Given a graph, a source vertex in the graph and a number k, find if there is a simple path (without any cycle) starting from given source and ending at any other vertex such that the distance from source to that vertex is atleast ‘k’ length.
Example:
Input : Source s = 0, k = 58
Output : True
There exists a simple path 0 -> 7 -> 1
-> 2 -> 8 -> 6 -> 5 -> 3 -> 4
Which has a total distance of 60 km which
is more than 58.
Input : Source s = 0, k = 62
Output : False
In the above graph, the longest simple
path has distance 61 (0 -> 7 -> 1-> 2
-> 3 -> 4 -> 5-> 6 -> 8, so output
should be false for any input greater
than 61.
We strongly recommend you to minimize your browser and try this yourself first.
One important thing to note is, simply doing BFS or DFS and picking the longest edge at every step would not work. The reason is, a shorter edge can produce longer path due to higher weight edges connected through it.
The idea is to use Backtracking. We start from given source, explore all paths from current vertex. We keep track of current distance from source. If distance becomes more than k, we return true. If a path doesn’t produces more than k distance, we backtrack.
How do we make sure that the path is simple and we don’t loop in a cycle? The idea is to keep track of current path vertices in an array. Whenever we add a vertex to path, we check if it already exists or not in current path. If it exists, we ignore the edge.
Below is implementation of above idea.
// Java Program to find if there is a simple path with // weight more than k import java.util.*;
public class GFG{
static class AdjListNode {
int v;
int weight;
AdjListNode( int _v, int _w)
{
v = _v;
weight = _w;
}
int getV() { return v; }
int getWeight() { return weight; }
}
// This class represents a dipathted graph using
// adjacency list representation
static class Graph
{
int V; // No. of vertices
// In a weighted graph, we need to store vertex
// and weight pair for every edge
ArrayList<ArrayList<AdjListNode>> adj;
// Allocates memory for adjacency list
Graph( int V)
{
this .V = V;
adj = new ArrayList<ArrayList<AdjListNode>>(V);
for ( int i = 0 ; i < V; i++)
{
adj.add( new ArrayList<AdjListNode>());
}
}
// Utility function to an edge (u, v) of weight w
void addEdge( int u, int v, int weight)
{
AdjListNode node1 = new AdjListNode(v, weight);
adj.get(u).add(node1); // Add v to u's list
AdjListNode node2 = new AdjListNode(u, weight);
adj.get(v).add(node2); // Add u to v's list
}
// Returns true if graph has path more than k length
boolean pathMoreThanK( int src, int k)
{
// Create a path array with nothing included
// in path
boolean path[] = new boolean [V];
Arrays.fill(path, false );
// Add source vertex to path
path[src] = true ;
return pathMoreThanKUtil(src, k, path);
}
// Prints shortest paths from src to all other vertices
boolean pathMoreThanKUtil( int src, int k, boolean [] path)
{
// If k is 0 or negative, return true;
if (k <= 0 )
return true ;
// Get all adjacent vertices of source vertex src and
// recursively explore all paths from src.
ArrayList<AdjListNode> it = adj.get(src);
int index = 0 ;
for ( int i = 0 ; i < adj.get(src).size(); i++)
{
AdjListNode vertex = adj.get(src).get(i);
// Get adjacent vertex and weight of edge
int v = vertex.v;
int w = vertex.weight;
// increase theindex
index++;
// If vertex v is already there in path, then
// there is a cycle (we ignore this edge)
if (path[v] == true )
continue ;
// If weight of is more than k, return true
if (w >= k)
return true ;
// Else add this vertex to path
path[v] = true ;
// If this adjacent can provide a path longer
// than k, return true.
if (pathMoreThanKUtil(v, k-w, path))
return true ;
// Backtrack
path[v] = false ;
}
// If no adjacent could produce longer path, return
// false
return false ;
}
}
// Driver program to test methods of graph class
public 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 );
int src = 0 ;
int k = 62 ;
if (g.pathMoreThanK(src, k))
System.out.println( "YES" );
else
System.out.println( "NO" );
k = 60 ;
if (g.pathMoreThanK(src, k))
System.out.println( "YES" );
else
System.out.println( "NO" );
}
} // This code is contributed by adityapande88. |
# Program to find if there is a simple path with # weight more than k # This class represents a dipathted graph using # adjacency list representation class Graph:
# Allocates memory for adjacency list
def __init__( self , V):
self .V = V
self .adj = [[] for i in range (V)]
# Returns true if graph has path more than k length
def pathMoreThanK( self ,src, k):
# Create a path array with nothing included
# in path
path = [ False ] * self .V
# Add source vertex to path
path[src] = 1
return self .pathMoreThanKUtil(src, k, path)
# Prints shortest paths from src to all other vertices
def pathMoreThanKUtil( self ,src, k, path):
# If k is 0 or negative, return true
if (k < = 0 ):
