【国科大卜算】Truck History 最小生成树Prim

Truck History

problem description

Advanced Cargo Movement, Ltd. uses trucks of different types. Some trucks are used for vegetable delivery, other for furniture, or for bricks. The company has its own code describing each type of a truck. The code is simply a string of exactly seven lowercase letters (each letter on each position has a very special meaning but that is unimportant for this task). At the beginning of company’s history, just a single truck type was used but later other types were derived from it, then from the new types another types were derived, and so on.

Today, ACM is rich enough to pay historians to study its history. One thing historians tried to find out is so called derivation plan – i.e. how the truck types were derived. They defined the distance of truck types as the number of positions with different letters in truck type codes. They also assumed that each truck type was derived from exactly one other truck type (except for the first truck type which was not derived from any other type). The quality of a derivation plan was then defined as
$$\frac{1}{{\Sigma }_{(t_o,t_d)d(t_o,t_d)}}$$
where the sum goes over all pairs of types in the derivation plan such that $t_o$ is the original type and $t_d$ the type derived from it and $d(t_o,t_d) $ is the distance of the types.

Since historians failed, you are to write a program to help them. Given the codes of truck types, your program should find the highest possible quality of a derivation plan.

Input

The input consists of several test cases. Each test case begins with a line containing the number of truck types, N, 2 <= N <= 2 000. Each of the following N lines of input contains one truck type code (a string of seven lowercase letters). You may assume that the codes uniquely describe the trucks, i.e., no two of these N lines are the same. The input is terminated with zero at the place of number of truck types.

Output

For each test case, your program should output the text “The highest possible quality is 1/Q.”, where 1/Q is the quality of the best derivation plan.

Sample

Input	Output
4 aaaaaaa baaaaaa abaaaaa aabaaaa 0	The highest possible quality is 1/3.

个人理解

用一个由7个字符组成的字符串代表一个卡车，其中两个不同的字符串之间各个相同位置的不同的字符数目代表其距离$d(t_o,t_d)$，其中$t_d$为从$t_o$中派生出来的类型。其中第一个输入的字符串无父类型。所以可以理解为树形结构，其中第一个输入的字符串为这颗树的父节点，且每一个节点到其衍生节点到距离可以视为该段的权值，所以求最小生成树。

#include
#include
#include
#include
using namespace std;
int a[2010][2010],book[2010],dis[2020];
char c[2010][2010];
int n,m,t;//n个点，m条边
const int inf=0x3f3f3f;
int sum;
 
int prim()
{
	int u;sum=0;
	
	for(int i=1;i<=n;i++)
	{
		book[i]=0;
		dis[i]=a[1][i];
	}
	book[1]=1;
	dis[1]=0;
	for(int i=1;i<n;i++)
	{
		int minn=inf;
		for(int j=1;j<=n;j++)
		{
			if(dis[j]<minn&&book[j]==0)
			{
				u=j;
				minn=dis[j];
			}
		}
		book[u]=1;
		sum+=dis[u];
		
		for(int v=1;v<=n;v++)
		{
			if(!book[v]&&dis[v]>a[u][v])
			dis[v]=a[u][v];
		}
	}
	return sum;
}
int main()
{
	
	while(cin>>n,n)
	{
		memset(a,inf,sizeof(a));
		
		for(int i=1;i<=n;i++)
		{
			cin>>c[i];
		}
		
		for(int i=1;i<=n;i++)
		{
			for(int j=i+1;j<=n;j++)
			{
				t=0;
				for(int k=0;k<7;k++)
				{
					if(c[i][k]!=c[j][k])
					{
						t++;
					}
				}
				a[i][j]=a[j][i]=t;
			}
		}
		prim();
		printf("The highest possible quality is 1/%d.\n",sum);
		
	}
	
	return 0;
}