【HDU - 5873】Football Games（兰道定理，知识点总结）

题干：

A mysterious country will hold a football world championships---Abnormal Cup, attracting football teams and fans from all around the world. This country is so mysterious that none of the information of the games will be open to the public till the end of all the matches. And finally only the score of each team will be announced.
At the first phase of the championships, teams are divided into MM groups using the single round robin rule where one and only one game will be played between each pair of teams within each group. The winner of a game scores 2 points, the loser scores 0, when the game is tied both score 1 point. The schedule of these games are unknown, only the scores of each team in each group are available.
When those games finished, some insider revealed that there were some false scores in some groups. This has aroused great concern among the pubic, so the the Association of Credit Management (ACM) asks you to judge which groups' scores must be false.

Input

Multiple test cases, process till end of the input.
For each case, the first line contains a positive integers MM, which is the number of groups.
The ii-th of the next MM lines begins with a positive integer BiBi representing the number of teams in the ii-th group, followed by BiBi nonnegative integers representing the score of each team in this group.
number of test cases <= 10
M<= 100
B[i]<= 20000
score of each team <= 20000

Output

For each test case, output MM lines. Output ``F" (without quotes) if the scores in the i-th group must be false, output ``T" (without quotes) otherwise. See samples for detail.

Sample Input

2
3 0 5 1
2 1 1

Sample Output

F
T

题目大意：

每组样例占一行，先给定一个数n，紧接着有n个数。

有n支足球队，每两支队伍都要比一场，赢了得2分，输了得0分，平局双方各得1分。给定n个球队的得分情况，问是否合法。

解题报告：

首先来说明一下兰道定理

兰道定理又称竞赛图定理，是一个定义在有向图上的概念，顾名思义，它可以想象成n个人两两对决，赢得向输的连边，其实就是给一副完全图的无向边定了方向。

定义一个竞赛图的比分序列(score sequence)，是把竞赛图的每一个点的出度从小到大排列得到的序列。

一个长度为n的序列

是合法的比分序列当且仅当：且k==n的时候必须取等。

但是兰道定理是赢的一方得一分，输的一方不得分，似乎跟这道题不是很符合，但是你把它改为赢的一方得两分，输的一方不得分，平局各得一分，只需要把公式后面给C（2,K）乘2就好了。

AC代码：代码来自

#include <bits/stdc++.h>
using namespace std;
const int INF = 0x3f3f3f3f;
const int MAXN = 20010;
int a[MAXN];
int main() {int t;while(scanf("%d",&t) != EOF) {while(t--) {memset(a,0,sizeof(a));int n,i,sum = 0;bool flag = true;scanf("%d",&n);for(int i = 0; i < n; i++) {scanf("%d",&a[i]);sum += a[i];}if(sum != n*(n-1)) {flag = false;puts("F");continue;}sort(a,a+n);sum = 0;for(int i = 0; i < n-1; i++) {sum += a[i];if(sum >= (i + 1) * i) continue;else {flag = false;puts("F");break;}}if(flag) puts("T");}}return 0;
}

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