הנתיב הארוך ביותר בגרף אציקלי מכוון | סט 2
בהינתן גרף מכוון אציקלי משוקלל (DAG) וקודקוד מקור בו מצא את המרחקים הארוכים ביותר מקודקוד המקור לכל שאר הקודקודים בגרף הנתון.
כבר דנו כיצד נוכל למצוא הנתיב הארוך ביותר בגרף אציקלי מכוון (DAG) בסט 1. בפוסט זה נדון בפתרון מעניין נוסף למציאת הנתיב הארוך ביותר של DAG שמשתמש באלגוריתם למציאת הדרך הקצרה ביותר ב-DAG .
הרעיון הוא לשלול את משקלי הנתיב ולמצוא את הנתיב הקצר ביותר בגרף . המסלול הארוך ביותר בין שני קודקודים נתונים s ו-t בגרף G משוקלל זהה למסלול הקצר ביותר בגרף G' הנגזר מ-G על ידי שינוי כל משקל לשלילתו. לכן אם ניתן למצוא את השבילים הקצרים ביותר ב-G' אז ניתן למצוא את השבילים הארוכים ביותר גם ב-G.
להלן התהליך שלב אחר שלב של מציאת הנתיבים הארוכים ביותר -
אנו משנים משקל של כל קצה של גרף נתון לשלילה שלו ומאתחלים מרחקים לכל הקודקודים כאינסופיים ומרחק למקור כ-0 ואז נמצא מיון טופולוגי של הגרף המייצג סדר ליניארי של הגרף. כאשר אנו רואים קודקוד u בסדר טופולוגי, מובטח שחשבנו על כל קצה נכנס אליו. כלומר, כבר מצאנו את הנתיב הקצר ביותר לאותו קודקוד ונוכל להשתמש במידע הזה כדי לעדכן נתיב קצר יותר של כל הקודקודים הסמוכים לו. ברגע שיש לנו סדר טופולוגי אנחנו מעבדים בזה אחר זה את כל הקודקודים בסדר טופולוגי. עבור כל קודקוד שמעובד אנו מעדכנים מרחקים של הקודקוד הסמוך לו תוך שימוש במרחק הקצר ביותר של הקודקוד הנוכחי מקודקוד המקור ומשקל הקצה שלו. כְּלוֹמַר
for every adjacent vertex v of every vertex u in topological order if (dist[v] > dist[u] + weight(u v)) dist[v] = dist[u] + weight(u v)
ברגע שמצאנו את כל הנתיבים הקצרים ביותר מקודקוד המקור הנתיבים הארוכים ביותר יהיו רק שלילה של הנתיבים הקצרים ביותר.
להלן יישום הגישה לעיל:
C++ // A C++ program to find single source longest distances // in a DAG #include using namespace std ; // Graph is represented using adjacency list. Every node of // adjacency list contains vertex number of the vertex to // which edge connects. It also contains weight of the edge class AdjListNode { int v ; int weight ; public : AdjListNode ( int _v int _w ) { v = _v ; weight = _w ; } int getV () { return v ; } int getWeight () { return weight ; } }; // Graph class represents a directed graph using adjacency // list representation class Graph { int V ; // No. of vertices // Pointer to an array containing adjacency lists list < AdjListNode >* adj ; // This function uses DFS void longestPathUtil ( int vector < bool > & stack < int > & ); public : Graph ( int ); // Constructor ~ Graph (); // Destructor // function to add an edge to graph void addEdge ( int int int ); void longestPath ( int ); }; Graph :: Graph ( int V ) // Constructor { this -> V = V ; adj = new list < AdjListNode > [ V ]; } Graph ::~ Graph () // Destructor { delete [] adj ; } void Graph :: addEdge ( int u int v int weight ) { AdjListNode node ( v weight ); adj [ u ]. push_back ( node ); // Add v to u's list } // A recursive function used by longestPath. See below // link for details. // https://www.geeksforgeeks.org/dsa/topological-sorting/ void Graph :: longestPathUtil ( int v vector < bool > & visited stack < int > & Stack ) { // Mark the current node as visited visited [ v ] = true ; // Recur for all the vertices adjacent to this vertex for ( AdjListNode node : adj [ v ]) { if ( ! visited [ node . getV ()]) longestPathUtil ( node . getV () visited Stack ); } // Push current vertex to stack which stores topological // sort Stack . push ( v ); } // The function do Topological Sort and finds longest // distances from given source