Minimum Product Spanning Tree

• Difficulty Level : Medium
• Last Updated : 16 Jun, 2021

Given a connected and undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. A minimum product spanning tree for a weighted, connected and undirected graph is a spanning tree with weight product less than or equal to the weight product of every other spanning tree. The weight product of a spanning tree is the product of weights corresponding to each edge of the spanning tree. All weights of the given graph will be positive for simplicity.

Examples: Minimum Product that we can obtain is
180 for above graph by choosing edges
0-1, 1-2, 0-3 and 1-4

This problem can be solved using standard minimum spanning tree algorithms like krushkal and prim’s algorithm, but we need to modify our graph to use these algorithms. Minimum spanning tree algorithms tries to minimize total sum of weights, here we need to minimize total product of weights. We can use property of logarithms to overcome this problem.
As we know,

log(w1* w2 * w3 * …. * wN) =
log(w1) + log(w2) + log(w3) ….. + log(wN)

We can replace each weight of graph by its log value, then we apply any minimum spanning tree algorithm which will try to minimize sum of log(wi) which in-turn minimizes weight product.
For example graph, steps are shown in below diagram, In below code first we have constructed the log graph from given input graph, then that graph is given as input to prim’s MST algorithm, which will minimize the total sum of weights of tree. Since weight of modified graph are logarithms of actual input graph, we actually minimize the product of weights of spanning tree.

C++

 // A C++ program for getting minimum product // spanning tree The program is for adjacency matrix // representation of the graph #include   // Number of vertices in the graph #define V 5   // A utility function to find the vertex with minimum // key value, from the set of vertices not yet included // in MST int minKey(int key[], bool mstSet[]) {     // Initialize min value     int min = INT_MAX, min_index;       for (int v = 0; v < V; v++)         if (mstSet[v] == false && key[v] < min)             min = key[v], min_index = v;       return min_index; }   // A utility function to print the constructed MST // stored in parent[] and print Minimum Obtaiable // product int printMST(int parent[], int n, int graph[V][V]) {     printf("Edge   Weight\n");     int minProduct = 1;     for (int i = 1; i < V; i++) {         printf("%d - %d    %d \n",                parent[i], i, graph[i][parent[i]]);           minProduct *= graph[i][parent[i]];     }     printf("Minimum Obtainable product is %d\n",            minProduct); }   // Function to construct and print MST for a graph // represented using adjacency matrix representation // inputGraph is sent for printing actual edges and // logGraph is sent for actual MST operations void primMST(int inputGraph[V][V], double logGraph[V][V]) {     int parent[V]; // Array to store constructed MST     int key[V]; // Key values used to pick minimum     // weight edge in cut     bool mstSet[V]; // To represent set of vertices not     // yet included in MST       // Initialize all keys as INFINITE     for (int i = 0; i < V; i++)         key[i] = INT_MAX, mstSet[i] = false;       // Always include first 1st vertex in MST.     key = 0; // Make key 0 so that this vertex is     // picked as first vertex     parent = -1; // First node is always root of MST       // The MST will have V vertices     for (int count = 0; count < V - 1; count++) {         // Pick the minimum key vertex from the set of         // vertices not yet included in MST         int u = minKey(key, mstSet);           // Add the picked vertex to the MST Set         mstSet[u] = true;           // Update key value and parent index of the         // adjacent vertices of the picked vertex.         // Consider only those vertices which are not yet         // included in MST         for (int v = 0; v < V; v++)               // logGraph[u][v] is non zero only for             // adjacent vertices of m mstSet[v] is false             // for vertices not yet included in MST             // Update the key only if logGraph[u][v] is             // smaller than key[v]             if (logGraph[u][v] > 0 && mstSet[v] == false && logGraph[u][v] < key[v])                   parent[v] = u, key[v] = logGraph[u][v];     }       // print the constructed MST     printMST(parent, V, inputGraph); }   // Method to get minimum product spanning tree void minimumProductMST(int graph[V][V]) {     double logGraph[V][V];       // Constructing logGraph from original graph     for (int i = 0; i < V; i++) {         for (int j = 0; j < V; j++) {             if (graph[i][j] > 0)                 logGraph[i][j] = log(graph[i][j]);             else                 logGraph[i][j] = 0;         }     }       // Applyting standard Prim's MST algorithm on     // Log graph.     primMST(graph, logGraph); }   // driver program to test above function int main() {     /* Let us create the following graph            2    3        (0)--(1)--(2)         |   / \   |        6| 8/   \5 |7         | /     \ |        (3)-------(4)              9          */     int graph[V][V] = {         { 0, 2, 0, 6, 0 },         { 2, 0, 3, 8, 5 },         { 0, 3, 0, 0, 7 },         { 6, 8, 0, 0, 9 },         { 0, 5, 7, 9, 0 },     };       // Print the solution     minimumProductMST(graph);       return 0; }

