给定图的最小生成树成本

给定一个名为 V ₁ , V ₂ , V ₃ , ..., V _n的V节点 (V > 2) 的无向图。两个节点V _i和V _j相互连接当且仅当0 < |我 - j | ≤ 2 。任何顶点对(V _i , V _j )之间的每条边都被分配了一个权重i + j 。任务是找到具有V个节点的此类图的最小生成树的成本。
例子：

Input: V = 4

Output: 13
Input: V = 5
Output: 21

编程需要懂一点英语

方法：从具有最小节点（即3个节点）的图开始，最小生成树的成本将为7。现在对于可以添加到该图中的从第四个节点开始的每个节点i ，第i^个节点只能是连接到第 (i – 1 ⁾和( ⁱ – 2)节点，最小生成树将仅包含具有最小权重的节点，因此新添加的边将具有权重i + (i – 2) 。

So addition of fourth node will increase the overall weight as 7 + (4 + 2) = 13
Similarly adding fifth node, weight = 13 + (5 + 3) = 21
…
For n^th node, weight = weight + (n + (n – 2)).

编程需要懂一点英语

这可以概括为权重 = V ² – V + 1 ，其中V是图中的节点总数。
下面是上述方法的实现：

C++

// C++ implementation of the approach
#include 
using namespace std;
 
// Function that returns the minimum cost
// of the spanning tree for the required graph
int getMinCost(int Vertices)
{
    int cost = 0;
 
    // Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1;
 
    return cost;
}
 
// Driver code
int main()
{
    int V = 5;
    cout << getMinCost(V);
 
    return 0;
}

Java

// Java implementation of the approach
class GfG
{
 
// Function that returns the minimum cost
// of the spanning tree for the required graph
static int getMinCost(int Vertices)
{
    int cost = 0;
 
    // Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1;
 
    return cost;
}
 
// Driver code
public static void main(String[] args)
{
    int V = 5;
    System.out.println(getMinCost(V));
}
}
 
// This code is contributed by
// Prerna Saini.

C#

// C# implementation of the above approach
using System;
 
class GfG
{
 
    // Function that returns the minimum cost
    // of the spanning tree for the required graph
    static int getMinCost(int Vertices)
    {
        int cost = 0;
     
        // Calculating cost of MST
        cost = (Vertices * Vertices) - Vertices + 1;
     
        return cost;
    }
     
    // Driver code
    public static void Main()
    {
        int V = 5;
        Console.WriteLine(getMinCost(V));
    }
}
 
// This code is contributed by Ryuga

Python3

# python3 implementation of the approach
  
# Function that returns the minimum cost
# of the spanning tree for the required graph
def getMinCost( Vertices):
    cost = 0
  
    # Calculating cost of MST
    cost = (Vertices * Vertices) - Vertices + 1
  
    return cost
  
# Driver code
if __name__ == "__main__":
 
    V = 5
    print (getMinCost(V))

PHP

Javascript

输出：