Product of minimum edge weight between all pairs of a Tree

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Given a tree with N vertices and N-1 Edges. Let’s define a function F(a, b) which is equal to the minimum edge weight in the path between node a & b. The task is to calculate the product of all such F(a, b). Here a&b are unordered pairs and a!=b.
So, basically, we need to find the value of:

                           where 0<=i<j<=n-1.

In the input, we will be given the value of N and then N-1 lines follow. Each line contains 3 integers u, v, w denoting edge between node u and v and it’s weight w. Since the product will be very large, output it modulo 10^9+7.
Examples:

Input :
N = 4
1 2 1
1 3 3
4 3 2
Output : 12
Given tree is:
          1
      (1)/  \(3)
       2     3
              \(2)
               4
We will calculate the minimum edge weight between all the pairs:
F(1, 2) = 1         F(2, 3) = 1
F(1, 3) = 3         F(2, 4) = 1
F(1, 4) = 2         F(3, 4) = 2
Product of all F(i, j) = 1*3*2*1*1*2 = 12 mod (10^9 +7) =12

Input :
N = 5
1 2 1
1 3 3
4 3 2
1 5 4
Output :
288

If we observe carefully then we will see that if there is a set of nodes in which minimum edge weight is w and if we add a node to this set that connects the node with the whole set by an edge of weight W such that W<w then path formed between recently added node to all nodes present in the set will have minimum weight W.
So, here we can apply Disjoint-Set Union concept to solve the problem.
First, sort the data structure according to decreasing weight. Initially assign all nodes as a single set. Now when we merge two sets then do the following:-

      Product=(present weight)^(size of set1*size of set2).

We will multiply this product value for all edges of the tree.
Below is the implementation of the above approach:

C++

// C++ Implementation of above approach
#include <bits/stdc++.h>
using namespace std;
#define mod 1000000007
 
// Function to return  (x^y) mod p
int power(int x, unsigned int y, int p)
{
    int res = 1;
 
    x = x % p;
 
    while (y > 0) {
 
        if (y & 1)
            res = (res * x) % p;
        y = y >> 1;
        x = (x * x) % p;
    }
    return res;
}
 
// Declaring size array globally
int size[300005];
int freq[300004];
vector<pair<int, pair<int, int> > > edges;
 
// Initializing DSU data structure
void initialize(int Arr[], int N)
{
    for (int i = 0; i < N; i++) {
        Arr[i] = i;
        size[i] = 1;
    }
}
 
// Function to find the root of ith
// node in the disjoint set
int root(int Arr[], int i)
{
    while (Arr[i] != i) {
        i = Arr[i];
    }
    return i;
}
 
// Weighted union using Path Compression
void weighted_union(int Arr[],
                    int size[], int A, int B)
{
    int root_A = root(Arr, A);
    int root_B = root(Arr, B);
 
    // size of set A is small than size of set B
    if (size[root_A] < size[root_B]) {
        Arr[root_A] = Arr[root_B];
        size[root_B] += size[root_A];
    }
 
    // size of set B is small than size of set A
    else {
        Arr[root_B] = Arr[root_A];
        size[root_A] += size[root_B];
    }
}
 
// Function to add an edge in the tree
void AddEdge(int a, int b, int w)
{
    edges.push_back({ w, { a, b } });
}
 
// Build the tree
void MakeTree()
{
    AddEdge(1, 2, 1);
    AddEdge(1, 3, 3);
    AddEdge(3, 4, 2);
}
 
// Function to return the required product
int MinProduct()
{
    int result = 1;
 
    // Sorting the edges with respect to its weight
    sort(edges.begin(), edges.end());
 
    // Start iterating in decreasing order of weight
    for (int i = edges.size() - 1; i >= 0; i--) {
 
        // Determine Current edge values
        int curr_weight = edges[i].first;
        int Node1 = edges[i].second.first;
        int Node2 = edges[i].second.second;
 
        // Calculate root of each node
        // and size of each set
        int Root1 = root(freq, Node1);
        int Set1_size = size[Root1];
        int Root2 = root(freq, Node2);
        int Set2_size = size[Root2];
 
        // Using the formula
        int prod = Set1_size * Set2_size;
        int Product = power(curr_weight, prod, mod);
 
