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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Python3





# 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





Output:


1






          
          
          
          

            

    

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