Minimum path need to reverse to reach each node in a tree from each vertex

Given a directed tree consisting of N nodes valued from [0, N - 1] and M edges, the task is to find the minimum number of edges that need to reverse for each node X such that there is a path from node X to each vertex of the given Tree.

Examples:

Input: N = 6, edges[][] = {{0, 1}, {1, 3}, {2, 3}, {4, 0}, {4, 5}}
Output: 2 2 2 3 1 2
Explanation:
The answer for node 0 is 2, which can be calculated as:

From 0 to 0: No edges are required to reverse to reach 0 from 0.
From 0 to 1: Can be reached directly using edge 0 -> 1.
From 0 to 2: The edge 2 -> 3 must be reversed for the path 0 -> 1 -> 3 -> 2 to reach 2 from node 0.
From 0 to 3: Can be reached directly as 0 -> 1 -> 3.
From 0 to 4: The edge 4 -> 0 must be reversed to reach 4 from node 0.
From 0 to 5: The edge 4 -> 0 must be reversed to reach 5 from node 0 as 0 -> 4 -> 5

To reach every node from the node 0, edge 2 -> 3 and edge 4 -> 0 is reversed. So, a total of 2 edges is reversed for node 0. Similarly, the ans for all the nodes can be calculated.

Input: N = 5, edges[][] = {{1, 0}, {1, 2}, {3, 2}, {3, 4}}
Output: 2 1 2 1 2

Approach: To solve the above problem, the idea is to store the directed edge in the adjacency list along with the reversed directed edge with the negative sign i.e. for directed edge a -> b store the edge a -> b and b -> -a. Then, for each node X of the tree, the answer can be calculated as the number of negative edges encountered in the simple Depth For Search(DFS) from that node X.Follow the steps below to solve the problem:

• Initialize a 2-dimensional vector, say graph[][], to store the edges of the graph.
• Traverse the array edges[][] and for each pair (a, b), push the directed edge b in graph[a] and reversed directed edge -a in graph[b].
• Iterate over the range N, and for each node:
  • Initialize a variable, say count = 0, to count the required number of edges to reverse.
  • Call a recursive function, say reorderPaths(node, graph, count, visited) to perform the DFS on the tree.
  • Increase the value of count for each of the negative edges traversed in the DFS.
  • Print the value of the count, for each iteration of the node.

Below is the implementation of the above approach:

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to perform the DFS traversal
// of the tree with reordered paths
void reorderPaths(
    int s, vector<vector<int> > graph,
    int& count, vector<bool>& visited)
{
    visited[s] = true;

    // Traverse the adjacency list of
    // the source node
    for (auto i : graph[s]) {

        if (!visited[abs(i)]) {

            // Reorder the path
            if (i < 0)
                count++;

            // Recursively Call DFS
            reorderPaths(abs(i), graph,
                         count, visited);
        }
    }
}

// Function to find minimum edges to
// reverse to make the tree vertices
// reachable for each node
void minReorder(int n, vector<vector<int> > edges)
{
    // Stores the edges
    vector<vector<int> > graph(n);

    // Traversing the childs
    for (int i = 0; i < edges.size(); i++) {
        int a = edges[i][0];
        int b = edges[i][1];

        // Storing the direct edge
        graph[a].push_back(b);

        // Storing the reverse edge
        graph[b].push_back(-a);
    }

    // Finding ans for each node
    for (int i = 0; i < n; i++) {
        vector<bool> visited(n, false);
        int count = 0;

        // Function Call
        reorderPaths(i, graph, count, visited);
        cout << count << " ";
    }
}

// Driver Code
int main()
{
    int N = 6;
    vector<vector<int> > edges
        = { { 0, 1 }, { 1, 3 }, { 2, 3 }, { 4, 0 }, { 4, 5 } };
    minReorder(N, edges);

    return 0;
}

// Java program for the above approach
import java.util.*;

public class GFG{
static int count;
  
// Function to perform the DFS traversal
// of the tree with reordered paths
static void reorderPaths(
    int s, Vector<Integer>[] graph, boolean[] visited)
{
    visited[s] = true;

    // Traverse the adjacency list of
    // the source node
    for (int i : graph[s]) {

        if (!visited[(Math.abs(i))]) {

            // Reorder the path
            if (i < 0)
                count++;

            // Recursively Call DFS
            reorderPaths(Math.abs(i), graph, visited);
        }
    }
}

// Function to find minimum edges to
// reverse to make the tree vertices
// reachable for each node
static void minReorder(int n, int [][] edges)
{
  
    // Stores the edges
    Vector<Integer>[] graph = new Vector[n];
    for (int i = 0; i < n; i++) 
        graph[i] = new Vector<>();
  
    // Traversing the childs
    for (int i = 0; i < edges.length; i++) {

        int a = edges[i][0];
        int b = edges[i][1];

        // Storing the direct edge
        graph[a].add(b);

        // Storing the reverse edge
        graph[b].add(-a);
    }

    // Finding ans for each node
    for (int i = 0; i < n; i++) {
        boolean []visited = new boolean[n];
        count = 0;