return True
# Get all adjacent vertices of source vertex src and
# recursively explore all paths from src.
i = 0
while i ! = len ( self .adj[src]):
# Get adjacent vertex and weight of edge
v = self .adj[src][i][ 0 ]
w = self .adj[src][i][ 1 ]
i + = 1
# If vertex v is already there in path, then
# there is a cycle (we ignore this edge)
if (path[v] = = True ):
continue
# If weight of is more than k, return true
if (w > = k):
return True
# Else add this vertex to path
path[v] = True
# If this adjacent can provide a path longer
# than k, return true.
if ( self .pathMoreThanKUtil(v, k - w, path)):
return True
# Backtrack
path[v] = False
# If no adjacent could produce longer path, return
# false
return False
# Utility function to an edge (u, v) of weight w
def addEdge( self ,u, v, w):
self .adj[u].append([v, w])
self .adj[v].append([u, w])
# Driver program to test methods of graph class 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 )
src = 0
k = 62
if g.pathMoreThanK(src, k):
print ( "Yes" )
else :
print ( "No" )
k = 60
if g.pathMoreThanK(src, k):
print ( "Yes" )
else :
print ( "No" )
|
// C# Program to find if there is a simple path with // weight more than k using System;
using System.Collections.Generic;
class Program
{ // Driver program to test methods of graph class
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);
int src = 0;
int k = 62;
if (g.pathMoreThanK(src, k))
Console.WriteLine( "YES" );
else
Console.WriteLine( "NO" );
k = 60;
if (g.pathMoreThanK(src, k))
Console.WriteLine( "YES" );
else
Console.WriteLine( "NO" );
}
} class AdjListNode {
public int v;
public int weight;
public AdjListNode( int _v, int _w)
{
v = _v;
weight = _w;
}
public int getV() { return v; }
public int getWeight() { return weight; }
} // This class represents a dipathted graph using // adjacency list representation class Graph {
public int V; // No. of vertices
// In a weighted graph, we need to store vertex
// and weight pair for every edge
public List<List<AdjListNode> > adj
= new List<List<AdjListNode> >();
// Allocates memory for adjacency list
public Graph( int V)
{
this .V = V;
for ( int i = 0; i < V; i++)
adj.Add( new List<AdjListNode>());
}
// Utility function to an edge (u, v) of weight w
public void addEdge( int u, int v, int weight)
{
AdjListNode node1 = new AdjListNode(v, weight);
adj[u].Add(node1); // Add v to u's list
AdjListNode node2 = new AdjListNode(u, weight);
adj[v].Add(node2); // Add u to v's list
}
// Prints shortest paths from src to all other vertices
bool pathMoreThanKUtil( int src, int k, bool [] path)
{
// If k is 0 or negative, return true;
if (k <= 0)
return true ;
// Get all adjacent vertices of source vertex src
// and recursively explore all paths from src.
List<AdjListNode> it = adj[src];
for ( int i = 0; i < adj[src].Count; i++) {
AdjListNode vertex = adj[src][i];
// Get adjacent vertex and weight of edge
int v = vertex.v;
int w = vertex.weight;
// If vertex v is already there in path, then
// there is a cycle (we ignore this edge)
if (path[v] == true )
continue ;
// If weight of is more than k, return true
if (w >= k)
return true ;
// Else add this vertex to path
path[v] = true ;
// If this adjacent can provide a path longer
// than k, return true.
if (pathMoreThanKUtil(v, k - w, path))
return true ;
// Backtrack
path[v] = false ;
}
// If no adjacent could produce longer path, return
// false
return false ;
}
// Returns true if graph has path more than k length
public bool pathMoreThanK( int src, int k)
{
// Create a path array with nothing included
// in path
bool [] path = new bool [V];
for ( int i = 0; i < V; i++)
path[i] = false ;
// Add source vertex to path
path[src] = true ;
return pathMoreThanKUtil(src, k, path);
}
} // This code is contributed by Tapesh (tapeshdua420) |
// JavaScript Program to find if there is a simple path with // weight more than k class AdjListNode { constructor(v, weight) {
this .v = v;
this .weight = weight;
}
} // This class represents a dipathted graph using // adjacency list representation class Graph { // Allocates memory for adjacency list
constructor(V) {
this .V = V; // No. of vertices
// In a weighted graph, we need to store vertex
// and weight pair for every edge
this .adj = [];
for ( var i = 0; i < V; i++) {
this .adj[i] = [];
}
}
// Utility function to an edge (u, v) of weight w
addEdge(u, v, weight) {
var node1 = new AdjListNode(v, weight);
this .adj[u].push(node1); // Add v to u's list
var node2 = new AdjListNode(u, weight);
this .adj[v].push(node2); // Add u to v's list
}
// Prints shortest paths from src to all other vertices
pathMoreThanKUtil(src, k, path) {
// If k is 0 or negative, return true;
if (k <= 0)
return true ;
// Get all adjacent vertices of source vertex src and
// recursively explore all paths from src.
