Minimálny produktový strom
Ak je daný súvislý a neusmernený graf, kostra tohto grafu je podgrafom, ktorý je stromom a spája všetky vrcholy dohromady. Jeden graf môže mať mnoho rôznych kostrových stromov. Minimálna kostra súčinu pre vážený spojený a neusmernený graf je kostra s váhovým súčinom menším alebo rovným súčinu hmotnosti každej inej kostry. Hmotnostný súčin kostry je súčin váh zodpovedajúcich každej hrane kostry. Všetky váhy daného grafu budú pre jednoduchosť kladné.
Príklady:
Minimum Product that we can obtain is 180 for above graph by choosing edges 0-1 1-2 0-3 and 1-4
Tento problém je možné vyriešiť pomocou štandardných algoritmov minimálneho spanning tree ako Kruskal ( https://www.geeksforgeeks.org/dsa/kruskals-minimum-spanning-tree-algorithm-greedy-algo-2/ )a prim 's algoritmus, ale musíme upraviť náš graf, aby sme tieto algoritmy používali. Algoritmy minimálneho kostrového stromu sa snažia minimalizovať celkový súčet váh, tu musíme minimalizovať celkový súčin váh. Môžeme použiť vlastnosť logaritmy na prekonanie tohto problému.
Ako vieme
log(w1* w2 * w3 * …. * wN) = log(w1) + log(w2) + log(w3) ….. + log(wN)
Každú váhu grafu môžeme nahradiť jej logaritmickou hodnotou, potom použijeme ľubovoľný algoritmus minimálneho kostrového stromu, ktorý sa pokúsi minimalizovať súčet log(wi), čo zase minimalizuje súčin hmotnosti.
Napríklad graf krokov je znázornený na obrázku nižšie
V nižšie uvedenom kóde sme najprv vytvorili log graf z daného vstupného grafu a potom sme tento graf dali ako vstup do primovho MST algoritmu, ktorý minimalizuje celkový súčet váh stromu. Keďže váhy upraveného grafu sú logaritmy aktuálneho vstupného grafu, v skutočnosti minimalizujeme súčin váh kostry.
// 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 Obtainable // 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 ] = 0 ; // Make key 0 so that this vertex is // picked as first vertex parent [ 0 ] = -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 ; } } // Applying 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 Obtainable // product static void printMST ( int parent [] int n int graph [][] ) { System . out . printf ( 'Edge Weightn' ); 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 %dn' 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 ] = 0 ; // Make key 0 so that this vertex is // picked as first vertex parent [ 0 ] = - 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 ; } } } // Applying 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 # Obtainable 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 ] = 0 # Picked as first vertex # First node is always root of MST parent [ 0 ] = - 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 # Applying 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 Obtainable // product static void printMST ( int [] parent int n int [ ] graph ) { Console . Write ( 'Edge Weightn' ); 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 ] = 0 ; // Make key 0 so that this vertex is // picked as first vertex parent [ 0 ] = - 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 ; } } } // Applying 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 < script > // A Javascript program for getting minimum product // spanning tree The program is for adjacency matrix // representation of the graph // Number of vertices in the graph let V = 5 ; // A utility function to find the vertex with minimum // key value from the set of vertices not yet included // in MST function minKey ( key mstSet ) { // Initialize min value let min = Number . MAX_VALUE min_index = 0 ; for ( let 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 Obtainable // product function printMST ( parent n graph ) { document . write ( 'Edge Weight
' ); let minProduct = 1 ; for ( let i = 1 ; i < V ; i ++ ) { document . write ( parent [ i ] + ' - ' + i + ' ' + graph [ i ][ parent [ i ]] + '
' ); minProduct *= graph [ i ][ parent [ i ]]; } document . write ( 'Minimum Obtainable product is ' 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 function primMST ( inputGraph logGraph ) { let parent = new Array ( V ); // Array to store constructed MST let key = new Array ( V ); // Key values used to pick minimum // weight edge in cut let mstSet = new Array ( V ); // To represent set of vertices not // yet included in MST // Initialize all keys as INFINITE for ( let i = 0 ; i < V ; i ++ ) { key [ i ] = Number . MAX_VALUE ; mstSet [ i ] = false ; } // Always include first 1st vertex in MST. key [ 0 ] = 0 ; // Make key 0 so that this vertex is // picked as first vertex parent [ 0 ] = - 1 ; // First node is always root of MST // The MST will have V vertices for ( let count = 0 ; count < V - 1 ; count ++ ) { // Pick the minimum key vertex from the set of // vertices not yet included in MST let 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 ( let 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 function minimumProductMST ( graph ) { let logGraph = new Array ( V ); // Constructing logGraph from original graph for ( let i = 0 ; i < V ; i ++ ) { logGraph [ i ] = new Array ( V ); for ( let j = 0 ; j < V ; j ++ ) { if ( graph [ i ][ j ] > 0 ) { logGraph [ i ][ j ] = Math . log ( graph [ i ][ j ]); } else { logGraph [ i ][ j ] = 0 ; } } } // Applying standard Prim's MST algorithm on // Log graph. primMST ( graph logGraph ); } // Driver code /* Let us create the following graph 2 3 (0)--(1)--(2) | / | 6| 8/ 5 |7 | / | (3)-------(4) 9 */ let 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 rag2127 < /script>
výstup:
Edge Weight 0 - 1 2 1 - 2 3 0 - 3 6 1 - 4 5 Minimum Obtainable product is 180
The časová zložitosť tohto algoritmu je O(V2), pretože existujú dve vnorené slučky for, ktoré iterujú cez všetky vrcholy.
The priestorovú zložitosť tohto algoritmu je O(V2), pretože na uloženie vstupného grafu používame 2-D pole veľkosti V x V.