Java

 // A Java program for getting minimum product // spanning tree The program is for adjacency matrix // representation of the graph import java.util.*;   class GFG {       // Number of vertices in the graph     static int V = 5;       // A utility function to find the vertex with minimum     // key value, from the set of vertices not yet included     // in MST     static int minKey(int key[], boolean[] mstSet)     {         // Initialize min value         int min = Integer.MAX_VALUE, min_index = 0;           for (int v = 0; v < V; v++) {             if (mstSet[v] == false && key[v] < min) {                 min = key[v];                 min_index = v;             }         }         return min_index;     }       // A utility function to print the constructed MST     // stored in parent[] and print Minimum Obtaiable     // product     static void printMST(int parent[], int n, int graph[][])     {         System.out.printf("Edge Weight\n");         int minProduct = 1;         for (int i = 1; i < V; i++) {             System.out.printf("%d - %d %d \n",                               parent[i], i, graph[i][parent[i]]);               minProduct *= graph[i][parent[i]];         }         System.out.printf("Minimum Obtainable product is %d\n",                           minProduct);     }       // Function to construct and print MST for a graph     // represented using adjacency matrix representation     // inputGraph is sent for printing actual edges and     // logGraph is sent for actual MST operations     static void primMST(int inputGraph[][], double logGraph[][])     {         int[] parent = new int[V]; // Array to store constructed MST         int[] key = new int[V]; // Key values used to pick minimum         // weight edge in cut         boolean[] mstSet = new boolean[V]; // To represent set of vertices not         // yet included in MST           // Initialize all keys as INFINITE         for (int i = 0; i < V; i++) {             key[i] = Integer.MAX_VALUE;             mstSet[i] = false;         }         // Always include first 1st vertex in MST.         key = 0; // Make key 0 so that this vertex is         // picked as first vertex         parent = -1; // First node is always root of MST           // The MST will have V vertices         for (int count = 0; count < V - 1; count++) {             // Pick the minimum key vertex from the set of             // vertices not yet included in MST             int u = minKey(key, mstSet);               // Add the picked vertex to the MST Set             mstSet[u] = true;               // Update key value and parent index of the             // adjacent vertices of the picked vertex.             // Consider only those vertices which are not yet             // included in MST             for (int v = 0; v < V; v++) // logGraph[u][v] is non zero only for             // adjacent vertices of m mstSet[v] is false             // for vertices not yet included in MST             // Update the key only if logGraph[u][v] is             // smaller than key[v]             {                 if (logGraph[u][v] > 0                     && mstSet[v] == false                     && logGraph[u][v] < key[v]) {                       parent[v] = u;                     key[v] = (int)logGraph[u][v];                 }             }         }           // print the constructed MST         printMST(parent, V, inputGraph);     }       // Method to get minimum product spanning tree     static void minimumProductMST(int graph[][])     {         double[][] logGraph = new double[V][V];           // Constructing logGraph from original graph         for (int i = 0; i < V; i++) {             for (int j = 0; j < V; j++) {                 if (graph[i][j] > 0) {                     logGraph[i][j] = Math.log(graph[i][j]);                 }                 else {                     logGraph[i][j] = 0;                 }             }         }           // Applyting standard Prim's MST algorithm on         // Log graph.         primMST(graph, logGraph);     }       // Driver code     public static void main(String[] args)     {         /* Let us create the following graph         2 3     (0)--(1)--(2)         | / \ |     6| 8/ \5 |7         | /     \ |     (3)-------(4)             9         */         int graph[][] = {             { 0, 2, 0, 6, 0 },             { 2, 0, 3, 8, 5 },             { 0, 3, 0, 0, 7 },             { 6, 8, 0, 0, 9 },             { 0, 5, 7, 9, 0 },         };           // Print the solution         minimumProductMST(graph);     } }   // This code has been contributed by 29AjayKumar

