# Assign weights to edges such that longest path in terms of weights is minimized

• Last Updated : 13 Sep, 2022

Given the edges of a tree and a sum S. The task is to assign weights to all the edges of the tree such that the longest path in terms of weights is minimized and the total sum of weights assigned should be S and print the longest path’s weight.

Note: Edges can be assigned any weights in range [0, S] and can be fractional also.

Examples:

```Input:
1
/     |     \
2      3      4
S = 3
Output: 2
All the edges can be assigned weights of 1, so
the longest path will in terms of weight will be
2--1--4 or 2--1--3

Input:   1
/
2
/      \
3        4
/   \
5      6
S = 1
Output: 0.50
Assign the given below weights to edges.
1--2: 0.25
2--3: 0.25
2--4: 0
4--5: 0.25
4--6: 0.25

Hence the longest path in terms of weight is 1--2--3
or 1--2--4--5 or 1--2--4--6. ```

Approach: The property of a tree that a path can have a maximum of two leaf nodes in it can be used to solve the above problem. So if we assign weights only to the edges connecting the leaf nodes, and assign other edges to 0. Then every edge connecting to the leaf nodes will be assigned s/(number of leaf nodes). Since a path can contain a maximum of two leaf nodes, hence the longest path will be 2 * (s/number of leaf nodes).

Below is the implementation of the above approach:

## C++

 `// C++ program to assign weights to edges to``// minimize the longest path in terms of weight``#include ``using` `namespace` `std;` `// Function to add edges``void` `addEdges(``int` `u, ``int` `v, vector<``int``> adj[])``{``    ``adj[u].push_back(v);``    ``adj[v].push_back(u);``}` `// Function that assigns weights``// and returns the longest path``long` `double` `longestPath(vector<``int``> adj[],``                              ``int` `s, ``int` `n)``{``    ``int` `cnt = 0;``    ``for` `(``int` `i = 1; i <= n; i++) {` `        ``if` `(adj[i].size() == 1)``            ``cnt++;``    ``}` `    ``long` `double` `ans =``       ``2.0 * (``long` `double``)(s / (``long` `double``)(cnt));``    ``return` `ans;``}` `// Driver Code``int` `main()``{``    ``int` `n = 4;` `    ``// Create an adjacency list``    ``// to store tree``    ``vector<``int``> adj[n + 1];` `    ``// Add edges``    ``addEdges(1, 2, adj);``    ``addEdges(1, 3, adj);``    ``addEdges(1, 4, adj);` `    ``// Given Sum``    ``int` `s = 3;` `    ``// Function that prints the``    ``// longest path in terms of weights``    ``cout << longestPath(adj, s, n);``}`

## Java

 `// Java program to assign weights to edges to``// minimize the longest path in terms of weight``import` `java.util.*;` `class` `GFG``{` `    ``// Function to add edges``    ``static` `void` `addEdges(``int` `u, ``int` `v, Vector adj[])``    ``{``        ``adj[u].add(v);``        ``adj[v].add(u);``    ``}` `    ``// Function that assigns weights``    ``// and returns the longest path``    ``static` `double` `longestPath(Vector adj[], ``int` `s, ``int` `n)``    ``{``        ``int` `cnt = ``0``;``        ``for` `(``int` `i = ``1``; i <= n; i++)``        ``{` `            ``if` `(adj[i].size() == ``1``)``                ``cnt++;``        ``}` `        ``double` `ans = ``2.0` `* (``double``) (s / (``double``) (cnt));``        ``return` `ans;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `n = ``4``;` `        ``// Create an adjacency list``        ``// to store tree``        ``Vector[] adj = ``new` `Vector[n + ``1``];``        ``for` `(``int` `i = ``0``; i < n + ``1``; i++)``            ``adj[i] = ``new` `Vector();` `        ``// Add edges``        ``addEdges(``1``, ``2``, adj);``        ``addEdges(``1``, ``3``, adj);``        ``addEdges(``1``, ``4``, adj);` `        ``// Given Sum``        ``int` `s = ``3``;` `        ``// Function that prints the``        ``// longest path in terms of weights``        ``System.out.print(longestPath(adj, s, n));``    ``}``}` `// This code is contributed by Rajput-Ji`

## Python3

 `# Python3 program to assign weights to``# edges to minimize the longest path``# in terms of weight` `# Function to add edges``def` `addEdges(u, v, adj):` `    ``adj[u].append(v)``    ``adj[v].append(u)` `# Function that assigns weights``# and returns the longest path``def` `longestPath(adj, s, n):` `    ``cnt ``=` `0``    ``for` `i ``in` `range``(``1``, n ``+` `1``):` `        ``if` `len``(adj[i]) ``=``=` `1``:``            ``cnt ``+``=` `1` `    ``ans ``=` `2` `*` `(s ``/` `cnt)``    ``return` `ans` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:` `    ``n ``=` `4` `    ``# Create an adjacency list``    ``# to store tree``    ``adj ``=` `[[] ``for` `i ``in` `range``(n ``+` `1``)]` `    ``# Add edges``    ``addEdges(``1``, ``2``, adj)``    ``addEdges(``1``, ``3``, adj)``    ``addEdges(``1``, ``4``, adj)` `    ``# Given Sum``    ``s ``=` `3` `    ``# Function that prints the``    ``# longest path in terms of weights``    ``print``(longestPath(adj, s, n))` `# This code is contributed by Rituraj Jain`

## C#

 `// C# program to assign weights to edges to``// minimize the longest path in terms of weight``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{` `    ``// Function to add edges``    ``static` `void` `addEdges(``int` `u, ``int` `v, List<``int``> []adj)``    ``{``        ``adj[u].Add(v);``        ``adj[v].Add(u);``    ``}` `    ``// Function that assigns weights``    ``// and returns the longest path``    ``static` `double` `longestPath(List<``int``> []adj, ``int` `s, ``int` `n)``    ``{``        ``int` `cnt = 0;``        ``for` `(``int` `i = 1; i <= n; i++)``        ``{` `            ``if` `(adj[i].Count == 1)``                ``cnt++;``        ``}` `        ``double` `ans = 2.0 * (``double``) (s / (``double``) (cnt));``        ``return` `ans;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``int` `n = 4;` `        ``// Create an adjacency list``        ``// to store tree``        ``List<``int``>[] adj = ``new` `List<``int``>[n + 1];``        ``for` `(``int` `i = 0; i < n + 1; i++)``            ``adj[i] = ``new` `List<``int``>();` `        ``// Add edges``        ``addEdges(1, 2, adj);``        ``addEdges(1, 3, adj);``        ``addEdges(1, 4, adj);` `        ``// Given Sum``        ``int` `s = 3;` `        ``// Function that prints the``        ``// longest path in terms of weights``        ``Console.Write(longestPath(adj, s, n));``    ``}``}` `// This code is contributed by Rajput-Ji`

## Javascript

 ``

Output

`2`

Complexity Analysis:

• Time Complexity: O(N), as we are using a loop to traverse N times, where N is the number of nodes in the tree.
• Auxiliary Space: O(N), as we are using extra space for adj array.

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