Dynamic Connectivity | Set 1 (Incremental)

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

  1. Incremental Connectivity : Edges are only added to the graph.
  2. Decremental Connectivity : Edges are only deleted from the graph.
  3. 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.

  1. An edge is added to the graph.
  2. 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.

Implementation is given below here we are using union by rank and path compression

C++

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// 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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Python3

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# Python3 implementation of
# incremental connectivity 
  
# Finding the root of node i 
def root(arr, i):
    while (arr[i] != i):
        arr[i] = arr[arr[i]] 
        i = arr[i]
    return i
  
# union of two nodes a and b 
def weighted_union(arr, rank, a, b):
    root_a = root (arr, a) 
    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 
def areSame(arr, a, b):
    return (root(arr, a) == root(arr, b))
  
# Performing an operation according
# to query type 
def query(type, x, y, arr, 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): 
            print("Yes"
        else:
            print("No")
  
    # type 2 query refers union of
    # x and y 
    elif (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 Code
if __name__ == '__main__':
  
    # No.of nodes 
    n = 7
  
    # The following two arrays are used to 
    # implement disjoset data structure. 
    # arr[] holds the parent nodes while rank 
    # array holds the rank of subset 
    arr = [None] * n
    rank = [None] * n
  
    # initializing both array 
    # and rank
    for i in range(n):
        arr[i] =
        rank[i] = 1
  
    # number of queries 
    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)
  
# This code is contributed by PranchalK

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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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Improved By : PranchalK