Dynamic Disjoint Set Data Structure for large range values

Prerequisites:

Disjoint Set data structure is used to keeps track of a set of elements partitioned into a number of disjoint (non-overlapping) subsets.

In this article, we will learn about constructing the same Data Structure dynamically. This data structure basically helps in situation where we cannot simply use arrays for creating disjoint sets because of large inputs of order 109.



To illustrate this, consider the following problem. We need to find the total number of connected components in the Graph when the total Number of Vertices can be up to 10^9.

Examples:

Input : Edges : { { 1, 2 }, 
                  { 2, 3 }, 
                  { 4, 5 } }
Output : 2
Explanation: {1, 2, 3} forms a component and 
{4, 5} forms another component.

The idea to solve this problem is, we will maintain two hash tables (implemented using unordered_maps in C++). One for parent and other for degree. Parent[V] will give the parent of the component which the Vertex V is part of and Degree will give the number of vertices in that component.

Initially, both Parent and Degree will be empty. We will keep inserting vertices to the maps as sequentially.

See the code and the explanation simultaneously for a better understanding. Below are the methods used in the code to solve the above problem:

  1. getParent(V): This method will give the parent of the vertex V. Here we recursively find the parent of the vertex V( see code), meanwhile we assign all the vertex in that component to have the same parent.( In disjoint set data structure all the vertex in the same component have the same parent.)
  2. Union(): When we add a edge and the two vertexes are of different components we call the Union() method to join both the component. Here the parent of the component formed after joining both the components will be the parent of the component among the two which had more number of vertexes before the union. The degree of the new component is updated accordingly.
  3. getTotalComponent(): Vertex in the same component will have the same parent.
    We use unordered_set (STL) to count the total number of components. As we have maintained the Data Structure as Dynamic so, there can be any vertex which has not been added to any of the components hence they are different component alone. So the total number of components will be given by,

    Total no of Component =  Total Vertices - Number of Vertices
                             in parent (Map)  + Number of Component 
                             formed from the Vertexes inserted 
                             in the graph.
    

Below is the implementation of above idea:

C++

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// Dynamic Disjoint Set Data Structure
// Union-Find
  
#include <bits/stdc++.h>
using namespace std;
  
int N;
int Edges[3][2];
  
// Dynamic Disjoint Set Data Structure
struct DynamicDisjointSetDS {
  
    // We will add the vertex to the edge
    // only when it is asked to i.e. maintain
    // a dynamic DS.
    unordered_map<int, int> parent, degree;
  
    // Total nomber of Vertex in the Graph
    int N;
  
    // Constructor
    DynamicDisjointSetDS(int n)
    {
        N = n;
    }
  
    // Get Parent of vertex V
    int getParent(int vertex)
    {
  
        // If the vertex is already inserted 
        // in the graph
        if (parent.find(vertex) != parent.end()) {
  
            if (parent[vertex] != vertex) {
                parent[vertex] = 
                      getParent(parent[vertex]);
                return parent[vertex];
            }
        }
  
        // if the vertex is operated for the first
        // time
        else {
  
            // insert the vertex and assign its 
            // parent to itself
            parent.insert(make_pair(vertex, vertex));
  
            // Degree of the vertex
            degree.insert(make_pair(vertex, 1));
        }
  
        return vertex;
    }
  
    // Union by Rank
    void Union(int vertexA, int vertexB)
    {
        // Parent of Vertex A
        int x = getParent(vertexA);
  
        // Parent of Vertex B
        int y = getParent(vertexB);
  
        // if both have same parent
        // Do Nothing
        if (x == y)
            return;
  
        // Merging the componet
        // Assigning the parent of smaller Component
        // as the parent of the bigger Component.
        if (degree[x] > degree[y]) {
            parent[y] = x;
            degree[x] = degree[x] + degree[y];
        }
        else {
            parent[x] = y;
            degree[y] = degree[y] + degree[x];
        }
    }
  
    // Count total Component in the Graph
    int GetTotalComponent()
    {
        // To count the total Component formed 
        // from the inserted vertex in the Graph
        unordered_set<int> total;
  
        // Iterate through the parent
        for (auto itr = parent.begin(); 
            itr != parent.end(); itr++) {
  
            // Add the parent of each Vertex
            // to the set
            total.insert(getParent(itr->first));
        }
  
        // Total Component = Total Vertexes -
        // Number of Vertex in the parent + 
        // Number of Component formed from 
        // the Vertexes inserted in the Graph
        return N - parent.size() + total.size();
    }
};
  