vertex void Graph :: longestPath ( int s ) { // Initialize distances to all vertices as infinite and // distance to source as 0 int dist [ V ]; for ( int i = 0 ; i < V ; i ++ ) dist [ i ] = INT_MAX ; dist [ s ] = 0 ; stack < int > Stack ; // Mark all the vertices as not visited vector < bool > visited ( V false ); for ( int i = 0 ; i < V ; i ++ ) if ( visited [ i ] == false ) longestPathUtil ( i visited Stack ); // Process vertices in topological order while ( ! Stack . empty ()) { // Get the next vertex from topological order int u = Stack . top (); Stack . pop (); if ( dist [ u ] != INT_MAX ) { // Update distances of all adjacent vertices // (edge from u -> v exists) for ( AdjListNode v : adj [ u ]) { // consider negative weight of edges and // find shortest path if ( dist [ v . getV ()] > dist [ u ] + v . getWeight () * -1 ) dist [ v . getV ()] = dist [ u ] + v . getWeight () * -1 ; } } } // Print the calculated longest distances for ( int i = 0 ; i < V ; i ++ ) { if ( dist [ i ] == INT_MAX ) cout < < 'INT_MIN ' ; else cout < < ( dist [ i ] * -1 ) < < ' ' ; } } // Driver code int main () { Graph g ( 6 ); g . addEdge ( 0 1 5 ); g . addEdge ( 0 2 3 ); g . addEdge ( 1 3 6 ); g . addEdge ( 1 2 2 ); g . addEdge ( 2 4 4 ); g . addEdge ( 2 5 2 ); g . addEdge ( 2 3 7 ); g . addEdge ( 3 5 1 ); g . addEdge ( 3 4 -1 ); g . addEdge ( 4 5 -2 ); int s = 1 ; cout < < 'Following are longest distances from ' < < 'source vertex ' < < s < < ' n ' ; g . longestPath ( s ); return 0 ; }
Python3 # A Python3 program to find single source # longest distances in a DAG import sys def addEdge ( u v w ): global adj adj [ u ] . append ([ v w ]) # A recursive function used by longestPath. # See below link for details. # https:#www.geeksforgeeks.org/topological-sorting/ def longestPathUtil ( v ): global visited adj Stack visited [ v ] = 1 # Recur for all the vertices adjacent # to this vertex for node in adj [ v ]: if ( not visited [ node [ 0 ]]): longestPathUtil ( node [ 0 ]) # Push current vertex to stack which # stores topological sort Stack . append ( v ) # The function do Topological Sort and finds # longest distances from given source vertex def longestPath ( s ): # Initialize distances to all vertices # as infinite and global visited Stack adj V dist = [ sys . maxsize for i in range ( V )] # for (i = 0 i < V i++) # dist[i] = INT_MAX dist [ s ] = 0 for i in range ( V ): if ( visited [ i ] == 0 ): longestPathUtil ( i ) # print(Stack) while ( len ( Stack ) > 0 ): # Get the next vertex from topological order u = Stack [ - 1 ] del Stack [ - 1 ] if ( dist [ u ] != sys . maxsize ): # Update distances of all adjacent vertices # (edge from u -> v exists) for v in adj [ u ]: # Consider negative weight of edges and # find shortest path if ( dist [ v [ 0 ]] > dist [ u ] + v [ 1 ] * - 1 ): dist [ v [ 0 ]] = dist [ u ] + v [ 1 ] * - 1 # Print the calculated longest distances for i in range ( V ): if ( dist [ i ] == sys . maxsize ): print ( 'INT_MIN ' end = ' ' ) else : print ( dist [ i ] * ( - 1 ) end = ' ' ) # Driver code if __name__ == '__main__' : V = 6 visited = [ 0 for i in range ( 7 )] Stack = [] adj = [[] for i in range ( 7 )] addEdge ( 0 1 5 ) addEdge ( 0 2 3 ) addEdge ( 1 3 6 ) addEdge ( 1 2 2 ) addEdge ( 2 4 4 ) addEdge ( 2 5 2 ) addEdge ( 2 3 7 ) addEdge ( 3 5 1 ) addEdge ( 3 4 - 1 ) addEdge ( 4 5 - 2 ) s = 1 print ( 'Following are longest distances from source vertex' s ) longestPath ( s ) # This code is contributed by mohit kumar 29