Python3

 # A Python3 program for getting minimum # product spanning tree The program is # for adjacency matrix representation # of the graph import math   # Number of vertices in the graph V = 5   # A utility function to find the vertex # with minimum key value, from the set # of vertices not yet included in MST def minKey(key, mstSet):       # Initialize min value     min = 10000000     min_index = 0           for v in range(V):               if (mstSet[v] == False and                key[v] < min):             min = key[v]             min_index = v       return min_index   # A utility function to print the constructed # MST stored in parent[] and print Minimum # Obtaiable product def printMST(parent, n, graph):           print("Edge Weight")     minProduct = 1           for i in range(1, V):         print("{} - {}   {} ".format(parent[i], i,                             graph[i][parent[i]]))         minProduct *= graph[i][parent[i]]           print("Minimum Obtainable product is {}".format(           minProduct))   # Function to construct and print MST for # a graph represented using adjacency # matrix representation inputGraph is # sent for printing actual edges and # logGraph is sent for actual MST # operations def primMST(inputGraph, logGraph):           # Array to store constructed MST     parent = [0 for i in range(V)]           # Key values used to pick minimum     key = [10000000 for i in range(V)]           # weight edge in cut     # To represent set of vertices not     mstSet = [False for i in range(V)]           # Yet included in MST     # Always include first 1st vertex in MST           # Make key 0 so that this vertex is     key = 0           # Picked as first vertex           # First node is always root of MST     parent = -1       # The MST will have V vertices     for count in range(0, V - 1):               # Pick the minimum key vertex from         # the set of vertices not yet         # included in MST         u = minKey(key, mstSet)           # Add the picked vertex to the MST Set         mstSet[u] = True           # Update key value and parent index         # of the adjacent vertices of the         # picked vertex. Consider only those         # vertices which are not yet         # included in MST         for v in range(V):               # logGraph[u][v] is non zero only             # for adjacent vertices of m             # mstSet[v] is false for vertices             # not yet included in MST. Update             # the key only if logGraph[u][v] is             # smaller than key[v]             if (logGraph[u][v] > 0 and                 mstSet[v] == False and                 logGraph[u][v] < key[v]):                 parent[v] = u                 key[v] = logGraph[u][v]           # Print the constructed MST     printMST(parent, V, inputGraph)   # Method to get minimum product spanning tree def minimumProductMST(graph):       logGraph = [[0 for j in range(V)]                    for i in range(V)]       # Constructing logGraph from     # original graph     for i in range(V):         for j in range(V):             if (graph[i][j] > 0):                 logGraph[i][j] = math.log(graph[i][j])             else:                 logGraph[i][j] = 0               # Applyting standard Prim's MST algorithm     # on Log graph.     primMST(graph, logGraph)   # Driver code if __name__=='__main__':           ''' Let us create the following graph         2 3     (0)--(1)--(2)         | / \ |     6| 8/ \5 |7         | /     \ |     (3)-------(4)             9         '''     graph = [ [ 0, 2, 0, 6, 0 ],               [ 2, 0, 3, 8, 5 ],               [ 0, 3, 0, 0, 7 ],               [ 6, 8, 0, 0, 9 ],               [ 0, 5, 7, 9, 0 ], ]       # Print the solution     minimumProductMST(graph)   # This code is contributed by rutvik_56

