Minimum distance to visit all the nodes of an undirected weighted tree

Given a weighted tree with N nodes starting from 1 to N. The distance between any two nodes is given by the edge weight. Node 1 is the source, the task is to visit all the nodes of the tree with minimum distance travelled.

Examples:

Input:
u[] = {1, 1, 2, 2, 1}
v[] = {2, 3, 5, 6, 4}
w[] = {1, 4, 2, 50, 5}
Output: 73



Input:
u[] = {1, 2}
v[] = {2, 3}
w[] = {3, 1}
Output: 4

Approach: Let’s suppose there are n leaf l1, l2, l3, ……, ln and the cost of the path from root to each leaf is c1, c2, c3, ……, cn.

To travel from l1 to l2 some of the edges will be visited twice ( till the LCA of l1 and l2 all the edges will be visited twice ), for l2 to l3 and some of the edges will be visited ( till the LCA of l2 and l3 all the edges will be visited twice ) twice and similary every edge of the tree will be visited twice ( observation ).

To minimise the cost of travelling, the maximum cost path from the root to some leaf should be avoided.

Hence the cost = (c1 + c2 + c3 + …… + cn) – max(c1, c2, c3, ……, cn)

min cost = (2 * sum of edge weight)max(c1, c2, c3, ……, cn)

DFS can be used with some modification to find the largest distance.

Below is the implementation of the above approach:

Java

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// Java implementation of the approach
import java.util.LinkedList;
import java.util.Scanner;
  
class Graph {
  
    class Edge {
  
        // from - The source of an edge
        // to - destination of an edge
        // wt - distance between two nodes
        int from;
        int to;
        long wt;
        Edge(int a, int b, long w)
        {
            from = a;
            to = b;
            wt = w;
        }
    }
  
    // adj_lis - It is used to
    // make the adjacency list of a tree
  
    // V - Total number of nodes in a tree
  
    // val - This array stores the
    // distance from node 1 to node 'n'
    static LinkedList<Edge>[] adj_lis;
    static int V;
    static long val[];
  
    Graph(int v)
    {
        this.V = v;
        adj_lis = new LinkedList[V];
        for (int i = 0; i < V; i++)
            adj_lis[i] = new LinkedList<>();
    }
  
    // Method to add an edge between two nodes
    void add_edge(int to, int from, long wt)
    {
        adj_lis[from].add(
            new Edge(from, to, wt));
        adj_lis[to].add(
            new Edge(to, from, wt));
    }
  
    // DFS method to find distance
    // between node 1 to other nodes
    void dfs(int v,
             int par,
             long sum,
             boolean[] visited)
    {
        val[v] = sum;
        visited[v] = true;
        for (Edge e : adj_lis[v]) {
            if (!visited[e.to])
                dfs(e.to,
                    v,
                    sum + e.wt,
                    visited);
        }
    }
  
    // Driver code
    public static void main(String a[])
    {
  
        // Number of nodes
        int v = 6;
        Graph obj = new Graph(v);
        val = new long[v];
        boolean[] visited
            = new boolean[v];
  
        int sum = 0;
  
        // Edge from a node to another
        // node with some weight
        int from[] = { 2, 3, 5, 6, 4 };
        int to[] = { 1, 1, 2, 2, 1 };
        int wt[] = { 1, 4, 2, 50, 5 };
  
        for (int i = 0; i < v - 1; i++) {
            sum += 2 * wt[i];
            obj.add_edge(to[i] - 1,
                         from[i] - 1,
                         wt[i]);
        }
  
        obj.dfs(0, -1, 0, visited);
        long large = Integer.MIN_VALUE;
  
        // Loop to find largest
        // distance in a val.
        for (int i = 1; i < val.length;
             i++)
            if (val[i] > large)
                large = val[i];
  
        System.out.println(sum - large);
    }
}

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C#

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// C# program to implement 
// the above appraoch
using System;
using System.Collections.Generic;
class Graph
{
    public class Edge
    {
  
        // from - The source of an edge
        // to - destination of an edge
        // wt - distance between two nodes
        public int from;
        public int to;
        public long wt;
        public Edge(int a, int b, long w)
        {
            from = a;
            to = b;
            wt = w;
        }
    }
  
    // adj_lis - It is used to
    // make the adjacency list of a tree
  
    // V - Total number of nodes in a tree
  
    // val - This array stores the
    // distance from node 1 to node 'n'
    public static List<Edge>[] adj_lis;
    public static int V;
    public static long []val;
  
    public Graph(int v)
    {
        V = v;
        adj_lis = new List<Edge>[V];
        for (int i = 0; i < V; i++)
            adj_lis[i] = new List<Edge>();
    }
  
    // Method to add an edge between two nodes
    void add_edge(int to, int from, long wt)
    {
        adj_lis[from].Add(
                 new Edge(from, to, wt));
        adj_lis[to].Add(
               new Edge(to, from, wt));
    }
  
    // DFS method to find distance
    // between node 1 to other nodes
    void dfs(int v,
            int par,
            long sum,
            bool[] visited)
    {
        val[v] = sum;
        visited[v] = true;
        foreach (Edge e in adj_lis[v]) 
        {
            if (!visited[e.to])
                dfs(e.to, v, 
                    sum + e.wt, visited);
        }
    }
  
    // Driver code
    public static void Main(String []a)
    {
  
        // Number of nodes
        int v = 6;
        Graph obj = new Graph(v);
        val = new long[v];
        bool []visited = new bool[v];
  
        int sum = 0;
  
        // Edge from a node to another
        // node with some weight
        int []from = { 2, 3, 5, 6, 4 };
        int []to = { 1, 1, 2, 2, 1 };
        int []wt = { 1, 4, 2, 50, 5 };
  
        for (int i = 0; i < v - 1; i++) 
        {
            sum += 2 * wt[i];
            obj.add_edge(to[i] - 1,
                       from[i] - 1, wt[i]);
        }
  
        obj.dfs(0, -1, 0, visited);
        long large = int.MinValue;
  
        // Loop to find largest
        // distance in a val.
        for (int i = 1; i < val.Length;
            i++)
            if (val[i] > large)
                large = val[i];
  
        Console.WriteLine(sum - large);
    }
}
  
// This code is contributed by Princi Singh

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Output:

73


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