Minimum edges to be added in a directed graph so that any node can be reachable from a given node

Given a directed graph and a node X. The task is to find the minimum number of edges that must be added to the graph such that any node can be reachable from the given node.

Examples:

Input: X = 0

Output: 3



Input: X = 4

Output: 1

Approach: First, let’s mark all the vertices reachable from X as good, using a simple DFS. Then, for each bad vertex (vertices which are not reachable from X) v, count the number of bad vertices reachable from v (it also can be done by simple DFS). Let this number be cntv. Now, iterate over all bad vertices in non-increasing order of cntv. For the current bad vertex v, if it is still not marked as good, run a DFS from it, marking all the reachable vertices as good, and increase the answer by 1 (in fact, we are implicitly adding the edge (s, v)). It can be proved that this solution gives an optimal answer.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
  
#include <bits/stdc++.h>
using namespace std;
  
const int N = 5010;
  
int n, x;
  
vector<int> g[N];
  
// To check if the vertex has been
// visited or not
bool vis[N];
  
// To store if vertex is reachable
// from source or not
bool good[N];
  
int cnt;
  
void ADD_EDGE(int u, int v)
{
    g[u].push_back(v);
}
  
// Function to find all good vertices
void dfs1(int v)
{
    good[v] = true;
    for (auto to : g[v])
        if (!good[to])
            dfs1(to);
}
  
// Function to find cnt of all unreachable vertices
void dfs2(int v)
{
    vis[v] = true;
    ++cnt;
    for (auto to : g[v])
        if (!vis[to] && !good[to])
            dfs2(to);
}
  
// Function to return the minimum edges required
int Minimum_Edges()
{
  
    // Find all vertices reachable from the source
    dfs1(x);
  
    // To store all vertices with their cnt value
    vector<pair<int, int> > val;
  
    for (int i = 0; i < n; ++i) {
  
        // If vertex is bad i.e. not reachable
        if (!good[i]) {
            cnt = 0;
            memset(vis, false, sizeof(vis));
  
            // Find cnt of this vertex
            dfs2(i);
            val.push_back(make_pair(cnt, i));
        }
    }
  
    // Sort all unreachable vertices in
    // non-decreasing order of their cnt values
    sort(val.begin(), val.end());
    reverse(val.begin(), val.end());
  
    // Find the minimum number of edges
    // needed to be added
    int ans = 0;
    for (auto it : val) {
        if (!good[it.second]) {
            ++ans;
            dfs1(it.second);
        }
    }
  
    return ans;
}
  
// Driver code
int main()
{
    // Number of nodes and source node
    n = 5, x = 4;
  
    // Add edges to the graph
    ADD_EDGE(0, 1);
    ADD_EDGE(1, 2);
    ADD_EDGE(2, 3);
    ADD_EDGE(3, 0);
  
    cout << Minimum_Edges();
  
    return 0;
}

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Java

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// Java implementation of the approach 
import java.util.*;
  
class GFG
{
      
// pair
static class pair
{
    int first,second;
    pair(int a,int b)
    {
        first = a;
        second = b;
    }
}
  
static int N = 5010
  
static int n, x; 
  
static Vector<Vector<Integer>> g = new Vector<Vector<Integer>>(); 
  
// To check if the vertex has been 
// visited or not 
static boolean vis[] = new boolean[N]; 
  
// To store if vertex is reachable 
// from source or not 
static boolean good[] = new boolean[N]; 
  
static int cnt; 
  
static void ADD_EDGE(int u, int v) 
    g.get(u).add(v); 
  
// Function to find all good vertices 
static void dfs1(int v) 
    good[v] = true
    for (int to = 0; to < g.get(v).size(); to++) 
        if (!good[g.get(v).get(to)]) 
            dfs1(g.get(v).get(to)); 
  
