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

Floyd Warshall Algorithm All-Pairs Shortest Path

Problem Statement

The Floyd Warshall Algorithm solves the problem of finding the shortest path between all pairs of vertices in a weighted, directed graph.

The graph is given as an n × n adjacency matrix matrix, where matrix[i][j] represents the weight of the edge from node i to node j. A value of -1 indicates that there is no direct edge between the two nodes.

Your task is to compute the shortest distances between every pair of nodes. If there is no path between two nodes, the corresponding distance should remain -1.

Examples

Input Matrix Output Matrix Description
[[0, 3, -1], [-1, 0, 1], [4, -1, 0]] [[0, 3, 4], [5, 0, 1], [4, 7, 0]] Shortest paths computed between all vertex pairs
[[0, 5, -1], [-1, 0, 2], [3, -1, 0]] [[0, 5, 7], [5, 0, 2], [3, 8, 0]] Path from 0 to 2 via 1 with weight 5+2=7
[[0, -1, -1], [-1, 0, -1], [-1, -1, 0]] [[0, -1, -1], [-1, 0, -1], [-1, -1, 0]] No edges means no shorter paths can be found

Solution

Understanding the Problem

The Floyd Warshall Algorithm is used to find the shortest distances between every pair of nodes in a weighted graph. This is also called the All-Pairs Shortest Path problem. The graph may be directed or undirected and can contain positive or negative edge weights—but no negative weight cycles.

You are given a matrix where matrix[i][j] represents the weight of the edge from node i to node j. If there is no direct edge between i and j, the value is -1, which we will interpret as "infinity" (no connection). Our goal is to fill this matrix with the shortest path values between all pairs of nodes.

Step-by-Step Solution with Example

Step 1: Convert -1 to Infinity

Since -1 means there's no direct path between two nodes, we replace it with a large number (e.g., 1e9 or Infinity) so that it doesn't interfere in the minimum comparisons during the algorithm.

Step 2: Initialize the Graph

Create a distance matrix from the input graph. The diagonal entries (i.e., distance from node to itself) should be 0, and all other entries should be as per the input (or converted to infinity if there’s no edge).

Step 3: Run Floyd Warshall Triple Loop

We now go through each node k as an intermediate node. For each pair of nodes (i, j), we check whether going from i → k → j is shorter than the direct path i → j. If it is, we update the distance.

The idea is to allow all nodes to be potential "intermediate stations" between any two other nodes.

Step 4: Convert Infinity Back to -1

Once the algorithm completes, any value in the distance matrix that is still Infinity means no path exists. So, we convert those back to -1 to match the input format.

Example:


// Input matrix
[
  [0,  3, -1],
  [-1, 0,  1],
  [4, -1, 0]
]

// After converting -1 to Infinity
[
  [0,  3,  INF],
  [INF, 0, 1],
  [4, INF, 0]
]

// Run Floyd Warshall updates:
→ After considering node 0:
  no changes

→ After considering node 1:
  matrix[0][2] = min(INF, 3 + 1) = 4
  matrix[2][1] = min(INF, 4 + 3) = 7

→ After considering node 2:
  matrix[1][0] = min(INF, 1 + 4) = 5

// Final result after converting back INF to -1
[
  [0,  3,  4],
  [5,  0,  1],
  [4,  7,  0]
]

Edge Cases

  • No path between nodes: Represented as -1 initially and should stay -1 in the final matrix.
  • Self loops: Distance from a node to itself is always 0.
  • Disconnected Graph: If a node is completely disconnected, all entries to and from that node (except to itself) remain -1.
  • Negative edge weights: Allowed, as long as there are no negative weight cycles.

Finally

The Floyd Warshall Algorithm is elegant and powerful for dense graphs and small-to-medium size node counts. It ensures that by the end of the triple-loop process, all indirect paths are checked, and the shortest ones are stored.

This algorithm runs in O(n³) time, so it is not ideal for very large graphs. But when you need shortest paths between every pair of vertices, this algorithm is often the go-to solution.

Algorithm Steps

  1. Let matrix be the input adjacency matrix of size n × n.
  2. Initialize a distance matrix dist with the same values as matrix.
  3. Replace all -1 values (except diagonals) in dist with Infinity.
  4. For each node k from 0 to n-1:
    1. For each node i from 0 to n-1:
    2. For each node j from 0 to n-1:
    3. If dist[i][k] + dist[k][j] < dist[i][j], update dist[i][j].
  5. Convert all Infinity values in dist back to -1.

Code

JavaScript
function floydWarshall(matrix) {
  const n = matrix.length;
  const dist = Array.from({ length: n }, (_, i) =>
    matrix[i].map(val => (val === -1 && i !== matrix[i].indexOf(val) ? Infinity : val))
  );

  for (let k = 0; k < n; k++) {
    for (let i = 0; i < n; i++) {
      for (let j = 0; j < n; j++) {
        if (dist[i][k] !== Infinity && dist[k][j] !== Infinity) {
          dist[i][j] = Math.min(dist[i][j], dist[i][k] + dist[k][j]);
        }
      }
    }
  }

  // Convert Infinity back to -1
  for (let i = 0; i < n; i++) {
    for (let j = 0; j < n; j++) {
      if (dist[i][j] === Infinity) dist[i][j] = -1;
    }
  }

  return dist;
}

const matrix = [
  [0, 3, -1],
  [-1, 0, 1],
  [4, -1, 0]
];

console.log("Shortest distances:", floydWarshall(matrix));

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