# Find if there is a path between two vertices in a directed graph | Set 2

• Difficulty Level : Easy
• Last Updated : 30 Jun, 2021

Given a Directed Graph and two vertices in it, check whether there is a path from the first given vertex to second.

Example:

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Consider the following Graph: Input : (u, v) = (1, 3)
Output: Yes
Explanation:
There is a path from 1 to 3, 1 -> 2 -> 3

Input : (u, v) = (3, 6)
Output: No
Explanation:
There is no path from 3 to 6

A BFS or DFS based solution of this problem is discussed here.
Approach: Here we will discuss a Dynamic Programming based solution using Floyd Warshall Algorithm

• Create a boolean 2D matrix mat where mat[i][j] will be true if there is a path from vertex i to j.
• For every starting vertex i and ending vertex j iterate over all intermediate vertex k and do check if there is a path for i to j through k then mark mat[i][j] as true.
• Finally, check if mat[u][v] is true then return true else return false.

Below is the implementation of the above approach:

## C++

 `// C++ program to find if there is a``// path between two vertices in a``// directed graph using Dynamic Programming` `#include ``using` `namespace` `std;``#define X 6``#define Z 2` `// function to find if there is a``// path between two vertices in a``// directed graph``bool` `existPath(``int` `V, ``int` `edges[X][Z],``               ``int` `u, ``int` `v)``{``    ``// dp matrix``    ``bool` `mat[V][V];``    ``memset``(mat, ``false``, ``sizeof``(mat));` `    ``// set dp[i][j]=true if there is``    ``// edge between i to j``    ``for` `(``int` `i = 0; i < X; i++)``        ``mat[edges[i]][edges[i]] = ``true``;` `    ``// check for all intermediate vertex``    ``for` `(``int` `k = 0; k < V; k++) {``        ``for` `(``int` `i = 0; i < V; i++) {``            ``for` `(``int` `j = 0; j < V; j++) {` `                ``mat[i][j] = mat[i][j]``                            ``|| mat[i][k]``                                   ``&& mat[k][j];``            ``}``        ``}``    ``}` `    ``// if vertex is invalid``    ``if` `(u >= V || v >= V) {``        ``return` `false``;``    ``}` `    ``// if there is a path``    ``if` `(mat[u][v])``        ``return` `true``;``    ``return` `false``;``}` `// Driver function``int` `main()``{``    ``int` `V = 4;``    ``int` `edges[X][Z]``        ``= { { 0, 2 }, { 0, 1 },``            ``{ 1, 2 }, { 2, 3 },``            ``{ 2, 0 }, { 3, 3 } };``    ``int` `u = 1, v = 3;` `    ``if` `(existPath(V, edges, u, v))``        ``cout << ``"Yes\n"``;``    ``else``        ``cout << ``"No\n"``;``    ``return` `0;``}`

## Java

 `// Java program to find if there is a path``// between two vertices in a directed graph``// using Dynamic Programming``import` `java.util.*;` `class` `GFG{``    ` `static` `final` `int` `X = ``6``;``static` `final` `int` `Z = ``2``;` `// Function to find if there is a``// path between two vertices in a``// directed graph``static` `boolean` `existPath(``int` `V, ``int` `edges[][],``                         ``int` `u, ``int` `v)``{``    ` `    ``// mat matrix``    ``boolean` `[][]mat = ``new` `boolean``[V][V];` `    ``// set mat[i][j]=true if there is``    ``// edge between i to j``    ``for` `(``int` `i = ``0``; i < X; i++)``        ``mat[edges[i][``0``]][edges[i][``1``]] = ``true``;` `    ``// Check for all intermediate vertex``    ``for``(``int` `k = ``0``; k < V; k++)``    ``{``        ``for``(``int` `i = ``0``; i < V; i++)``        ``{``            ``for``(``int` `j = ``0``; j < V; j++)``            ``{``                ``mat[i][j] = mat[i][j] ||``                            ``mat[i][k] &&``                            ``mat[k][j];``            ``}``        ``}``    ``}` `    ``// If vertex is invalid``    ``if` `(u >= V || v >= V)``    ``{``        ``return` `false``;``    ``}` `    ``// If there is a path``    ``if` `(mat[u][v])``        ``return` `true``;``    ``return` `false``;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `V = ``4``;``    ``int` `edges[][] = { { ``0``, ``2` `}, { ``0``, ``1` `},``                      ``{ ``1``, ``2` `}, { ``2``, ``3` `},``                      ``{ ``2``, ``0` `}, { ``3``, ``3` `} };``    ``int` `u = ``1``, v = ``3``;` `    ``if` `(existPath(V, edges, u, v))``        ``System.out.print(``"Yes\n"``);``    ``else``        ``System.out.print(``"No\n"``);``}``}` `// This code is contributed by Princi Singh`