C#

 // C# program for getting minimum product // spanning tree The program is for adjacency matrix // representation of the graph using System;   class GFG {       // Number of vertices in the graph     static int V = 5;       // A utility function to find the vertex with minimum     // key value, from the set of vertices not yet included     // in MST     static int minKey(int[] key, Boolean[] mstSet)     {         // Initialize min value         int min = int.MaxValue, min_index = 0;           for (int v = 0; v < V; v++) {             if (mstSet[v] == false && key[v] < min) {                 min = key[v];                 min_index = v;             }         }         return min_index;     }       // A utility function to print the constructed MST     // stored in parent[] and print Minimum Obtaiable     // product     static void printMST(int[] parent, int n, int[, ] graph)     {         Console.Write("Edge Weight\n");         int minProduct = 1;         for (int i = 1; i < V; i++) {             Console.Write("{0} - {1} {2} \n",                           parent[i], i, graph[i, parent[i]]);               minProduct *= graph[i, parent[i]];         }         Console.Write("Minimum Obtainable product is {0}\n",                       minProduct);     }       // Function to construct and print MST for a graph     // represented using adjacency matrix representation     // inputGraph is sent for printing actual edges and     // logGraph is sent for actual MST operations     static void primMST(int[, ] inputGraph, double[, ] logGraph)     {         int[] parent = new int[V]; // Array to store constructed MST         int[] key = new int[V]; // Key values used to pick minimum         // weight edge in cut         Boolean[] mstSet = new Boolean[V]; // To represent set of vertices not         // yet included in MST           // Initialize all keys as INFINITE         for (int i = 0; i < V; i++) {             key[i] = int.MaxValue;             mstSet[i] = false;         }         // Always include first 1st vertex in MST.         key = 0; // Make key 0 so that this vertex is         // picked as first vertex         parent = -1; // First node is always root of MST           // The MST will have V vertices         for (int count = 0; count < V - 1; count++) {             // Pick the minimum key vertex from the set of             // vertices not yet included in MST             int u = minKey(key, mstSet);               // Add the picked vertex to the MST Set             mstSet[u] = true;               // Update key value and parent index of the             // adjacent vertices of the picked vertex.             // Consider only those vertices which are not yet             // included in MST             for (int v = 0; v < V; v++) // logGraph[u, v] is non zero only for             // adjacent vertices of m mstSet[v] is false             // for vertices not yet included in MST             // Update the key only if logGraph[u, v] is             // smaller than key[v]             {                 if (logGraph[u, v] > 0                     && mstSet[v] == false                     && logGraph[u, v] < key[v]) {                       parent[v] = u;                     key[v] = (int)logGraph[u, v];                 }             }         }           // print the constructed MST         printMST(parent, V, inputGraph);     }       // Method to get minimum product spanning tree     static void minimumProductMST(int[, ] graph)     {         double[, ] logGraph = new double[V, V];           // Constructing logGraph from original graph         for (int i = 0; i < V; i++) {             for (int j = 0; j < V; j++) {                 if (graph[i, j] > 0) {                     logGraph[i, j] = Math.Log(graph[i, j]);                 }                 else {                     logGraph[i, j] = 0;                 }             }         }           // Applyting standard Prim's MST algorithm on         // Log graph.         primMST(graph, logGraph);     }       // Driver code     public static void Main(String[] args)     {         /* Let us create the following graph         2 3     (0)--(1)--(2)         | / \ |     6| 8/ \5 |7         | /     \ |     (3)-------(4)             9         */         int[, ] graph = {             { 0, 2, 0, 6, 0 },             { 2, 0, 3, 8, 5 },             { 0, 3, 0, 0, 7 },             { 6, 8, 0, 0, 9 },             { 0, 5, 7, 9, 0 },         };           // Print the solution         minimumProductMST(graph);     } } /* This code contributed by PrinciRaj1992 */

Javascript



Output:

Edge   Weight
0 - 1    2
1 - 2    3
0 - 3    6
1 - 4    5
Minimum Obtainable product is 180

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