        // Calculating final result
        result = ((result % mod) * 
                             (Product % mod)) % mod;
 
        // Weighted union using Path Compression
        weighted_union(freq, size, Node1, Node2);
    }
    return result % mod;
}
 
// Driver code
int main()
{
    int n = 4;
 
    initialize(freq, n);
 
    MakeTree();
 
    cout << MinProduct();
}

Java

// Java Implementation of above approach
 
import java.util.ArrayList;
import java.util.Collections;
 
public class Product {
 
    // to store first vertex, second vertex and weight of
    // the edge
    static class Edge implements Comparable<Edge> {
        int first, second, weight;
        Edge(int x, int y, int w)
        {
            this.first = x;
            this.second = y;
            this.weight = w;
        }
 
        @Override public int compareTo(Edge edge)
        {
            return this.weight - edge.weight;
        }
    }
 
    static int mod = 1000000007;
 
    // Function to return (x^y) mod p
    static int power(int x, int y, int p)
    {
        int res = 1;
        x = x % p;
        while (y > 0) {
            if ((y & 1) == 1)
                res = (res * x) % p;
            y = y >> 1;
            x = (x * x) % p;
        }
        return res;
    }
 
    // Declaring size array globally
    static int size[] = new int[300005];
    static int freq[] = new int[300004];
    static ArrayList<Edge> edges = new ArrayList<Edge>();
 
    // Initializing DSU data structure
    static void initialize(int Arr[], int N)
    {
        for (int i = 0; i < N; i++) {
            Arr[i] = i;
            size[i] = 1;
        }
    }
 
    // Function to find the root of ith
    // node in the disjoint set
    static int root(int Arr[], int i)
    {
        while (Arr[i] != i) {
            i = Arr[i];
        }
        return i;
    }
 
    // Weighted union using Path Compression
    static void weighted_union(int Arr[], int size[], int A,
                               int B)
    {
        int root_A = root(Arr, A);
        int root_B = root(Arr, B);
 
        // size of set A is small than size of set B
        if (size[root_A] < size[root_B]) {
            Arr[root_A] = Arr[root_B];
            size[root_B] += size[root_A];
        }
 
        // size of set B is small than size of set A
        else {
            Arr[root_B] = Arr[root_A];
            size[root_A] += size[root_B];
        }
    }
 
    // Function to add an edge in the tree
    static void AddEdge(int a, int b, int w)
    {
        edges.add(new Edge(a, b, w));
    }
 
    // Build the tree
    static void MakeTree()
    {
        AddEdge(1, 2, 1);
        AddEdge(1, 3, 3);
        AddEdge(3, 4, 2);
    }
 
    // Function to return the required product
    static int MinProduct()
    {
        int result = 1;
 
        // Sorting the edges with respect to its weight
        // ascending order
        Collections.sort(edges);
 
        // Start iterating in decreasing order of weight
        for (int i = edges.size() - 1; i >= 0; i--) {
 
            // Determine Current edge values
            int curr_weight = edges.get(i).weight;
            int Node1 = edges.get(i).first;
            int Node2 = edges.get(i).second;
 
            // Calculate root of each node
            // and size of each set
            int Root1 = root(freq, Node1);
            int Set1_size = size[Root1];
            int Root2 = root(freq, Node2);
            int Set2_size = size[Root2];
 
            // Using the formula
            int prod = Set1_size * Set2_size;
            int Product = power(curr_weight, prod, mod);
 
            // Calculating final result
            result
                = ((result % mod) * (Product % mod)) % mod;
 
            // Weighted union using Path Compression
            weighted_union(freq, size, Node1, Node2);
        }
        return result % mod;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int n = 4;
        initialize(freq, n);
        MakeTree();
        System.out.println(MinProduct());
    }
}
 
// This code is contributed by jainlovely450

Python3

# Python3 implementation of the approach
mod = 1000000007
 
# Function to return (x^y) mod p
def power(x: int, y: int, p: int) -> int:
    res = 1
    x %= p
    while y > 0:
        if y & 1:
            res = (res * x) % p
        y = y // 2
        x = (x * x) % p
    return res
 