C# // C# program to find single source longest distances // in a DAG using System ; using System.Collections.Generic ; // Graph is represented using adjacency list. Every node of // adjacency list contains vertex number of the vertex to // which edge connects. It also contains weight of the edge class AdjListNode { private int v ; private int weight ; public AdjListNode ( int _v int _w ) { v = _v ; weight = _w ; } public int getV () { return v ; } public int getWeight () { return weight ; } } // Graph class represents a directed graph using adjacency // list representation class Graph { private int V ; // No. of vertices // Pointer to an array containing adjacency lists private List < AdjListNode > [] adj ; public Graph ( int v ) // Constructor { V = v ; adj = new List < AdjListNode > [ v ]; for ( int i = 0 ; i < v ; i ++ ) adj [ i ] = new List < AdjListNode > (); } public void AddEdge ( int u int v int weight ) { AdjListNode node = new AdjListNode ( v weight ); adj [ u ]. Add ( node ); // Add v to u's list } // A recursive function used by longestPath. See below // link for details. // https://www.geeksforgeeks.org/dsa/topological-sorting/ private void LongestPathUtil ( int v bool [] visited Stack < int > stack ) { // Mark the current node as visited visited [ v ] = true ; // Recur for all the vertices adjacent to this // vertex foreach ( AdjListNode node in adj [ v ]) { if ( ! visited [ node . getV ()]) LongestPathUtil ( node . getV () visited stack ); } // Push current vertex to stack which stores // topological sort stack . Push ( v ); } // The function do Topological Sort and finds longest // distances from given source vertex public void LongestPath ( int s ) { // Initialize distances to all vertices as infinite // and distance to source as 0 int [] dist = new int [ V ]; for ( int i = 0 ; i < V ; i ++ ) dist [ i ] = Int32 . MaxValue ; dist [ s ] = 0 ; Stack < int > stack = new Stack < int > (); // Mark all the vertices as not visited bool [] visited = new bool [ V ]; for ( int i = 0 ; i < V ; i ++ ) { if ( visited [ i ] == false ) LongestPathUtil ( i visited stack ); } // Process vertices in topological order while ( stack . Count > 0 ) { // Get the next vertex from topological order int u = stack . Pop (); if ( dist [ u ] != Int32 . MaxValue ) { // Update distances of all adjacent vertices // (edge from u -> v exists) foreach ( AdjListNode v in adj [ u ]) { // consider negative weight of edges and // find shortest path if ( dist [ v . getV ()] > dist [ u ] + v . getWeight () * - 1 ) dist [ v . getV ()] = dist [ u ] + v . getWeight () * - 1 ; } } } // Print the calculated longest distances for ( int i = 0 ; i < V ; i ++ ) { if ( dist [ i ] == Int32 . MaxValue ) Console . Write ( 'INT_MIN ' ); else Console . Write ( '{0} ' dist [ i ] * - 1 ); } Console . WriteLine (); } } public class GFG { // Driver code static void Main ( string [] args ) { Graph g = new Graph ( 6 ); g . AddEdge ( 0 1 5 ); g . AddEdge ( 0 2 3 ); g . AddEdge ( 1 3 6 ); g . AddEdge ( 1 2 2 ); g . AddEdge ( 2 4 4 ); g . AddEdge ( 2 5 2 ); g . AddEdge ( 2 3 7 ); g . AddEdge ( 3 5 1 ); g . AddEdge ( 3 4 - 1 ); g . AddEdge ( 4 5 - 2 ); int s = 1 ; Console . WriteLine ( 'Following are longest distances from source vertex {0} ' s ); g . LongestPath ( s ); } } // This code is contributed by cavi4762.
