Dynamic connectivity is a data structure that dynamically maintains the information about thee connected components of graph. In simple words suppose there is a graph G(V, E) in which no. of vertices V is constant but no. of edges E is variable. There are three ways in which we can change the number of edges

- Incremental Connectivity : Edges are only added to the graph.
- Decremental Connectivity : Edges are only deleted from the graph.
- Fully Dynamic Connectivity : Edges can both be deleted and added to the graph.

In this article only **Incremental connectivity **is discussed. There are mainly two operations that need to be handled.

- An edge is added to the graph.
- Information about two nodes x and y whether they are in the same connected components or not.

Example:

Input : V = 7 Number of operations = 11 1 0 1 2 0 1 2 1 2 1 0 2 2 0 2 2 2 3 2 3 4 1 0 5 2 4 5 2 5 6 1 2 6 Note: 7 represents number of nodes, 11 represents number of queries. There are two types of queries Type 1 : 1 x y in this if the node x and y are connected print Yes else No Type 2 : 2 x y in this add an edge between node x and y Output : No Yes No Yes Explanation : Initially there are no edges so node 0 and 1 will be disconnected so answer will be No Node 0 and 2 will be connected through node 1 so answer will be Yes similarly for other queries we can find whether two nodes are connected or not

To solve the problems of incremental connectivity disjoint data structure is used. Here each connected component represents a set and if the two nodes belong to the same set it means that they are connected.

C++ implementation is given below here we are using union by rank and path compression

`// C++ implementation of incremental connectivity ` `#include<bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Finding the root of node i ` `int` `root(` `int` `arr[], ` `int` `i) ` `{ ` ` ` `while` `(arr[i] != i) ` ` ` `{ ` ` ` `arr[i] = arr[arr[i]]; ` ` ` `i = arr[i]; ` ` ` `} ` ` ` `return` `i; ` `} ` ` ` `// union of two nodes a and b ` `void` `weighted_union(` `int` `arr[], ` `int` `rank[], ` ` ` `int` `a, ` `int` `b) ` `{ ` ` ` `int` `root_a = root (arr, a); ` ` ` `int` `root_b = root (arr, b); ` ` ` ` ` `// union based on rank ` ` ` `if` `(rank[root_a] < rank[root_b]) ` ` ` `{ ` ` ` `arr[root_a] = arr[root_b]; ` ` ` `rank[root_b] += rank[root_a]; ` ` ` `} ` ` ` `else` ` ` `{ ` ` ` `arr[root_b] = arr[root_a]; ` ` ` `rank[root_a] += rank[root_b]; ` ` ` `} ` `} ` ` ` `// Returns true if two nodes have same root ` `bool` `areSame(` `int` `arr[], ` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `(root(arr, a) == root(arr, b)); ` `} ` ` ` `// Performing an operation according to query type ` `void` `query(` `int` `type, ` `int` `x, ` `int` `y, ` `int` `arr[], ` `int` `rank[]) ` `{ ` ` ` `// type 1 query means checking if node x and y ` ` ` `// are connected or not ` ` ` `if` `(type == 1) ` ` ` `{ ` ` ` `// If roots of x and y is same then yes ` ` ` `// is the answer ` ` ` `if` `(areSame(arr, x, y) == ` `true` `) ` ` ` `cout << ` `"Yes"` `<< endl; ` ` ` `else` ` ` `cout << ` `"No"` `<< endl; ` ` ` `} ` ` ` ` ` `// type 2 query refers union of x and y ` ` ` `else` `if` `(type == 2) ` ` ` `{ ` ` ` `// If x and y have different roots then ` ` ` `// union them ` ` ` `if` `(areSame(arr, x, y) == ` `false` `) ` ` ` `weighted_union(arr, rank, x, y); ` ` ` `} ` `} ` ` ` `// Driver function ` `int` `main() ` `{ ` ` ` `// No.of nodes ` ` ` `int` `n = 7; ` ` ` ` ` `// The following two arrays are used to ` ` ` `// implement disjoint set data structure. ` ` ` `// arr[] holds the parent nodes while rank ` ` ` `// array holds the rank of subset ` ` ` `int` `arr[n], rank[n]; ` ` ` ` ` `// initializing both array and rank ` ` ` `for` `(` `int` `i=0; i<n; i++) ` ` ` `{ ` ` ` `arr[i] = i; ` ` ` `rank[i] = 1; ` ` ` `} ` ` ` ` ` `// number of queries ` ` ` `int` `q = 11; ` ` ` `query(1, 0, 1, arr, rank); ` ` ` `query(2, 0, 1, arr, rank); ` ` ` `query(2, 1, 2, arr, rank); ` ` ` `query(1, 0, 2, arr, rank); ` ` ` `query(2, 0, 2, arr, rank); ` ` ` `query(2, 2, 3, arr, rank); ` ` ` `query(2, 3, 4, arr, rank); ` ` ` `query(1, 0, 5, arr, rank); ` ` ` `query(2, 4, 5, arr, rank); ` ` ` `query(2, 5, 6, arr, rank); ` ` ` `query(1, 2, 6, arr, rank); ` ` ` `return` `0; ` `} ` |

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Output:

No Yes No Yes

**Time Complexity:**

The amortized time complexity is O(alpha(n)) per operation where alpha is inverse ackermann function which is nearly constant.

**Reference:**

https://en.wikipedia.org/wiki/Dynamic_connectivity

This article is contributed by **Ayush Jha**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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