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:

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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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Output:

1


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