var it = this .adj[src];
for ( var i = 0; i < this .adj[src].length; i++) {
var vertex = this .adj[src][i];
// Get adjacent vertex and weight of edge
var v = vertex.v;
var w = vertex.weight;
// If vertex v is already there in path, then
// there is a cycle (we ignore this edge)
if (path[v] == true )
continue ;
// If weight of is more than k, return true
if (w >= k)
return true ;
// Else add this vertex to path
path[v] = true ;
// If this adjacent can provide a path longer
// than k, return true.
if ( this .pathMoreThanKUtil(v, k - w, path))
return true ;
// Backtrack
path[v] = false ;
}
// If no adjacent could produce longer path, return
// false
return false ;
}
// Returns true if graph has path more than k length
pathMoreThanK(src, k) {
// Create a path array with nothing included
// in path
var path = new Array(V);
for ( var i = 0; i < V; i++) {
path[i] = false ;
}
// Add source vertex to path
path[src] = true ;
return this .pathMoreThanKUtil(src, k, path);
}
} // Driver program to test methods of graph class // create the graph given in above figure var V = 9;
var 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); var src = 0;
var k = 62;
if (g.pathMoreThanK(src, k)) {
console.log( "YES" );
} else {
console.log( "NO" );
} k = 60 if (g.pathMoreThanK(src, k)) {
console.log( "YES" );
} else {
console.log( "NO" );
} // This code is contributed by Tapesh(tapeshdua420) |
// Program to find if there is a simple path with // weight more than k #include<bits/stdc++.h> using namespace std;
// iPair ==> Integer Pair typedef pair< int , int > iPair;
// This class represents a dipathted graph using // adjacency list representation class Graph
{ int V; // No. of vertices
// In a weighted graph, we need to store vertex
// and weight pair for every edge
list< pair< int , int > > *adj;
bool pathMoreThanKUtil( int src, int k, vector< bool > &path);
public :
Graph( int V); // Constructor
// function to add an edge to graph
void addEdge( int u, int v, int w);
bool pathMoreThanK( int src, int k);
}; // Returns true if graph has path more than k length bool Graph::pathMoreThanK( int src, int k)
{ // Create a path array with nothing included
// in path
vector< bool > path(V, false );
// Add source vertex to path
path[src] = 1;
return pathMoreThanKUtil(src, k, path);
} // Prints shortest paths from src to all other vertices bool Graph::pathMoreThanKUtil( int src, int k, vector< bool > &path)
{ // If k is 0 or negative, return true;
if (k <= 0)
return true ;
// Get all adjacent vertices of source vertex src and
// recursively explore all paths from src.
list<iPair>::iterator i;
for (i = adj[src].begin(); i != adj[src].end(); ++i)
{
// Get adjacent vertex and weight of edge
int v = (*i).first;
int w = (*i).second;
// If vertex v is already there in path, then
// there is a cycle (we ignore this edge)
if (path[v] == true )
continue ;
// If weight of is more than k, return true
if (w >= k)
return true ;
// Else add this vertex to path
path[v] = true ;
// If this adjacent can provide a path longer
// than k, return true.
if (pathMoreThanKUtil(v, k-w, path))
return true ;
// Backtrack
path[v] = false ;
}
// If no adjacent could produce longer path, return
// false
return false ;
} // Allocates memory for adjacency list Graph::Graph( int V)
{ this ->V = V;
adj = new list<iPair> [V];
} // Utility function to an edge (u, v) of weight w void Graph::addEdge( int u, int v, int w)
{ adj[u].push_back(make_pair(v, w));
adj[v].push_back(make_pair(u, w));
} // Driver program to test methods of graph class 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);
int src = 0;
int k = 62;
g.pathMoreThanK(src, k)? cout << "Yes\n" :
cout << "No\n" ;
k = 60;
g.pathMoreThanK(src, k)? cout << "Yes\n" :
cout << "No\n" ;
return 0;
} |
Output:
No
Yes
Exercise:
Modify the above solution to find weight of longest path from a given source.
Time Complexity: The time complexity of this algorithm is O((V-1)!) where V is the number of vertices.
Auxiliary Space: O(V)
Explanation:
From the source node, we one-by-one visit all the paths and check if the total weight is greater than k for each path. So, the worst case will be when the number of possible paths is maximum. This is the case when every node is connected to every other node.
Since the starting point is fixed, we need to find a path including all the other vertices. There can be (V-1) choices for the second node, (V-2) choices for the third node, and so on till, 1 choice for the last node.
Therefore the total number of possible paths are (V-1)! in the worst case.
In the recursive call, when we reach a node, we again do recursive call for all of its child. Changing the state of array(which stores whether the node is visited or not), takes O(1) time.
So the time complexity becomes O((V-1)!).
The explanation for time complexity is contributed by Pranav Nambiar, and Aayush Patel.