Algoritmo de eliminación inversa para árbol de expansión mínimo
Pruébalo en GfG Practice 
#practiceLinkDiv { mostrar: ninguno !importante; }
#practiceLinkDiv { mostrar: ninguno !importante; } El algoritmo de eliminación inversa está estrechamente relacionado con algoritmo de Kruskal . En el algoritmo de Kruskal lo que hacemos es: ordenar las aristas por orden creciente de sus pesos. Después de clasificar, seleccionamos uno por uno los bordes en orden creciente. Incluimos el borde seleccionado actual si al incluirlo en el árbol de expansión no se forma ningún ciclo hasta que haya bordes V-1 en el árbol de expansión donde V = número de vértices.
En el algoritmo de eliminación inversa ordenamos todos los bordes en decreciente orden de sus pesos. Después de ordenar, seleccionamos uno por uno los bordes en orden decreciente. Nosotros incluya el borde seleccionado actual si la exclusión del borde actual provoca la desconexión en el gráfico actual . La idea principal es eliminar el borde si su eliminación no conduce a la desconexión del gráfico.
El algoritmo:
- Ordene todos los bordes del gráfico en orden no creciente de pesos de borde.
- Inicialice MST como gráfico original y elimine los bordes adicionales siguiendo el paso 3.
- Elija el borde de mayor peso de los bordes restantes y compruebe si eliminar el borde desconecta el gráfico o no .
Si se desconecta, no eliminamos el borde.
De lo contrario eliminamos el borde y continuamos.
Ilustración:
Entendámoslo con el siguiente ejemplo:
Si eliminamos el borde de peso más alto del gráfico de peso 14 no se desconecta, por lo que lo eliminamos.
A continuación eliminamos 11 ya que eliminarlo no desconecta el gráfico.
A continuación eliminamos 10 ya que eliminarlo no desconecta el gráfico.
El siguiente es el 9. No podemos eliminar el 9 porque eliminarlo provoca la desconexión.
Seguimos así y los siguientes bordes quedan en el MST final.
Edges in MST
(3 4)
(0 7)
(2 3)
(2 5)
(0 1)
(5 6)
(2 8)
(6 7)
Nota : En el caso de bordes del mismo peso, podemos elegir cualquier borde del mismo peso.
Práctica recomendada Algoritmo de eliminación inversa para árbol de expansión mínimo ¡Pruébalo!Implementación:
C++Java// C++ program to find Minimum Spanning Tree // of a graph using Reverse Delete Algorithm #includeusing namespace std ; // Creating shortcut for an integer pair typedef pair < int int > iPair ; // Graph class represents a directed graph // using adjacency list representation class Graph { int V ; // No. of vertices list < int > * adj ; vector < pair < int iPair > > edges ; void DFS ( int v bool visited []); public : Graph ( int V ); // Constructor // function to add an edge to graph void addEdge ( int u int v int w ); // Returns true if graph is connected bool isConnected (); void reverseDeleteMST (); }; Graph :: Graph ( int V ) { this -> V = V ; adj = new list < int > [ V ]; } void Graph :: addEdge ( int u int v int w ) { adj [ u ]. push_back ( v ); // Add w to v’s list. adj [ v ]. push_back ( u ); // Add w to v’s list. edges . push_back ({ w { u v }}); } void Graph :: DFS ( int v bool visited []) { // Mark the current node as visited and print it visited [ v ] = true ; // Recur for all the vertices adjacent to // this vertex list < int >:: iterator i ; for ( i = adj [ v ]. begin (); i != adj [ v ]. end (); ++ i ) if ( ! visited [ * i ]) DFS ( * i visited ); } // Returns true if given graph is connected else false bool Graph :: isConnected () { bool visited [ V ]; memset ( visited false sizeof ( visited )); // Find all reachable vertices from first vertex DFS ( 0 visited ); // If set of reachable vertices includes all // return true. for ( int i = 1 ; i < V ; i ++ ) if ( visited [ i ] == false ) return false ; return true ; } // This function assumes that edge (u v) // exists in graph or not void Graph :: reverseDeleteMST () { // Sort edges in increasing order on basis of cost sort ( edges . begin () edges . end ()); int mst_wt = 0 ; // Initialize weight of MST cout < < 'Edges in MST n ' ; // Iterate through all sorted edges in // decreasing order of weights for ( int i = edges . size () -1 ; i >= 0 ; i -- ) { int u = edges [ i ]. second . first ; int v = edges [ i ]. second . second ; // Remove edge from undirected graph adj [ u ]. remove ( v ); adj [ v ]. remove ( u ); // Adding the edge back if removing it // causes disconnection. In this case this // edge becomes part of MST. if ( isConnected () == false ) { adj [ u ]. push_back ( v ); adj [ v ]. push_back ( u ); // This edge is part of MST cout < < '(' < < u < < ' ' < < v < < ') n ' ; mst_wt += edges [ i ]. first ; } } cout < < 'Total weight of MST is ' < < mst_wt ; } // Driver code int main () { // create the graph given in above figure int V = 9 ; Graph g ( V ); // making above shown graph g . addEdge ( 0 1 4 ); g . addEdge ( 0 7 8 ); g . addEdge ( 1 2 8 ); g . addEdge ( 1 7 11 ); g . addEdge ( 2 3 7 ); g . addEdge ( 2 8 2 ); g . addEdge ( 2 5 4 ); g . addEdge ( 3 4 9 ); g . addEdge ( 3 5 14 ); g . addEdge ( 4 5 10 ); g . addEdge ( 5 6 2 ); g . addEdge ( 6 7 1 ); g . addEdge ( 6 8 6 ); g . addEdge ( 7 8 7 ); g . reverseDeleteMST (); return 0 ; } Python3// Java program to find Minimum Spanning Tree // of a graph using Reverse Delete Algorithm import java.util.* ; // class to represent an edge class Edge implements Comparable < Edge > { int u v w ; Edge ( int u int v int w ) { this . u = u ; this . w = w ; this . v = v ; } public int compareTo ( Edge other ) { return ( this . w - other . w ); } } // Class to represent a graph using adjacency list // representation public class GFG { private int V ; // No. of vertices private List < Integer >[] adj ; private List < Edge > edges ; @SuppressWarnings ({ 'unchecked' 'deprecated' }) public GFG ( int v ) // Constructor { V = v ; adj = new ArrayList [ v ] ; for ( int i = 0 ; i < v ; i ++ ) adj [ i ] = new ArrayList < Integer > (); edges = new ArrayList < Edge > (); } // function to Add an edge public void AddEdge ( int u int v int w ) { adj [ u ] . add ( v ); // Add w to v’s list. adj [ v ] . add ( u ); // Add w to v’s list. edges . add ( new Edge ( u v w )); } // function to perform dfs private void DFS ( int v boolean [] visited ) { // Mark the current node as visited and print it visited [ v ] = true ; // Recur for all the vertices adjacent to // this vertex for ( int i : adj [ v ] ) { if ( ! visited [ i ] ) DFS ( i visited ); } } // Returns true if given graph is connected else false private boolean IsConnected () { boolean [] visited = new boolean [ V ] ; // Find all reachable vertices from first vertex DFS ( 0 visited ); // If set of reachable vertices includes all // return true. for ( int i = 1 ; i < V ; i ++ ) { if ( visited [ i ] == false ) return false ; } return true ; } // This function assumes that edge (u v) // exists in graph or not public void ReverseDeleteMST () { // Sort edges in increasing order on basis of cost Collections . sort ( edges ); int mst_wt = 0 ; // Initialize weight of MST System . out . println ( 'Edges in MST' ); // Iterate through all sorted edges in // decreasing order of weights for ( int i = edges . size () - 1 ; i >= 0 ; i -- ) { int u = edges . get ( i ). u ; int v = edges . get ( i ). v ; // Remove edge from undirected graph adj [ u ] . remove ( adj [ u ] . indexOf ( v )); adj [ v ] . remove ( adj [ v ] . indexOf ( u )); // Adding