Java // A Java program to find single source longest distances // in a DAG import java.util.* ; // Graph is represented using adjacency list. Every // node of adjacency list contains vertex number of // the vertex to which edge connects. It also // contains weight of the edge class AdjListNode { private int v ; private int weight ; AdjListNode ( int _v int _w ) { v = _v ; weight = _w ; } int getV () { return v ; } int getWeight () { return weight ; } } // Class to represent a graph using adjacency list // representation public class GFG { int V ; // No. of vertices' // Pointer to an array containing adjacency lists ArrayList < AdjListNode >[] adj ; @SuppressWarnings ( 'unchecked' ) GFG ( int V ) // Constructor { this . V = V ; adj = new ArrayList [ V ] ; for ( int i = 0 ; i < V ; i ++ ) { adj [ i ] = new ArrayList <> (); } } void addEdge ( int u int v int weight ) { AdjListNode node = new AdjListNode ( v weight ); adj [ u ] . add ( node ); // Add v to u's list } // A recursive function used by longestPath. See // below link for details https:// // www.geeksforgeeks.org/topological-sorting/ void topologicalSortUtil ( int v boolean visited [] Stack < Integer > stack ) { // Mark the current node as visited visited [ v ] = true ; // Recur for all the vertices adjacent to this // vertex for ( int i = 0 ; i < adj [ v ] . size (); i ++ ) { AdjListNode node = adj [ v ] . get ( i ); if ( ! visited [ node . getV () ] ) topologicalSortUtil ( node . getV () visited stack ); } // Push current vertex to stack which stores // topological sort stack . push ( v ); } // The function to find Smallest distances from a // given vertex. It uses recursive // topologicalSortUtil() to get topological sorting. void longestPath ( int s ) { Stack < Integer > stack = new Stack < Integer > (); int dist [] = new int [ V ] ; // Mark all the vertices as not visited boolean visited [] = new boolean [ V ] ; for ( int i = 0 ; i < V ; i ++ ) visited [ i ] = false ; // Call the recursive helper function to store // Topological Sort starting from all vertices // one by one for ( int i = 0 ; i < V ; i ++ ) if ( visited [ i ] == false ) topologicalSortUtil ( i visited stack ); // Initialize distances to all vertices as // infinite and distance to source as 0 for ( int i = 0 ; i < V ; i ++ ) dist [ i ] = Integer . MAX_VALUE ; dist [ s ] = 0 ; // Process vertices in topological order while ( stack . isEmpty () == false ) { // Get the next vertex from topological // order int u = stack . peek (); stack . pop (); // Update distances of all adjacent vertices if ( dist [ u ] != Integer . MAX_VALUE ) { for ( AdjListNode v : adj [ u ] ) { if ( dist [ v . getV () ] > dist [ u ] + v . getWeight () * - 1 ) dist [ v . getV () ] = dist [ u ] + v . getWeight () * - 1 ; } } } // Print the calculated longest distances for ( int i = 0 ; i < V ; i ++ ) if ( dist [ i ] == Integer . MAX_VALUE ) System . out . print ( 'INF ' ); else System . out . print ( dist [ i ] * - 1 + ' ' ); } // Driver program to test above functions public static void