给定一个具有N 个节点和E 个边的小图,任务是在给定图中找到最大的团。团是给定图的完整子图。这意味着所述子图中的所有节点都直接相互连接,或者子图中任意两个节点之间存在边。最大团是包含最大节点数的给定图的完整子图。
例子:
Input: N = 4, edges[][] = {{1, 2}, {2, 3}, {3, 1}, {4, 3}, {4, 1}, {4, 2}}
Output: 4
Input: N = 5, edges[][] = {{1, 2}, {2, 3}, {3, 1}, {4, 3}, {4, 5}, {5, 3}}
Output: 3
方法:思路是用递归来解决问题。
- 当一条边被添加到当前列表中时,检查是否通过将该边添加到当前列表中,它是否仍然形成一个集团。
- 添加顶点直到列表不形成团。然后,该列表被回溯以找到形成集团的更大子集。
下面是上述方法的实现:
C++
// C++ implementation of the approach
#include
using namespace std;
const int MAX = 100;
// Stores the vertices
int store[MAX], n;
// Graph
int graph[MAX][MAX];
// Degree of the vertices
int d[MAX];
// Function to check if the given set of
// vertices in store array is a clique or not
bool is_clique(int b)
{
// Run a loop for all set of edges
for (int i = 1; i < b; i++) {
for (int j = i + 1; j < b; j++)
// If any edge is missing
if (graph[store[i]][store[j]] == 0)
return false;
}
return true;
}
// Function to find all the sizes
// of maximal cliques
int maxCliques(int i, int l)
{
// Maximal clique size
int max_ = 0;
// Check if any vertices from i+1
// can be inserted
for (int j = i + 1; j <= n; j++) {
// Add the vertex to store
store[l] = j;
// If the graph is not a clique of size k then
// it cannot be a clique by adding another edge
if (is_clique(l + 1)) {
// Update max
max_ = max(max_, l);
// Check if another edge can be added
max_ = max(max_, maxCliques(j, l + 1));
}
}
return max_;
}
// Driver code
int main()
{
int edges[][2] = { { 1, 2 }, { 2, 3 }, { 3, 1 },
{ 4, 3 }, { 4, 1 }, { 4, 2 } };
int size = sizeof(edges) / sizeof(edges[0]);
n = 4;
for (int i = 0; i < size; i++) {
graph[edges[i][0]][edges[i][1]] = 1;
graph[edges[i][1]][edges[i][0]] = 1;
d[edges[i][0]]++;
d[edges[i][1]]++;
}
cout << maxCliques(0, 1);
return 0;
}
Java
// Java implementation of the approach
import java.util.*;
class GFG
{
static int MAX = 100, n;
// Stores the vertices
static int []store = new int[MAX];
// Graph
static int [][]graph = new int[MAX][MAX];
// Degree of the vertices
static int []d = new int[MAX];
// Function to check if the given set of
// vertices in store array is a clique or not
static boolean is_clique(int b)
{
// Run a loop for all set of edges
for (int i = 1; i < b; i++)
{
for (int j = i + 1; j < b; j++)
// If any edge is missing
if (graph[store[i]][store[j]] == 0)
return false;
}
return true;
}
// Function to find all the sizes
// of maximal cliques
static int maxCliques(int i, int l)
{
// Maximal clique size
int max_ = 0;
// Check if any vertices from i+1
// can be inserted
for (int j = i + 1; j <= n; j++)
{
// Add the vertex to store
store[l] = j;
// If the graph is not a clique of size k then
// it cannot be a clique by adding another edge
if (is_clique(l + 1))
{
// Update max
max_ = Math.max(max_, l);
// Check if another edge can be added
max_ = Math.max(max_, maxCliques(j, l + 1));
}
}
return max_;
}
// Driver code
public static void main(String[] args)
{
int [][]edges = { { 1, 2 }, { 2, 3 }, { 3, 1 },
{ 4, 3 }, { 4, 1 }, { 4, 2 } };
int size = edges.length;
n = 4;
for (int i = 0; i < size; i++)
{
graph[edges[i][0]][edges[i][1]] = 1;
graph[edges[i][1]][edges[i][0]] = 1;
d[edges[i][0]]++;
d[edges[i][1]]++;
}
System.out.print(maxCliques(0, 1));
}
}
// This code is contributed by 29AjayKumar
Python3
# Python3 implementation of the approach
MAX = 100;
n = 0;
# Stores the vertices
store = [0] * MAX;
# Graph
graph = [[0 for i in range(MAX)] for j in range(MAX)];
# Degree of the vertices
d = [0] * MAX;
# Function to check if the given set of
# vertices in store array is a clique or not
def is_clique(b):
# Run a loop for all set of edges
for i in range(1, b):
for j in range(i + 1, b):
# If any edge is missing
if (graph[store[i]][store[j]] == 0):
return False;
return True;
# Function to find all the sizes
# of maximal cliques
def maxCliques(i, l):
# Maximal clique size
max_ = 0;
# Check if any vertices from i+1
# can be inserted
for j in range(i + 1, n + 1):
# Add the vertex to store
store[l] = j;
# If the graph is not a clique of size k then
# it cannot be a clique by adding another edge
if (is_clique(l + 1)):
# Update max
max_ = max(max_, l);
# Check if another edge can be added
max_ = max(max_, maxCliques(j, l + 1));
return max_;
# Driver code
if __name__ == '__main__':
edges = [[ 1, 2 ],[ 2, 3 ],[ 3, 1 ],
[ 4, 3 ],[ 4, 1 ],[ 4, 2 ]];
size = len(edges);
n = 4;
for i in range(size):
graph[edges[i][0]][edges[i][1]] = 1;
graph[edges[i][1]][edges[i][0]] = 1;
d[edges[i][0]] += 1;
d[edges[i][1]] += 1;
print(maxCliques(0, 1));
# This code is contributed by PrinciRaj1992
C#
// C# implementation of the approach
using System;
class GFG
{
static int MAX = 100, n;
// Stores the vertices
static int []store = new int[MAX];
// Graph
static int [,]graph = new int[MAX,MAX];
// Degree of the vertices
static int []d = new int[MAX];
// Function to check if the given set of
// vertices in store array is a clique or not
static bool is_clique(int b)
{
// Run a loop for all set of edges
for (int i = 1; i < b; i++)
{
for (int j = i + 1; j < b; j++)
// If any edge is missing
if (graph[store[i],store[j]] == 0)
return false;
}
return true;
}
// Function to find all the sizes
// of maximal cliques
static int maxCliques(int i, int l)
{
// Maximal clique size
int max_ = 0;
// Check if any vertices from i+1
// can be inserted
for (int j = i + 1; j <= n; j++)
{
// Add the vertex to store
store[l] = j;
// If the graph is not a clique of size k then
// it cannot be a clique by adding another edge
if (is_clique(l + 1))
{
// Update max
max_ = Math.Max(max_, l);
// Check if another edge can be added
max_ = Math.Max(max_, maxCliques(j, l + 1));
}
}
return max_;
}
// Driver code
public static void Main(String[] args)
{
int [,]edges = { { 1, 2 }, { 2, 3 }, { 3, 1 },
{ 4, 3 }, { 4, 1 }, { 4, 2 } };
int size = edges.GetLength(0);
n = 4;
for (int i = 0; i < size; i++)
{
graph[edges[i, 0], edges[i, 1]] = 1;
graph[edges[i, 1], edges[i, 0]] = 1;
d[edges[i, 0]]++;
d[edges[i, 1]]++;
}
Console.Write(maxCliques(0, 1));
}
}
// This code is contributed by PrinciRaj1992
输出:
4
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