# Declaring size array globally
size = [0] * 300005
freq = [0] * 300004
edges = []
 
# Initializing DSU data structure
def initialize(arr: list, N: int):
    for i in range(N):
        arr[i] = i
        size[i] = 1
 
# Function to find the root of ith
# node in the disjoint set
def root(arr: list, i: int) -> int:
    while arr[i] != i:
        i = arr[i]
    return i
 
# Weighted union using Path Compression
def weighted_union(arr: list, size: list, A: int, B: int):
    root_A = root(arr, A)
    root_B = root(arr, B)
 
    # size of set A is small than size of set B
    if size[root_A] < size[root_B]:
        arr[root_A] = arr[root_B]
        size[root_B] += size[root_A]
 
    # size of set B is small than size of set A
    else:
        arr[root_B] = arr[root_A]
        size[root_A] += size[root_B]
 
# Function to add an edge in the tree
def AddEdge(a: int, b: int, w: int):
    edges.append((w, (a, b)))
 
# Build the tree
def makeTree():
    AddEdge(1, 2, 1)
    AddEdge(1, 3, 3)
    AddEdge(3, 4, 2)
 
# Function to return the required product
def minProduct() -> int:
    result = 1
 
    # Sorting the edges with respect to its weight
    edges.sort(key = lambda a: a[0])
 
    # Start iterating in decreasing order of weight
    for i in range(len(edges) - 1, -1, -1):
 
        # Determine Current edge values
        curr_weight = edges[i][0]
        node1 = edges[i][1][0]
        node2 = edges[i][1][1]
 
        # Calculate root of each node
        # and size of each set
        root1 = root(freq, node1)
        set1_size = size[root1]
        root2 = root(freq, node2)
        set2_size = size[root2]
 
        # Using the formula
        prod = set1_size * set2_size
        product = power(curr_weight, prod, mod)
 
        # Calculating final result
        result = ((result % mod) * (product % mod)) % mod
 
        # Weighted union using Path Compression
        weighted_union(freq, size, node1, node2)
 
    return result % mod
 
# Driver Code
if __name__ == "__main__":
 
    # Number of nodes and edges
    n = 4
    initialize(freq, n)
    makeTree()
    print(minProduct())
 
# This code is contributed by
# sanjeev2552

Javascript

<script>
 
// JavaScript implementation of the approach
const mod = 1000000007
 
// Function to return (x^y) mod p
function power(x, y, p){
    let res = 1
    x %= p
    while(y > 0){
        if(y & 1)
            res = (res * x) % p
        y = Math.floor(y / 2)
        x = (x * x) % p
    }
    return res
}
 
// Declaring size array globally
let size = new Array(300005).fill(0)
let freq = new Array(300004).fill(0)
let edges = []
 
// Initializing DSU data structure
function initialize(arr, N){
    for(let i=0;i<N;i++){
        arr[i] = i
        size[i] = 1
    }
}
 
// Function to find the root of ith
// node in the disjoint set
function root(arr, i){
    while(arr[i] != i)
        i = arr[i]
    return i
}
 
// Weighted union using Path Compression
function weighted_union(arr, size, A, B){
    let root_A = root(arr, A)
    let root_B = root(arr, B)
 
    // size of set A is small than size of set B
    if(size[root_A] < size[root_B]){
        arr[root_A] = arr[root_B]
        size[root_B] += size[root_A]
    }
 
    // size of set B is small than size of set A
    else{
        arr[root_B] = arr[root_A]
        size[root_A] += size[root_B]
    }
}
 
// Function to add an edge in the tree
function AddEdge(a, b, w){
    edges.push([w, [a, b]])
}
 
// Build the tree
function makeTree(){
    AddEdge(1, 2, 1)
    AddEdge(1, 3, 3)
    AddEdge(3, 4, 2)
}
 
// Function to return the required product
function minProduct(){
    let result = 1
 
    // Sorting the edges with respect to its weight
    edges.sort((a,b)=>a[0]-b[0])
 
    // Start iterating in decreasing order of weight
    for(let i=edges.length - 1;i>=0;i--){
 
        // Determine Current edge values
        let curr_weight = edges[i][0]
        let node1 = edges[i][1][0]
        let node2 = edges[i][1][1]
 