main ( String args [] ) { // Create a graph given in the above diagram. // Here vertex numbers are 0 1 2 3 4 5 with // following mappings: // 0=r 1=s 2=t 3=x 4=y 5=z GFG g = new GFG ( 6 ); g . addEdge ( 0 1 5 ); g . addEdge ( 0 2 3 ); g . addEdge ( 1 3 6 ); g . addEdge ( 1 2 2 ); g . addEdge ( 2 4 4 ); g . addEdge ( 2 5 2 ); g . addEdge ( 2 3 7 ); g . addEdge ( 3 5 1 ); g . addEdge ( 3 4 - 1 ); g . addEdge ( 4 5 - 2 ); int s = 1 ; System . out . print ( 'Following are longest distances from source vertex ' + s + ' n' ); g . longestPath ( s ); } } // This code is contributed by Prithi_Dey
JavaScript class AdjListNode { constructor ( v weight ) { this . v = v ; this . weight = weight ; } getV () { return this . v ; } getWeight () { return this . weight ; } } class GFG { constructor ( V ) { this . V = V ; this . adj = new Array ( V ); for ( let i = 0 ; i < V ; i ++ ) { this . adj [ i ] = new Array (); } } addEdge ( u v weight ) { let node = new AdjListNode ( v weight ); this . adj [ u ]. push ( node ); } topologicalSortUtil ( v visited stack ) { visited [ v ] = true ; for ( let i = 0 ; i < this . adj [ v ]. length ; i ++ ) { let node = this . adj [ v ][ i ]; if ( ! visited [ node . getV ()]) { this . topologicalSortUtil ( node . getV () visited stack ); } } stack . push ( v ); } longestPath ( s ) { let stack = new Array (); let dist = new Array ( this . V ); let visited = new Array ( this . V ); for ( let i = 0 ; i < this . V ; i ++ ) { visited [ i ] = false ; } for ( let i = 0 ; i < this . V ; i ++ ) { if ( ! visited [ i ]) { this . topologicalSortUtil ( i visited stack ); } } for ( let i = 0 ; i < this . V ; i ++ ) { dist [ i ] = Number . MAX_SAFE_INTEGER ; } dist [ s ] = 0 ; let u = stack . pop (); while ( stack . length > 0 ) { u = stack . pop (); if ( dist [ u ] !== Number . MAX_SAFE_INTEGER ) { for ( let v of this . adj [ u ]) { if ( dist [ v . getV ()] > dist [ u ] + v . getWeight () * - 1 ) { dist [ v . getV ()] = dist [ u ] + v . getWeight () * - 1 ; } } } } for ( let i = 0 ; i < this . V ; i ++ ) { if ( dist [ i ] === Number . MAX_SAFE_INTEGER ) { console . log ( 'INF' ); } else { console . log ( dist [ i ] * - 1 ); } } } } let g = new GFG ( 6 ); g . addEdge ( 0 1 5 ); g . addEdge ( 0 2 3 ); g . addEdge ( 1 3 6 ); g . addEdge ( 1 2 2 ); g . addEdge ( 2 4 4 ); g . addEdge ( 2 5 2 ); g . addEdge ( 2 3 7 ); g . addEdge ( 3 5 1 ); g . addEdge ( 3 4 - 1 ); g . addEdge ( 4 5 - 2 ); console . log ( 'Longest distances from the vertex 1 : ' ); g . longestPath ( 1 ); //this code is contributed by devendra
תְפוּקָה
Following are longest distances from source vertex 1 INT_MIN 0 2 9 8 10
מורכבות זמן : מורכבות הזמן של מיון טופולוגי היא O(V + E). לאחר מציאת סדר טופולוגי האלגוריתם מעבד את כל הקודקודים ולכל קודקוד הוא מפעיל לולאה עבור כל הקודקודים הסמוכים. מכיוון שסך הקודקודים הסמוכים בגרף הוא O(E), הלולאה הפנימית פועלת O(V + E) פעמים. לכן מורכבות הזמן הכוללת של אלגוריתם זה היא O(V + E).
מורכבות החלל:
מורכבות החלל של האלגוריתם לעיל היא O(V). אנו מאחסנים את מערך הפלט וערימה למיון טופולוגי.
