Given N dependencies of the form X Y, where X & Y represents two different tasks. The dependency X Y denotes dependency of the form Y -> X i.e, if task Y happens then task X will happen in other words task Y has to be completed first to initiate task X. Also given M tasks that will initiate first. The task is to print all the tasks that will get completed at the end in the lexicographical order.
Note that the tasks will be represented by upper case English letters only.
Input: dep[][] = {{A, B}, {C, B}, {D, A}, {D, C}, {B, E}}, tasks[] = {B, C}
Output: A B C D Task A happens after task B and task D can only happen after the completion of tasks A or C. So, the required order is A B C D.
Input: dep[][] = {{Q, P}, {S, Q}, {Q, R}}, tasks[] = {R}
Output: Q R S
Approach: DFS can be used to solve the problem. The dependencies of the form X Y (Y -> X) can be represented as an edge from node Y to node X in the graph. Initiate the DFS from each of the M initial nodes and mark the nodes that are encountered as visited using a boolean array. At last, print the nodes/tasks that are covered using DFS in lexicographical order. The approach works because DFS will cover all the nodes starting from the initial nodes in sequential manner. Consider the diagram below that represents the first example from the above:
The diagram shows the edges covered during DFS from initial tasks B and C as Red in color. The nodes thus visited were A, B, C and D. Below is the implementation of the above approach:
// C++ implementation of the approach #include <cstring> #include <iostream> #include <vector> using namespace std;
// Graph class represents a directed graph // using adjacency list representation class Graph {
// Number of vertices
int V;
// Pointer to an array containing
// adjacency lists
vector< int >* adj;
// Boolean array to mark tasks as visited
bool visited[26];
// A recursive function used by DFS
void DFSUtil( int v);
public :
// Constructor
Graph()
{
// There are only 26 English
// upper case letters
this ->V = 26;
adj = new vector< int >[26];
}
// Function to add an edge to the graph
void addEdge( char v, char w);
// DFS traversal of the vertices
// reachable from v
void DFS( char start[], int M);
void printTasks();
}; // Function to add an edge to the graph void Graph::addEdge( char v, char w)
{ // Add w to v's list
adj[v - 65].push_back(w - 65);
} void Graph::DFSUtil( int v)
{ // Mark the current node as visited and
// print it
visited[v] = true ;
// Recur for all the vertices adjacent
// to this vertex
vector< int >::iterator i;
for (i = adj[v].begin(); i != adj[v].end(); ++i)
if (!visited[*i])
DFSUtil(*i);
} // DFS traversal of the vertices reachable // from start nodes // It uses recursive DFSUtil() void Graph::DFS( char start[], int M)
{ // Mark all the vertices as not visited
for ( int i = 0; i < V; i++)
visited[i] = false ;
// Call the recursive helper function
// to print DFS traversal
for ( int i = 0; i < M; i++)
DFSUtil(start[i] - 65);
} // Helper function to print the tasks in // lexicographical order that are completed // at the end of the DFS void Graph::printTasks()
{ for ( int i = 0; i < 26; i++) {
if (visited[i])
cout << char (i + 65) << " " ;
}
cout << endl;
} // Driver code int main()
{ // Create the graph
Graph g;
g.addEdge( 'B' , 'A' );
g.addEdge( 'B' , 'C' );
g.addEdge( 'A' , 'D' );
g.addEdge( 'C' , 'D' );
g.addEdge( 'E' , 'B' );
// Initial tasks to be run
char start[] = { 'B' , 'C' };
int n = sizeof (start) / sizeof ( char );
// Start the dfs
g.DFS(start, n);
// Print the tasks that will get finished
g.printTasks();
return 0;
} |
// Java implementation of the approach import java.util.ArrayList;
import java.util.List;
public class Graph {
// Number of vertices
int V;
// Pointer to an array containing
// adjacency lists
List<Integer>[] adj;
// Boolean array to mark tasks as visited
boolean [] visited;
// A recursive function used by DFS
void DFSUtil( int v)
{
// Mark the current node as visited and
// print it
visited[v] = true ;
// Recur for all the vertices adjacent
// to this vertex
List<Integer> list = adj[v];
for ( int node : list) {
if (!visited[node])
DFSUtil(node);
}
}
// Constructor
Graph()
{
// There are only 26 English
// upper case letters
this .V = 26 ;
adj = new ArrayList[ 26 ];
for ( int i = 0 ; i < 26 ; i++)
adj[i] = new ArrayList<Integer>();
}
// Function to add an edge to the graph
void addEdge( char v, char w)
{
// Add w to v's list
adj[v - 65 ].add(w - 65 );
}
// DFS traversal of the vertices
// reachable from start nodes
void DFS( char start[], int M)
{
// Mark all the vertices as not visited
visited = new boolean [ 26 ];
for ( int i = 0 ; i < 26 ; i++)
visited[i] = false ;
// Call the recursive helper function
// to print DFS traversal
for ( int i = 0 ; i < M; i++)
DFSUtil(start[i] - 65 );
}
// Helper function to print the tasks in
// lexicographical order that are completed
// at the end of the DFS
void printTasks()
{
for ( int i = 0 ; i < 26 ; i++) {
if (visited[i])
System.out.print(( char )(i + 65 ) + " " );
}
System.out.println();
}