// Function to find cnt of all unreachable vertices 
static void dfs2(int v) 
    vis[v] = true
    ++cnt; 
    for (int to = 0; to < g.get(v).size(); to++) 
        if (!vis[g.get(v).get(to)] && !good[g.get(v).get(to)]) 
            dfs2(g.get(v).get(to)); 
  
// Function to return the minimum edges required 
static int Minimum_Edges() 
  
    // Find all vertices reachable from the source 
    dfs1(x); 
  
    // To store all vertices with their cnt value 
    Vector<pair> val = new Vector<pair>(); 
  
    for (int i = 0; i < n; ++i) 
    
  
        // If vertex is bad i.e. not reachable 
        if (!good[i]) 
        
            cnt = 0
            for(int j = 0; j < vis.length; j++)
                vis[j] = false;
  
            // Find cnt of this vertex 
            dfs2(i); 
            val.add(new pair(cnt, i)); 
        
    
  
    // Sort all unreachable vertices in 
    // non-decreasing order of their cnt values 
    Collections.sort(val,new Comparator<pair>()
    
            public int compare(pair p1, pair p2) 
            
                return p1.first - p2.first; 
            
    }); 
          
    Collections.reverse(val); 
  
    // Find the minimum number of edges 
    // needed to be added 
    int ans = 0
    for (int it = 0; it < val.size(); it++) 
    
        if (!good[val.get(it).second]) 
        
            ++ans; 
            dfs1(val.get(it).second); 
        
    
  
    return ans; 
  
// Driver code 
public static void main(String args[])
    // Number of nodes and source node 
    n = 5; x = 4
      
    for(int i = 0; i < N + 1; i++)
    g.add(new Vector<Integer>());
  
    // Add edges to the graph 
    ADD_EDGE(0, 1); 
    ADD_EDGE(1, 2); 
    ADD_EDGE(2, 3); 
    ADD_EDGE(3, 0); 
  
    System.out.println( Minimum_Edges()); 
}
  
// This code is contributed by Arnab Kundu

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Python3

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# Python3 implementation of the approach
N = 5010
g = [[] for i in range(N)]
  
# To check if the vertex 
# has been visited or not
vis = [False for i in range(N)]
  
# To store if vertex is reachable
# from source or not
good = [False for i in range(N)]
  
def ADD_EDGE(u, v):
  
    g[u].append(v)
  
# Function to find all good vertices
def dfs1(v):
  
    good[v] = True
    for to in g[v]:
        if not good[to]:
            dfs1(to)
  
# Function to find cnt of
# all unreachable vertices
def dfs2(v):
      
    global cnt
    vis[v] = True
    cnt += 1
    for to in g[v]:
        if not vis[to] and not good[to]:
            dfs2(to)
  
# Function to return  
# the minimum edges required
def Minimum_Edges():
  
    global vis, cnt
      
    # Find all vertices reachable
    # from the source
    dfs1(x)
  
    # To store all vertices 
    # with their cnt value
    val = []
  
    for i in range(0, n): 
  
        # If vertex is bad i.e. not reachable
        if not good[i]:
            cnt = 0
            vis = [False for i in range(N)]
  
            # Find cnt of this vertex
            dfs2(i)
            val.append((cnt, i))
  
    # Sort all unreachable vertices 
    # in non-decreasing order of
    # their cnt values
    val.sort(reverse = True)
  
    # Find the minimum number of edges
    # needed to be added
    ans = 0
    for it in val: 
        if not good[it[1]]: 
            ans += 1
            dfs1(it[1])
  
    return ans
  
# Driver code
if __name__ == "__main__":
  
    # Number of nodes and source node
    n, x = 5, 4
  
    # Add edges to the graph
    ADD_EDGE(0, 1)
    ADD_EDGE(1, 2)
    ADD_EDGE(2, 3)
    ADD_EDGE(3, 0)
  
    print(Minimum_Edges())
  
# This code is contributed by Rituraj Jain

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Output:

1


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