// Solve
void Solve()
{
  
    // Declaring the Dynamic Disjoint Set DS
    DynamicDisjointSetDS dsu(N);
  
    // Traversing through the Edges
    for (int i = 0; i < 3; i++) {
  
        // If the Vertexes in the Edges
        // have same parent do nothing
        if (dsu.getParent(Edges[i][0]) == 
            dsu.getParent(Edges[i][1])) {
            continue;
        }
  
        // else Do Union of both the Components.
        else {
            dsu.Union(Edges[i][0], Edges[i][1]);
        }
    }
  
    // Get total Components
    cout << dsu.GetTotalComponent();
}
  
// Driver Code
int main()
{
    // Total Number of Vertexes
    N = 5;
      
    /* Edges
    * 1 <--> 2 
    * 2 <--> 3 
    * 4 <--> 5    */
  
    Edges[0][0] = 1;
    Edges[0][1] = 2;
    Edges[1][0] = 2;
    Edges[1][1] = 3;
    Edges[2][0] = 4;
    Edges[2][1] = 3;
  
    // Solve
    Solve();
  
    return 0;
}

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Python3

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# Dynamic Disjoint Set Data Structure 
# Union-Find 
  
# Dynamic Disjoint Set Data Structure 
class DynamicDisjointSetDS: 
  
    # Constructor 
    def __init__(self, n): 
          
        # Total nomber of Vertex in the Graph 
        self.N = n
          
        # We will add the vertex to the edge 
        # only when it is asked to i.e. maintain 
        # a dynamic DS. 
        self.parent = {}
        self.degree = {}
  
    # Get Parent of vertex V 
    def getParent(self, vertex): 
      
        # If the vertex is already inserted 
        # in the graph 
        if vertex in self.parent: 
  
            if self.parent[vertex] != vertex: 
                self.parent[vertex] = \
                    self.getParent(self.parent[vertex]) 
                      
                return self.parent[vertex] 
  
        # if the vertex is operated 
        # for the first time 
        else:
  
            # insert the vertex and assign 
            # its parent to itself 
            self.parent[vertex] = vertex 
  
            # Degree of the vertex 
            self.degree[vertex] = 1
          
        return vertex 
      
    # Union by Rank 
    def Union(self, vertexA, vertexB): 
      
        # Parent of Vertex A 
        x = self.getParent(vertexA) 
  
        # Parent of Vertex B 
        y = self.getParent(vertexB) 
  
        # if both have same parent 
        # Do Nothing 
        if x == y: 
            return
  
        # Merging the componet 
        # Assigning the parent of smaller Component 
        # as the parent of the bigger Component. 
        if self.degree[x] > self.degree[y]: 
            self.parent[y] =
            self.degree[x] = (self.degree[x] + 
                              self.degree[y]) 
          
        else:
            self.parent[x] =
            self.degree[y] = (self.degree[y] + 
                              self.degree[x]) 
          
    # Count total Component in the Graph 
    def GetTotalComponent(self):
      
        # To count the total Component formed 
        # from the inserted vertex in the Graph 
        total = set() 
  
        # Iterate through the parent 
        for itr in self.parent: 
  
            # Add the parent of each Vertex 
            # to the set 
            total.add(self.getParent(itr)) 
          
        # Total Component = Total Vertexes - 
        # Number of Vertex in the parent + 
        # Number of Component formed from 
        # the Vertexes inserted in the Graph 
        return self.N - len(self.parent) + len(total) 
  
# Solve 
def Solve(N):
  
    # Declaring the Dynamic Disjoint Set DS 
    dsu = DynamicDisjointSetDS(N) 
  
    # Traversing through the Edges 
    for i in range(0, 3): 
  
        # If the Vertexes in the Edges 
        # have same parent do nothing 
        if (dsu.getParent(Edges[i][0]) ==
            dsu.getParent(Edges[i][1])): 
            continue
  
        # else Do Union of both the Components. 
        else:
            dsu.Union(Edges[i][0], Edges[i][1]) 
  
    # Get total Components 
    print(dsu.GetTotalComponent())
  
# Driver Code 
if __name__ == "__main__"
  
    # Total Number of Vertexes 
    N = 5
    Edges = [[1, 2], [2, 3], [4, 3]]
  
    # Solve 
    Solve(N) 
  
# This code is contributed by 
# Rituraj Jain

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Output:

2

Note: If the number of vertices are even larger, we can implement the same code just by changing the data type from int to long long.



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