鉴于无向图与从图中N个顶点和E边和两个顶点(U,V),任务是,如果这两个顶点之间存在路径,以检测。如果路径存在,则打印“是”,否则打印“否” 。
例子:
U = 1, V = 2
Output: No
Explanation:
There is no edge between the two points and hence its not possible to reach 2 from 1.
Input:
U = 1, V = 3
Output: Yes
Explanation: Vertex 3 from vertex 1 via vertices 2 or 4.
天真的方法:
这个想法是使用 Floyd Warshall 算法。为了解决这个问题,我们需要尝试[1, N]范围内的所有中间顶点并检查:
- 如果两个节点之间已经存在直接边。
- 或者我们有一条从节点i到中间节点k以及从节点k到节点j的路径。
下面是上述方法的实现:
C++
// C++ program to detect if a path
// exists between any two vertices
// for the given undirected graph
#include
using namespace std;
// Class representing a undirected
// graph using matrix
// representation
class Graph {
int V;
int** g;
public:
Graph(int V);
// Function to add an edge to graph
void addEdge(int v, int w);
// Function to check if
// there exists a path or not
bool isReachable(int s, int d);
// function to compute paths
// in the matrix using
// Floyd Warshall Algorithm
void computePaths();
};
Graph::Graph(int V)
{
this->V = V;
g = new int*[V + 1];
for (int i = 1; i < V + 1; i++) {
// Rows may not be contiguous
g[i] = new int[V + 1];
// Initialize all entries
// as false to indicate
// that there are
// no edges initially
memset(g[i], 0, (V + 1) * sizeof(int));
}
// Initializing node to itself
// as it is always reachable
for (int i = 1; i <= V; i++)
g[i][i] = 1;
}
// Function to add edge between nodes
void Graph::addEdge(int v, int w)
{
g[v][w] = 1;
g[w][v] = 1;
}
// Function to compute the path
void Graph::computePaths()
{
// Use Floyd Warshall algorithm
// to detect if a path exists
for (int k = 1; k <= V; k++) {
// Try every vertex as an
// intermediate vertex
// to check if a path exists
for (int i = 1; i <= V; i++) {
for (int j = 1; j <= V; j++)
g[i][j] = g[i][j]
| (g[i][k]
&& g[k][j]);
}
}
}
// Function to check if nodes are reachable
bool Graph::isReachable(int s, int d)
{
if (g[s][d] == 1)
return true;
else
return false;
}
// Driver code
int main()
{
Graph _g(4);
_g.addEdge(1, 2);
_g.addEdge(2, 3);
_g.addEdge(1, 4);
_g.computePaths();
int u = 4, v = 3;
if (_g.isReachable(u, v))
cout << "Yes\n";
else
cout << "No\n";
return 0;
}
Java
// Java program to detect if a path
// exists between any two vertices
// for the given undirected graph
import java.util.Arrays;
class GFG{
// Class representing a undirected
// graph using matrix representation
static class Graph
{
int V;
int[][] g;
public Graph(int V)
{
this.V = V;
// Rows may not be contiguous
g = new int[V + 1][V + 1];
for(int i = 0; i < V + 1; i++)
{
// Initialize all entries
// as false to indicate
// that there are
// no edges initially
Arrays.fill(g[i], 0);
}
// Initializing node to itself
// as it is always reachable
for(int i = 1; i <= V; i++)
g[i][i] = 1;
}
// Function to add edge between nodes
void addEdge(int v, int w)
{
g[v][w] = 1;
g[w][v] = 1;
}
// Function to check if nodes are reachable
boolean isReachable(int s, int d)
{
if (g[s][d] == 1)
return true;
else
return false;
}
// Function to compute the path
void computePaths()
{
// Use Floyd Warshall algorithm
// to detect if a path exists
for(int k = 1; k <= V; k++)
{
// Try every vertex as an
// intermediate vertex
// to check if a path exists
for(int i = 1; i <= V; i++)
{
for(int j = 1; j <= V; j++)
g[i][j] = g[i][j] | ((g[i][k] != 0 &&
g[k][j] != 0) ? 1 : 0);
}
}
}
};
// Driver code
public static void main(String[] args)
{
Graph _g = new Graph(4);
_g.addEdge(1, 2);
_g.addEdge(2, 3);
_g.addEdge(1, 4);
_g.computePaths();
int u = 4, v = 3;
if (_g.isReachable(u, v))
System.out.println("Yes");
else
System.out.println("No");
}
}
// This code is contributed by sanjeev2552
Python3
# Python3 program to detect if a path
# exists between any two vertices
# for the given undirected graph
# Class representing a undirected
# graph using matrix
# representation
class Graph:
def __init__(self, V):
self.V = V
# Initialize all entries
# as false to indicate
# that there are
# no edges initially
self.g = [[0 for j in range(self.V + 1)]
for i in range(self.V + 1)]
# Initializing node to itself
# as it is always reachable
for i in range(self.V + 1):
self.g[i][i] = 1
# Function to add edge between nodes
def addEdge(self, v, w):
self.g[v][w] = 1
self.g[w][v] = 1
# Function to compute the path
def computePaths(self):
# Use Floyd Warshall algorithm
# to detect if a path exists
for k in range(1, self.V + 1):
# Try every vertex as an
# intermediate vertex
# to check if a path exists
for i in range(1, self.V + 1):
for j in range(1, self.V + 1):
self.g[i][j] = (self.g[i][j] |
(self.g[i][k] and
self.g[k][j]))
# Function to check if nodes
# are reachable
def isReachable(self, s, d):
if (self.g[s][d] == 1):
return True
else:
return False
# Driver code
if __name__=='__main__':
_g = Graph(4)
_g.addEdge(1, 2)
_g.addEdge(2, 3)
_g.addEdge(1, 4)
_g.computePaths()
u = 4
v = 3
if (_g.isReachable(u, v)):
print('Yes')
else:
print('No')
# This code is contributed by rutvik_56
Javascript
C++
// C++ program to check if there is exist a path between
// two vertices of an undirected graph.