the edge back if removing it // causes disconnection. In this case this // edge becomes part of MST. if ( IsConnected () == false ) { adj [ u ] . add ( v ); adj [ v ] . add ( u ); // This edge is part of MST System . out . println ( '(' + u + ' ' + v + ')' ); mst_wt += edges . get ( i ). w ; } } System . out . println ( 'Total weight of MST is ' + mst_wt ); } // Driver code public static void main ( String [] args ) { // create the graph given in above figure int V = 9 ; GFG g = new GFG ( V ); // making above shown graph g . AddEdge ( 0 1 4 ); g . AddEdge ( 0 7 8 ); g . AddEdge ( 1 2 8 ); g . AddEdge ( 1 7 11 ); g . AddEdge ( 2 3 7 ); g . AddEdge ( 2 8 2 ); g . AddEdge ( 2 5 4 ); g . AddEdge ( 3 4 9 ); g . AddEdge ( 3 5 14 ); g . AddEdge ( 4 5 10 ); g . AddEdge ( 5 6 2 ); g . AddEdge ( 6 7 1 ); g . AddEdge ( 6 8 6 ); g . AddEdge ( 7 8 7 ); g . ReverseDeleteMST (); } } // This code is contributed by Prithi_DeyC## Python3 program to find Minimum Spanning Tree # of a graph using Reverse Delete Algorithm # Graph class represents a directed graph # using adjacency list representation class Graph : def __init__ ( self v ): # No. of vertices self . v = v self . adj = [ 0 ] * v self . edges = [] for i in range ( v ): self . adj [ i ] = [] # function to add an edge to graph def addEdge ( self u : int v : int w : int ): self . adj [ u ] . append ( v ) # Add w to v’s list. self . adj [ v ] . append ( u ) # Add w to v’s list. self . edges . append (( w ( u v ))) def dfs ( self v : int visited : list ): # Mark the current node as visited and print it visited [ v ] = True # Recur for all the vertices adjacent to # this vertex for i in self . adj [ v ]: if not visited [ i ]: self . dfs ( i visited ) # Returns true if graph is connected # Returns true if given graph is connected else false def connected ( self ): visited = [ False ] * self . v # Find all reachable vertices from first vertex self . dfs ( 0 visited ) # If set of reachable vertices includes all # return true. for i in range ( 1 self . v ): if not visited [ i ]: return False return True # This function assumes that edge (u v) # exists in graph or not def reverseDeleteMST ( self ): # Sort edges in increasing order on basis of cost self . edges . sort ( key = lambda a : a [ 0 ]) mst_wt = 0 # Initialize weight of MST print ( 'Edges in MST' ) # Iterate through all sorted edges in # decreasing order of weights for i in range ( len ( self . edges ) - 1 - 1 - 1 ): u = self . edges [ i ][ 1 ][ 0 ] v = self . edges [ i ][ 1 ][ 1 ] # Remove edge from undirected graph self . adj [ u ] . remove ( v ) self . adj [ v ] . remove ( u ) # Adding the edge back if removing it # causes disconnection. In this case this # edge becomes part of MST. if self . connected () == False : self . adj [ u ] . append ( v ) self . adj [ v ] . append ( u ) # This edge is part of MST print ( '( %d %d )' % ( u v )) mst_wt += self . edges [ i ][ 0 ] print ( 'Total weight of MST is' mst_wt ) # Driver Code if __name__ == '__main__' : # create the graph given in above figure V = 9 g = Graph ( V ) # making above shown graph g . addEdge ( 0 1 4 ) g . addEdge ( 0 7 8 ) g . addEdge ( 1 2 8 ) g . addEdge ( 1 7 11 ) g . addEdge ( 2 3 7 ) g . addEdge ( 2 8 2 ) g . addEdge ( 2 5 4 ) g . addEdge ( 3 4 9 ) g . addEdge ( 3 5 14 ) g . addEdge ( 4 5 10 ) g . addEdge ( 5 6 2 ) g . addEdge ( 6 7 1 ) g . addEdge ( 6 8 6 ) g . addEdge ( 7 8 7 ) g . reverseDeleteMST () # This code is contributed by # sanjeev2552JavaScript// C# program to find Minimum Spanning Tree // of a graph using Reverse Delete Algorithm using System ; using System.Collections.Generic ; // class to represent an edge public class Edge : IComparable < Edge > { public int u v w ; public Edge ( int u int v int w ) { this . u = u ; this . v = v ; this . w = w ; } public int CompareTo ( Edge other ) { return this . w . CompareTo ( other . w ); } } // Graph class represents a directed graph // using adjacency list representation public class Graph { private int V ; // No. of vertices private List < int > [] adj ; private List < Edge > edges ; public Graph ( int v ) // Constructor { V = v ; adj = new List < int > [ v ]; for ( int i = 0 ; i < v ; i ++ ) adj [ i ] = new List < int > (); edges = new List < Edge > (); } // function to Add an edge public void AddEdge ( int u int v int w ) { adj [ u ]. Add ( v ); // Add w to v’s list. adj [ v ]. Add ( u ); // Add w to v’s list. edges . Add ( new Edge ( u v w )); } // function to perform dfs private void DFS ( int v bool [] visited ) { // Mark the current node as visited and print it visited [ v ] = true ; // Recur for all the vertices adjacent to // this vertex foreach ( int i in adj [ v ]) { if ( ! visited [ i ]) DFS ( i visited ); } } // Returns true if given graph is connected else false private bool IsConnected () { bool [] visited = new bool [ V ]; // Find all reachable vertices from first vertex DFS ( 0 visited ); // If set of reachable vertices includes all // return true. for ( int i = 1 ; i < V ; i ++ ) { if ( visited [ i ] == false ) return false ; } return true ; } // This function assumes that edge (u v) // exists in graph or not public void ReverseDeleteMST () { // Sort edges in increasing order on basis of cost edges . Sort (); int mst_wt = 0 ; // Initialize weight of MST Console . WriteLine ( 'Edges in MST' ); // Iterate through all sorted edges in // decreasing order of weights for ( int i = edges . Count - 1 ; i >= 0 ; i -- ) { int u = edges [ i ]. u ; int v = edges [ i ]. v ; // Remove edge from undirected graph adj [ u ]. Remove ( v ); adj [ v ]. Remove ( u ); // Adding the edge back if removing it // causes disconnection. In this case this // edge becomes part of MST. if ( IsConnected () == false ) { adj [ u ]. Add ( v ); adj [ v ]. Add ( u ); // This edge is part of MST Console . WriteLine ( '({0} {1})' u v ); mst_wt += edges [ i ]. w ; } } Console . WriteLine ( 'Total weight of MST is {0}' mst_wt ); } } class GFG { // Driver code static void Main ( string [] args ) { // create the graph given in above figure int V = 9 ; Graph g = new Graph ( V ); // making above shown graph g . AddEdge ( 0 1 4 ); g . AddEdge ( 0 7 8 ); g . AddEdge ( 1 2 8 ); g . AddEdge ( 1 7 11 ); g . AddEdge ( 2 3 7 ); g . AddEdge ( 2 8 2 ); g . AddEdge ( 2 5 4 ); g . AddEdge ( 3 4 9 ); g . AddEdge ( 3 5 14 ); g . AddEdge ( 4 5 10 ); g . AddEdge ( 5 6 2 ); g . AddEdge ( 6 7 1 ); g . AddEdge ( 6 8 6 ); g . AddEdge ( 7 8 7 ); g . ReverseDeleteMST (); } } // This code is contributed by cavi4762
ProducciónEdges in MST (3 4) (0 7) (2 3) (2 5) (0 1) (5 6) (2 8) (6 7) Total weight of MST is 37Complejidad del tiempo: O((E*(V+E)) + E log E) donde E es el número de aristas.
Complejidad espacial: O(V+E) donde V es el número de vértices y E es el número de aristas. Estamos usando una lista de adyacencia para almacenar el gráfico, por lo que necesitamos un espacio proporcional a O (V+E).
Notas:
- La implementación anterior es una implementación simple/ingenua del algoritmo de eliminación inversa y se puede optimizar para O(E log V (log log V) 3 ) [Fuente : Una semana ]. Pero esta complejidad de tiempo optimizada es aún menor que Remilgado y Kruskal Algoritmos para MST.
- La implementación anterior modifica el gráfico original. Podemos crear una copia del gráfico si es necesario conservar el gráfico original.
Crear cuestionario