        // Function Call
        reorderPaths(i, graph,  visited);
        System.out.print(count+ " ");
    }
}

// Driver Code
public static void main(String[] args)
{
    int N = 6;
    int [][] edges
        = { { 0, 1 }, { 1, 3 }, { 2, 3 }, { 4, 0 }, { 4, 5 } };
    minReorder(N, edges);

}
}

// This code is contributed by 29AjayKumar 

# Python 3 program for the above approach
count = 0
visited = []

# Function to perform the DFS traversal
# of the tree with reordered paths
def reorderPaths(s, graph):
    global count
    visited[s] = True

    # Traverse the adjacency list of
    # the source node
    for i in graph[s]:
        if(visited[abs(i)]==False):
            # Reorder the path
            if (i < 0):
                count += 1

            # Recursively Call DFS
            reorderPaths(abs(i), graph)

# Function to find minimum edges to
# reverse to make the tree vertices
# reachable for each node
def minReorder(n, edges):
    global count
    global visited
    # Stores the edges
    graph = [[] for i in range(n)]

    # Traversing the childs
    for i in range(len(edges)):
        a = edges[i][0]
        b = edges[i][1]

        # Storing the direct edge
        graph[a].append(b)

        # Storing the reverse edge
        graph[b].append(-a)

    # Finding ans for each node
    for i in range(n):
        visited = [False for i in range(n)]
        count = 0

        # Function Call
        reorderPaths(i, graph)
        print(count,end = " ")

# Driver Code
if __name__ == '__main__':
    N = 6
    edges = [[0, 1],[1, 3],[2, 3],[4, 0],[4, 5]]
    minReorder(N, edges)
    
    # This code is contributed by SURENDRA_GANGWAR.

// C# code for above approach
using System;
using System.Collections.Generic;

public class GFG {
    static int count;

    // Function to perform the DFS traversal
    // of the tree with reordered paths
    static void ReorderPaths(int s, List<int>[] graph,
                             bool[] visited)
    {
        visited[s] = true;

        // Traverse the adjacency list of
        // the source node
        foreach(int i in graph[s])
        {

            if (!visited[(Math.Abs(i))]) {

                // Reorder the path
                if (i < 0)
                    count++;

                // Recursively Call DFS
                ReorderPaths(Math.Abs(i), graph, visited);
            }
        }
    }

    // Function to find minimum edges to
    // reverse to make the tree vertices
    // reachable for each node
    static void MinReorder(int n, int[][] edges)
    {

        // Stores the edges
        List<int>[] graph = new List<int>[ n ];
        for (int i = 0; i < n; i++)
            graph[i] = new List<int>();

        // Traversing the childs
        for (int i = 0; i < edges.Length; i++) {

            int a = edges[i][0];
            int b = edges[i][1];

            // Storing the direct edge
            graph[a].Add(b);

            // Storing the reverse edge
            graph[b].Add(-a);
        }

        // Finding ans for each node
        for (int i = 0; i < n; i++) {
            bool[] visited = new bool[n];
            count = 0;

            // Function Call
            ReorderPaths(i, graph, visited);
            Console.Write(count + " ");
        }
    }

    // Driver Code
    public static void Main(string[] args)
    {
        int N = 6;
        int[][] edges
            = { new int[] { 0, 1 }, new int[] { 1, 3 },
                new int[] { 2, 3 }, new int[] { 4, 0 },
                new int[] { 4, 5 } };

        MinReorder(N, edges);
    }
}

// This code is contributed by Vaibhav.

<script>
    // Javascript program for the above approach
    
    // Function to perform the DFS traversal
    // of the tree with reordered paths
    
    let graph;
    let edges;
    let count;
    let visited;
    
    function reorderPaths(s)
    {
        visited[s] = true;

        // Traverse the adjacency list of
        // the source node
        for(let i  = 0; i < graph[s].length; i++) {

            if (!visited[(Math.abs(graph[s][i]))]) {

                // Reorder the path
                if (graph[s][i] < 0)
                    count++;

                // Recursively Call DFS
                reorderPaths(Math.abs(graph[s][i]));
            }
        }
    }

    // Function to find minimum edges to
    // reverse to make the tree vertices
    // reachable for each node
    function minReorder(n)
    {
        // Stores the edges
        graph = [];
        for(let i = 0; i < n; i++)
        {
            graph.push([]);
        }

        // Traversing the childs
        for (let i = 0; i < edges.length; i++) {
            let a = edges[i][0];
            let b = edges[i][1];

            // Storing the direct edge
            graph[a].push(b);

            // Storing the reverse edge
            graph[b].push(-a);
        }

        // Finding ans for each node
        for (let i = 0; i < n; i++) {
            visited = new Array(n);
            visited.fill(false);
            count = 0;

            // Function Call
            reorderPaths(i);
            document.write(count + " ");
        }
    }
    
    let N = 6;
    edges = [ [ 0, 1 ], [ 1, 3 ], [ 2, 3 ], [ 4, 0 ], [ 4, 5 ] ];
    minReorder(N);

// This code is contributed by decode2207.
</script>

2 2 2 3 1 2

Time Complexity: O(N²)
Auxiliary Space: O(N)