public static void main(String[] args)
{
// Create the graph
Graph g = new Graph();
g.addEdge( 'B' , 'A' );
g.addEdge( 'B' , 'C' );
g.addEdge( 'A' , 'D' );
g.addEdge( 'C' , 'D' );
g.addEdge( 'E' , 'B' );
// Initial tasks to be run
char start[] = { 'B' , 'C' };
int n = start.length;
// Start the dfs
g.DFS(start, n);
// Print the tasks that will get finished
g.printTasks();
}
} // This code is contributed by ishankhandelwals. |
# Python3 implementation of the approach from collections import defaultdict
# This class represents a directed graph # using adjacency list representation class Graph:
# Constructor
def __init__( self ):
# Default dictionary to store the graph
self .graph = defaultdict( list )
self .visited = [ False ] * 26
# Function to add an edge to the graph
def addEdge( self , u, v):
self .graph[ ord (u) - 65 ].append( ord (v) - 65 )
# A function used by DFS
def DFSUtil( self , v):
# Mark the current node as visited
# and print it
self .visited[v] = True
# Recur for all the vertices adjacent
# to this vertex
for i in self .graph[v]:
if self .visited[i] = = False :
self .DFSUtil(i)
# Function to perform the DFS traversal
# It uses recursive DFSUtil()
def DFS( self , start, M):
# Total vertices
V = len ( self .graph)
# Call the recursive helper function
# to print the DFS traversal starting
# from all vertices one by one
for i in range (M):
self .DFSUtil( ord (start[i]) - 65 )
def printOrder( self ):
for i in range ( 26 ):
if self .visited[i] = = True :
print ( chr (i + 65 ), end = " " )
print ( "\n" )
# Driver code g = Graph()
g.addEdge( 'B' , 'A' )
g.addEdge( 'B' , 'C' )
g.addEdge( 'A' , 'D' )
g.addEdge( 'C' , 'D' )
g.addEdge( 'E' , 'B' )
g.DFS([ 'B' , 'C' ], 2 )
g.printOrder() |
// JavaScript implementation of the approach class Graph { // Number of vertices
constructor() {
this .V = 26;
this .adj = new Array(26).fill([]);
this .visited = new Array(26).fill( false );
}
// Function to add an edge to the graph
addEdge(v, w) {
// Add w to v's list
this .adj[v.charCodeAt(0)-65].push(w.charCodeAt(0) - 65);
}
// A recursive function used by DFS
DFSUtil(v) {
// Mark the current node as visited and
// print it
this .visited[v] = true ;
// Recur for all the vertices adjacent
// to this vertex
for (let i = 0; i < this .adj[v].length; i++)
if (! this .visited[ this .adj[v][i]])
this .DFSUtil( this .adj[v][i]);
}
// DFS traversal of the vertices reachable
// from start nodes
// It uses recursive DFSUtil()
DFS(start, M) {
// Mark all the vertices as not visited
for (let i = 0; i < this .V; i++)
this .visited[i] = false ;
// Call the recursive helper function
// to print DFS traversal
for (let i = 0; i < M; i++)
this .DFSUtil(start[i].charCodeAt(0) - 65);
}
// Helper function to print the tasks in
// lexicographical order that are completed
// at the end of the DFS
printTasks() {
for (let i = 0; i < 26; i++) {
if ( this .visited[i])
console.log(String.fromCharCode(i + 65) + " " );
}
console.log( "\n" );
}
} // Driver code let g = new Graph();
g.addEdge('B ', ' A ');
g.addEdge(' B ', ' C ');
g.addEdge(' A ', ' D ');
g.addEdge(' C ', ' D ');
g.addEdge(' E ', ' B ');
// Initial tasks to be run let start = [' B ', ' C'];
let n = start.length; // Start the dfs g.DFS(start, n); // Print the tasks that will get finished g.printTasks(); // This code is contributed by ishankhandelwals. |
using System;
using System.Collections.Generic;
class Graph
{ // Number of vertices
private int V = 26;
// Adjacency list to store the graph
private List< int >[] adj;
// Boolean array to mark tasks as visited
private bool [] visited = new bool [26];
// Constructor
public Graph()
{
adj = new List< int >[26];
for ( int i = 0; i < 26; i++)
{
adj[i] = new List< int >();
}
}
// Function to add an edge to the graph
public void AddEdge( char v, char w)
{
adj[v - 65].Add(w - 65);
}
// Recursive function used by DFS
private void DFSUtil( int v)
{
visited[v] = true ;
foreach ( int i in adj[v])
{
if (!visited[i])
{
DFSUtil(i);
}
}
}
// DFS traversal of the vertices reachable from start nodes
public void DFS( char [] start)
{
// Mark all the vertices as not visited
for ( int i = 0; i < V; i++)
{
visited[i] = false ;
}
// Call the recursive helper function for DFS traversal
foreach ( char c in start)
{
DFSUtil(c - 65);
}
}
// Helper function to print the tasks in lexicographical order
public void PrintTasks()
{
for ( int i = 0; i < 26; i++)
{
if (visited[i])
{
Console.Write(( char )(i + 65) + " " );
}
}
Console.WriteLine();
}
} class Program
{ static void Main( string [] args)
{
// Create the graph
Graph g = new Graph();
g.AddEdge( 'B' , 'A' );
g.AddEdge( 'B' , 'C' );
g.AddEdge( 'A' , 'D' );
g.AddEdge( 'C' , 'D' );
g.AddEdge( 'E' , 'B' );
// Initial tasks to be run
char [] start = { 'B' , 'C' };
// Start the DFS
g.DFS(start);
// Print the tasks that will get finished
g.PrintTasks();
}
} |
A B C D
Time Complexity: O(V + E) where V is the number of nodes in the graph and E is the number of edges or dependencies. In this case, since V is always 26, so the time complexity is O(26 + E) or just O(E) in the worst case.
Space Complexity: O(V + E)