给定非负整数arr []的序列,任务是检查是否存在与该度序列相对应的简单图。请注意,简单图是没有自环和平行边的图。
例子:
Input: arr[] = {3, 3, 3, 3}
Output: Yes
This is actually a complete graph(K4)
Input: arr[] = {3, 2, 1, 0}
Output: No
A vertex has degree n-1 so it’s connected to all the other n-1 vertices.
But another vertex has degree 0 i.e. isolated. It’s a contradiction.
方法:一种检查简单图形是否存在的方法是使用Havel-Hakimi算法,如下所示:
- 以非递增顺序对非负整数序列进行排序。
- 删除第一个元素(例如V)。从下一个V元素中减去1。
- 重复1和2,直到满足其中一个停止条件。
停止条件:
- 剩余的所有元素都等于0(存在简单图)。
- 减后遇到负数(不存在简单图形)。
- 没有足够的元素可用于减法步骤(不存在简单图形)。
下面是上述方法的实现:
C++
// C++ implementation of the approach
#include
using namespace std;
// Function that returns true if
// a simple graph exists
bool graphExists(vector &a, int n)
{
// Keep performing the operations until one
// of the stopping condition is met
while (1)
{
// Sort the list in non-decreasing order
sort(a.begin(), a.end(), greater<>());
// Check if all the elements are equal to 0
if (a[0] == 0)
return true;
// Store the first element in a variable
// and delete it from the list
int v = a[0];
a.erase(a.begin() + 0);
// Check if enough elements
// are present in the list
if (v > a.size())
return false;
// Subtract first element from next v elements
for (int i = 0; i < v; i++)
{
a[i]--;
// Check if negative element is
// encountered after subtraction
if (a[i] < 0)
return false;
}
}
}
// Driver Code
int main()
{
vector a = {3, 3, 3, 3};
int n = a.size();
graphExists(a, n) ? cout << "Yes" :
cout << "NO" << endl;
return 0;
}
// This code is contributed by
// sanjeev2552
Java
// Java implementation of the approach
import java.util.*;
@SuppressWarnings("unchecked")
class GFG{
// Function that returns true if
// a simple graph exists
static boolean graphExists(ArrayList a, int n)
{
// Keep performing the operations until one
// of the stopping condition is met
while (true)
{
// Sort the list in non-decreasing order
Collections.sort(a, Collections.reverseOrder());
// Check if all the elements are equal to 0
if ((int)a.get(0) == 0)
return true;
// Store the first element in a variable
// and delete it from the list
int v = (int)a.get(0);
a.remove(a.get(0));
// Check if enough elements
// are present in the list
if (v > a.size())
return false;
// Subtract first element from
// next v elements
for(int i = 0; i < v; i++)
{
a.set(i, (int)a.get(i) - 1);
// Check if negative element is
// encountered after subtraction
if ((int)a.get(i) < 0)
return false;
}
}
}
// Driver Code
public static void main(String[] args)
{
ArrayList a = new ArrayList();
a.add(3);
a.add(3);
a.add(3);
a.add(3);
int n = a.size();
if (graphExists(a, n))
{
System.out.print("Yes");
}
else
{
System.out.print("NO");
}
}
}
// This code is contributed by pratham76
Python3
# Python3 implementation of the approach
# Function that returns true if
# a simple graph exists
def graphExists(a):
# Keep performing the operations until one
# of the stopping condition is met
while True:
# Sort the list in non-decreasing order
a = sorted(a, reverse = True)
# Check if all the elements are equal to 0
if a[0]== 0 and a[len(a)-1]== 0:
return True
# Store the first element in a variable
# and delete it from the list
v = a[0]
a = a[1:]
# Check if enough elements
# are present in the list
if v>len(a):
return False
# Subtract first element from next v elements
for i in range(v):
a[i]-= 1
# Check if negative element is
# encountered after subtraction
if a[i]<0:
return False
# Driver code
a = [3, 3, 3, 3]
if(graphExists(a)):
print("Yes")
else:
print("No")
C#
// C# implementation of the approach
using System;
using System.Collections;
class GFG{
// Function that returns true if
// a simple graph exists
static bool graphExists(ArrayList a, int n)
{
// Keep performing the operations until one
// of the stopping condition is met
while (true)
{
// Sort the list in non-decreasing order
a.Sort();
a.Reverse();
// Check if all the elements are equal to 0
if ((int)a[0] == 0)
return true;
// Store the first element in a variable
// and delete it from the list
int v = (int)a[0];
a.Remove(a[0]);
// Check if enough elements
// are present in the list
if (v > a.Count)
return false;
// Subtract first element from
// next v elements
for(int i = 0; i < v; i++)
{
a[i] = (int)a[i] - 1;
// Check if negative element is
// encountered after subtraction
if ((int)a[i] < 0)
return false;
}
}
}
// Driver Code
public static void Main(string[] args)
{
ArrayList a = new ArrayList(){ 3, 3, 3, 3 };
int n = a.Count;
if (graphExists(a, n))
{
Console.Write("Yes");
}
else
{
Console.Write("NO");
}
}
}
// This code is contributed by rutvik_56
输出:
Yes