist[u] + weight < dist[v], update dist[v] and add {dist[v], v} to the queue.
  4. Return dist[] as the shortest distances from source S.

Code

JavaScript
function dijkstra(V, adj, S) {
  const dist = new Array(V).fill(Infinity);
  dist[S] = 0;
  const minHeap = [[0, S]]; // [distance, vertex]

  while (minHeap.length > 0) {
    minHeap.sort((a, b) => a[0] - b[0]);
    const [d, u] = minHeap.shift();

    for (const [v, wt] of adj[u]) {
      if (d + wt < dist[v]) {
        dist[v] = d + wt;
        minHeap.push([dist[v], v]);
      }
    }
  }

  return dist;
}

// Example:
const adjList = [
  [[1, 1], [2, 4]],
  [[0, 1], [2, 2]],
  [[0, 4], [1, 2]]
];
console.log("Shortest distances:", dijkstra(3, adjList, 0)); // Output: [0, 1, 3]

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