## Python3

 `# Python3 program to find if there``# is a path between two vertices in a``# directed graph using Dynamic Programming``X ``=` `6``Z ``=` `2`` ` `# Function to find if there is a``# path between two vertices in a``# directed graph``def` `existPath(V, edges, u, v):``    ` `    ``# dp matrix``    ``mat ``=` `[[``False` `for` `i ``in` `range``(V)]``                  ``for` `j ``in` `range``(V)]`` ` `    ``# Set dp[i][j]=true if there is``    ``# edge between i to j``    ``for` `i ``in` `range``(X):``        ``mat[edges[i][``0``]][edges[i][``1``]] ``=` `True`` ` `    ``# Check for all intermediate vertex``    ``for` `k ``in` `range``(V):``        ``for` `i ``in` `range``(V):``            ``for` `j ``in` `range``(V):``                ``mat[i][j] ``=` `(mat[i][j] ``or``                             ``mat[i][k] ``and``                             ``mat[k][j])`` ` `    ``# If vertex is invalid``    ``if` `(u >``=` `V ``or` `v >``=` `V):``        ``return` `False`` ` `    ``# If there is a path``    ``if` `(mat[u][v]):``        ``return` `True``        ` `    ``return` `False` `# Driver code``V ``=` `4``edges ``=` `[ [ ``0``, ``2` `], [ ``0``, ``1` `],``          ``[ ``1``, ``2` `], [ ``2``, ``3` `],``          ``[ ``2``, ``0` `], [ ``3``, ``3` `] ]``        ` `u, v ``=` `1``, ``3` `if` `(existPath(V, edges, u, v)):``    ``print``(``"Yes"``)``else``:``    ``print``(``"No"``)` `# This code is contributed by divyeshrabadiya07`

## C#

 `// C# program to find if there is a path``// between two vertices in a directed graph``// using Dynamic Programming``using` `System;``class` `GFG{``    ` `static` `readonly` `int` `X = 6;``static` `readonly` `int` `Z = 2;` `// Function to find if there is a``// path between two vertices in a``// directed graph``static` `bool` `existPath(``int` `V, ``int` `[,]edges,``                      ``int` `u, ``int` `v)``{``    ` `    ``// mat matrix``    ``bool` `[,]mat = ``new` `bool``[V, V];` `    ``// set mat[i,j]=true if there is``    ``// edge between i to j``    ``for` `(``int` `i = 0; i < X; i++)``        ``mat[edges[i, 0], edges[i, 1]] = ``true``;` `    ``// Check for all intermediate vertex``    ``for``(``int` `k = 0; k < V; k++)``    ``{``        ``for``(``int` `i = 0; i < V; i++)``        ``{``            ``for``(``int` `j = 0; j < V; j++)``            ``{``                ``mat[i, j] = mat[i, j] ||``                            ``mat[i, k] &&``                            ``mat[k, j];``            ``}``        ``}``    ``}` `    ``// If vertex is invalid``    ``if` `(u >= V || v >= V)``    ``{``        ``return` `false``;``    ``}` `    ``// If there is a path``    ``if` `(mat[u, v])``        ``return` `true``;``    ``return` `false``;``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `V = 4;``    ``int` `[,]edges = { { 0, 2 }, { 0, 1 },``                     ``{ 1, 2 }, { 2, 3 },``                     ``{ 2, 0 }, { 3, 3 } };``    ``int` `u = 1, v = 3;` `    ``if` `(existPath(V, edges, u, v))``        ``Console.Write(``"Yes\n"``);``    ``else``        ``Console.Write(``"No\n"``);``}``}` `// This code is contributed by sapnasingh4991`

## Javascript

 ``
Output:
`Yes`

Time Complexity : O ( V 3)
Auxiliary Space : O ( V 2)

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