Dynamic Connectivity | Set 1 (Incremental)

Dynamic connectivity is a data structure that dynamically maintains the information about the 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++

// 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;
}

Java

// Java implementation of
// incremental connectivity
import java.util.*;

class GFG
{

// Finding the root of node i
static 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
static 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
static boolean areSame(int arr[],
                       int a, int b)
{
    return (root(arr, a) == root(arr, b));
}

// Performing an operation
// according to query type
static 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)
            System.out.println("Yes");
        else
            System.out.println("No");
    }

    // 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 Code
public static void main(String[] args)
{
    // 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 = new int[n];
    int []rank = new int[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);
}
}

// This code is contributed by Rajput-Ji

Python3

# 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 disjoint set 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] = 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

C#

// C# implementation of
// incremental connectivity
using System;

class GFG
{

// Finding the root of node i
static 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
static 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
static Boolean areSame(int []arr,
                       int a, int b)
{
    return (root(arr, a) == root(arr, b));
}

// Performing an operation
// according to query type
static 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)
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }

    // 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 Code
public static void Main(String[] args)
{
    // 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 = new int[n];
    int []rank = new int[n];

    // initializing both array and rank
    for (int i = 0; i < n; i++)
    {
        arr[i] = i;
        rank[i] = 1;
    }

    // number of queries
    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 PrinciRaj1992

Javascript

//Javascript implementation of incremental connectivity

      // Finding the root of node i
      function root(arr, i) {
        while (arr[i] != i) {
          arr[i] = arr[arr[i]];
          i = arr[i];
        }
        return i;
      }

      // union of two nodes a and b
      function weighted_union(arr, rank, a, b) {
        let root_a = root(arr, a);
        let 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
      function areSame(arr, a, b) {
        return root(arr, a) == root(arr, b);
      }

      // Performing an operation according to query type
      function 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) console.log("Yes");
          else console.log("No");
        }

        // 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

      // No.of nodes
      let 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

      let arr = new Array(n);
      let rank = new Array(n);
      // initializing both array and rank
      for (let i = 0; i < n; i++) {
        arr[i] = i;
        rank[i] = 1;
      }

      // number of queries
      let 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);

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.

The space complexity is O(n) since we are creating two arrays: arr and rank with length n.

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. O(alpha(n))