#include
#include
using namespace std;
// This class represents an undirected graph
// using adjacency list representation
class Graph {
int V; // No. of vertices
// Pointer to an array containing adjacency lists
list* adj;
public:
Graph(int V); // Constructor
// function to add an edge to graph
void addEdge(int v, int w);
bool isReachable(int s, int d);
};
Graph::Graph(int V)
{
this->V = V;
adj = new list[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w);
adj[w].push_back(v);
}
// A BFS based function to check whether d is reachable from s.
bool Graph::isReachable(int s, int d)
{
// Base case
if (s == d)
return true;
// Mark all the vertices as not visited
bool* visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Create a queue for BFS
list queue;
// Mark the current node as visited and enqueue it
visited[s] = true;
queue.push_back(s);
// it will be used to get all adjacent vertices of a vertex
list::iterator i;
while (!queue.empty()) {
// Dequeue a vertex from queue and print it
s = queue.front();
queue.pop_front();
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
for (i = adj[s].begin(); i != adj[s].end(); ++i) {
// If this adjacent node is the destination node,
// then return true
if (*i == d)
return true;
// Else, continue to do BFS
if (!visited[*i]) {
visited[*i] = true;
queue.push_back(*i);
}
}
}
// If BFS is complete without visiting d
return false;
}
// Driver program to test methods of graph class
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
int u = 1, v = 3;
if (g.isReachable(u, v))
cout << "\n There is a path from " << u << " to " << v;
else
cout << "\n There is no path from " << u << " to " << v;
return 0;
}
Java
import java.util.ArrayList;
import java.util.LinkedList;
import java.util.Queue;
// Java program to check if there is exist a path between
// two vertices of an undirected graph.
public class Graph {
// This class represents an undirected graph
// using adjacency list representation
int V; // No. of vertices
// Pointer to an array containing adjacency lists
ArrayList> adj;
Graph(int V){
this.V = V;
adj = new ArrayList<>();
for(int i=0;i());
}
// function to add an edge to graph
void addEdge(int v, int w)
{
adj.get(v).add(w);
adj.get(w).add(v);
}
// A BFS based function to check whether d is reachable from s.
boolean isReachable(int s, int d)
{
// Base case
if (s == d)
return true;
// Mark all the vertices as not visited
boolean[] visited = new boolean[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Create a queue for BFS
Queue queue = new LinkedList<>();
// Mark the current node as visited and enqueue it
visited[s] = true;
queue.add(s);
while (!queue.isEmpty()) {
// Dequeue a vertex from queue and print it
s = queue.remove();
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
for (int i=0; i
Python3
# Python3 program to check if there is exist a path between
# two vertices of an undirected graph.
from collections import deque
def addEdge(v, w):
global adj
adj[v].append(w)
adj[w].append(v)
# A BFS based function to check whether d is reachable from s.
def isReachable(s, d):
# Base case
if (s == d):
return True
# Mark all the vertices as not visited
visited = [False for i in range(V)]
# Create a queue for BFS
queue = deque()
# Mark the current node as visited and enqueue it
visited[s] = True
queue.append(s)
while (len(queue) > 0):
# Dequeue a vertex from queue and prit
s = queue.popleft()
# queue.pop_front()
# Get all adjacent vertices of the dequeued vertex s
# If a adjacent has not been visited, then mark it
# visited and enqueue it
for i in adj[s]:
# If this adjacent node is the destination node,
# then return true
if (i == d):
return True
# Else, continue to do BFS
if (not visited[i]):
visited[i] = True
queue.append(i)
# If BFS is complete without visiting d
return False
# Driver program to test methods of graph class
if __name__ == '__main__':
# Create a graph given in the above diagram
V = 4
adj = [[] for i in range(V+1)]
addEdge(0, 1)
addEdge(0, 2)
addEdge(1, 2)
addEdge(2, 0)
addEdge(2, 3)
addEdge(3, 3)
u,v = 1, 3
if (isReachable(u, v)):
print("There is a path from",u,"to",v)
else:
print("There is no path from",u,"to",v)
# This code is contributed by mohit kumar 29.
输出:
Yes
时间复杂度: O(V 3 )
辅助空间: O(V 2 )
高效的解决方案
我们可以使用 BFS 或 DFS 来查找是否存在从 u 到 v 的路径。以下是基于 BFS 的解决方案
C++
// C++ program to check if there is exist a path between
// two vertices of an undirected graph.
