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

Minimum Operations to Make Network Connected

Problem Statement

You are given a network of n computers numbered from 0 to n-1, connected by a list of m edges. Each edge connects two computers directly. In one operation, you can remove any existing edge and reconnect it between any two disconnected computers.

Your task is to determine the minimum number of such operations required to ensure that all computers are directly or indirectly connected. If it is not possible to connect the entire network, return -1.

Examples

n Edges Output Description
4 [[0,1],[0,2],[1,2]] 1 We can remove the edge [1,2] and use it to connect node 3
6 [[0,1],[0,2],[0,3],[1,4]] -1 Not enough edges to connect all components
5 [[0,1],[0,2],[3,4],[2,3],[2,4]] 0 The network is already connected
3 [[0,1],[1,2],[2,0]] 0 All nodes form a cycle and are already connected
4 [[0,1]] -1 Only 1 edge is not enough to connect 4 nodes

Solution

Understanding the Problem

We are given n computers labeled from 0 to n - 1 and a list of connections where each connection connects two computers directly. Our goal is to determine the minimum number of operations required to make the entire network connected — meaning every computer can reach every other computer, either directly or indirectly.

In each operation, we are allowed to rewire an existing connection — that is, remove one connection and place it between any two disconnected computers. The challenge is to determine whether this is possible and if so, what the minimum number of such operations is.

Step-by-Step Solution with Example

Step 1: Basic Requirement Check

To connect n computers, we must have at least n - 1 connections. This is a fundamental rule in graph theory: to connect n nodes in a single connected component without cycles, you need at least n - 1 edges.

So, if the number of connections is less than n - 1, we immediately return -1 because it's impossible to connect all computers.

Step 2: Count Connected Components

If we have at least n - 1 edges, then it's possible — but we still need to figure out how many disconnected parts (components) are present in the network.

We can use either Depth-First Search (DFS) or the Union-Find (Disjoint Set Union) algorithm to count how many separate connected components exist in the network.

Step 3: Calculate Operations Needed

If there are c connected components, then we need at least c - 1 operations to connect them all. This is because each operation can connect two components into one. So if we have 3 components, we need 2 operations to make them into 1.

Step 4: Work Through Example

Let's say n = 6 and the connections are:


[[0,1], [0,2], [0,3], [1,4]]
This is 4 connections, but we need at least 6 - 1 = 5 connections. Since we have only 4, it’s impossible to connect all 6 computers. We return -1.

Now, consider another example:


n = 6
connections = [[0,1], [0,2], [0,3], [1,4], [2,5]]
Here, we have 5 connections which is exactly n - 1. We run Union-Find or DFS and see all computers are connected — so only 1 component exists. No operations needed. We return 0.

Edge Cases

  • Not Enough Edges: If the number of connections is less than n - 1, we must return -1 right away — impossible case.
  • Already Connected: If the network is already a single connected component, no operations are required — return 0.
  • Multiple Disconnected Components: Count the components and return components - 1 as the number of needed operations.

Finally

This problem is a classic application of graph connectivity. The key insight is recognizing that to fully connect a network of n computers, you need at least n - 1 edges. Once that condition is met, counting connected components helps determine how many rearrangements are needed.

Always begin by validating input constraints (like number of edges), then apply a graph traversal algorithm to understand the structure of the network.

Algorithm Steps

  1. If number of edges < n - 1, return -1 — not enough edges to connect the graph.
  2. Initialize n nodes and mark them unvisited.
  3. Use DFS or Union-Find to count the number of connected components c.
  4. Return c - 1 as the minimum number of operations required.

Code

JavaScript
function makeConnected(n, connections) {
  if (connections.length < n - 1) return -1; // Not enough edges

  const parent = Array.from({ length: n }, (_, i) => i);

  function find(x) {
    if (parent[x] !== x) {
      parent[x] = find(parent[x]);
    }
    return parent[x];
  }

  function union(x, y) {
    const rootX = find(x);
    const rootY = find(y);
    if (rootX !== rootY) {
      parent[rootY] = rootX;
      return true;
    }
    return false;
  }

  let components = n;
  for (const [a, b] of connections) {
    if (union(a, b)) {
      components--;
    }
  }

  return components - 1;
}

console.log(makeConnected(4, [[0,1],[0,2],[1,2]])); // 1
console.log(makeConnected(6, [[0,1],[0,2],[0,3],[1,4]])); // -1

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