        // Calculate root of each node
        // and size of each set
        let root1 = root(freq, node1)
        let set1_size = size[root1]
        let root2 = root(freq, node2)
        let set2_size = size[root2]
 
        // Using the formula
        let prod = set1_size * set2_size
        let product = power(curr_weight, prod, mod)
 
        // Calculating final result
        result = ((result % mod) * (product % mod)) % mod
 
        // Weighted union using Path Compression
        weighted_union(freq, size, node1, node2)
    }
 
    return result % mod
}
 
// Driver Code
 
// Number of nodes and edges
let n = 4
initialize(freq, n)
makeTree()
document.write(minProduct(),"</br>")
 
// This code is contributed by shinjanpatra
</script>

C#

// C# code for the above approach
using System;
using System.Collections.Generic;
 
namespace MinimumProductSpanningTree {
 
// to store first vertex, second vertex and weight of
// the edge
class Edge : IComparable<Edge> {
    public int First
    {
        get;
        set;
    }
    public int Second
    {
        get;
        set;
    }
    public int Weight
    {
        get;
        set;
    }
 
    public Edge(int x, int y, int w)
    {
        First = x;
        Second = y;
        Weight = w;
    }
 
    public int CompareTo(Edge edge)
    {
        return Weight - edge.Weight;
    }
}
 
class Program {
    static readonly int Mod = 1000000007;
 
    // Declaring size array globally
    static readonly int[] Size = new int[300005];
    static readonly int[] Freq = new int[300004];
    static readonly List<Edge> Edges = new List<Edge>();
    // Function to Inialize DSU
    static void Initialize(int[] arr, int n)
    {
        for (int i = 0; i < n; i++) {
            arr[i] = i;
            Size[i] = 1;
        }
    }
    // Function to find the root of ith
    // node in the disjoint set
    static int Root(int[] arr, int i)
    {
        while (arr[i] != i) {
            i = arr[i];
        }
 
        return i;
    }
 
    static void WeightedUnion(int[] arr, int[] size, int a,
                              int b)
    {
        int rootA = Root(arr, a);
        int rootB = Root(arr, b);
 
        if (size[rootA] < size[rootB]) {
            arr[rootA] = arr[rootB];
            size[rootB] += size[rootA];
        }
        else {
            arr[rootB] = arr[rootA];
            size[rootA] += size[rootB];
        }
    }
 
    static void AddEdge(int a, int b, int w)
    {
        Edges.Add(new Edge(a, b, w));
    }
    // Build the tree
    static void MakeTree()
    {
        AddEdge(1, 2, 1);
        AddEdge(1, 3, 3);
        AddEdge(3, 4, 2);
    }
    // Function to return the required product
    static int MinProduct()
    {
        int result = 1;
        // Sorting the edges with respect to its weight
        // ascending order
        Edges.Sort();
        // Start iterating in decreasing order of weight
        for (int i = Edges.Count - 1; i >= 0; i--) {
            // Determine Current edge values
            int currWeight = Edges[i].Weight;
            int node1 = Edges[i].First;
            int node2 = Edges[i].Second;
 
            // Calculate root of each node
            // and size of each set
            int root1 = Root(Freq, node1);
            int set1Size = Size[root1];
            int root2 = Root(Freq, node2);
            int set2Size = Size[root2];
            // Using the formula
            int prod = set1Size * set2Size;
            int product = Power(currWeight, prod, Mod);
            // Calculating final result
            result = (result * product) % Mod;
            // Weighted union using Path Compression
            WeightedUnion(Freq, Size, node1, node2);
        }
 
        return result;
    }
    // Function to return (x^y) mod p
    static int Power(int x, int y, int p)
    {
        int res = 1;
        x = x % p;
        while (y > 0) {
            if ((y & 1) == 1) {
                res = (res * x) % p;
            }
 
            y = y >> 1;
            x = (x * x) % p;
        }
 
        return res;
    }
    // Driver code
    static void Main(string[] args)
    {
        Initialize(Freq, 300004);
        MakeTree();
        Console.WriteLine(MinProduct());
    }
}
}
// This code is contributed by Potta Lokesh

Output:

Time Complexity : O(N*logN)

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