#include
#include
using namespace std;
// This class represents an undirected graph
// using adjacency list representation
class Graph {
int V; // No. of vertices
// Pointer to an array containing adjacency lists
list* adj;
public:
Graph(int V); // Constructor
// function to add an edge to graph
void addEdge(int v, int w);
bool isReachable(int s, int d);
};
Graph::Graph(int V)
{
this->V = V;
adj = new list[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w);
adj[w].push_back(v);
}
// A BFS based function to check whether d is reachable from s.
bool Graph::isReachable(int s, int d)
{
// Base case
if (s == d)
return true;
// Mark all the vertices as not visited
bool* visited = new bool[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Create a queue for BFS
list queue;
// Mark the current node as visited and enqueue it
visited[s] = true;
queue.push_back(s);
// it will be used to get all adjacent vertices of a vertex
list::iterator i;
while (!queue.empty()) {
// Dequeue a vertex from queue and print it
s = queue.front();
queue.pop_front();
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
for (i = adj[s].begin(); i != adj[s].end(); ++i) {
// If this adjacent node is the destination node,
// then return true
if (*i == d)
return true;
// Else, continue to do BFS
if (!visited[*i]) {
visited[*i] = true;
queue.push_back(*i);
}
}
}
// If BFS is complete without visiting d
return false;
}
// Driver program to test methods of graph class
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
int u = 1, v = 3;
if (g.isReachable(u, v))
cout << "\n There is a path from " << u << " to " << v;
else
cout << "\n There is no path from " << u << " to " << v;
return 0;
}
Java
import java.util.ArrayList;
import java.util.LinkedList;
import java.util.Queue;
// Java program to check if there is exist a path between
// two vertices of an undirected graph.
public class Graph {
// This class represents an undirected graph
// using adjacency list representation
int V; // No. of vertices
// Pointer to an array containing adjacency lists
ArrayList> adj;
Graph(int V){
this.V = V;
adj = new ArrayList<>();
for(int i=0;i());
}
// function to add an edge to graph
void addEdge(int v, int w)
{
adj.get(v).add(w);
adj.get(w).add(v);
}
// A BFS based function to check whether d is reachable from s.
boolean isReachable(int s, int d)
{
// Base case
if (s == d)
return true;
// Mark all the vertices as not visited
boolean[] visited = new boolean[V];
for (int i = 0; i < V; i++)
visited[i] = false;
// Create a queue for BFS
Queue queue = new LinkedList<>();
// Mark the current node as visited and enqueue it
visited[s] = true;
queue.add(s);
while (!queue.isEmpty()) {
// Dequeue a vertex from queue and print it
s = queue.remove();
// Get all adjacent vertices of the dequeued vertex s
// If a adjacent has not been visited, then mark it
// visited and enqueue it
for (int i=0; i
蟒蛇3
# Python3 program to check if there is exist a path between
# two vertices of an undirected graph.
from collections import deque
def addEdge(v, w):
global adj
adj[v].append(w)
adj[w].append(v)
# A BFS based function to check whether d is reachable from s.
def isReachable(s, d):
# Base case
if (s == d):
return True
# Mark all the vertices as not visited
visited = [False for i in range(V)]
# Create a queue for BFS
queue = deque()
# Mark the current node as visited and enqueue it
visited[s] = True
queue.append(s)
while (len(queue) > 0):
# Dequeue a vertex from queue and prit
s = queue.popleft()
# queue.pop_front()
# Get all adjacent vertices of the dequeued vertex s
# If a adjacent has not been visited, then mark it
# visited and enqueue it
for i in adj[s]:
# If this adjacent node is the destination node,
# then return true
if (i == d):
return True
# Else, continue to do BFS
if (not visited[i]):
visited[i] = True
queue.append(i)
# If BFS is complete without visiting d
return False
# Driver program to test methods of graph class
if __name__ == '__main__':
# Create a graph given in the above diagram
V = 4
adj = [[] for i in range(V+1)]
addEdge(0, 1)
addEdge(0, 2)
addEdge(1, 2)
addEdge(2, 0)
addEdge(2, 3)
addEdge(3, 3)
u,v = 1, 3
if (isReachable(u, v)):
print("There is a path from",u,"to",v)
else:
print("There is no path from",u,"to",v)
# This code is contributed by mohit kumar 29.
输出:
There is a path from 1 to 3
时间复杂度: O(V + E)
辅助空间: O(V)
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