hdu 4779 Tower Defense （思维+组合数学）

Posted 2020-09-11 yutingliuyl

Tower Defense

Time Limit: 4000/2000 MS (Java/Others)    Memory Limit: 132768/132768 K (Java/Others)

Total Submission(s): 474    Accepted Submission(s): 126

Problem Description

　　DRD loves playing computer games, especially Tower Defense games. Tower Defense is a famous computer game with a number of variations. In general, you are to build some defense towers to guard your territory in this game.
　　However, in most Tower Defense games, your defending towers will not attack each other. You will see the shells flying through your towers and finally hit the target on your screen. DRD thinks it to be absurd, and he designed a new tower defense game.
　　In DRD’s game, you have two kinds of defending tower, heavy tower and light tower. You can put the tower on a grid with N rows and M columns and each cell in the grid can hold one tower at most. Both two kinds of towers can attack the cells in the same column or the same row as it is located in, and your towers may attack each other. Moreover, light towers should not be attacked by other towers while heavy towers can be attacked by at most one other tower.
　　You can put some of your towers (at least one tower) in the grid to build a tower formation satisfying the restriction above. And now, DRD wants you to calculate that how many different tower formations could be designed. Note that all the towers of the same type are considered to be identical. While the answer could be quite large, you should output the number mod (10⁹ + 7).

 

Input

　　There are multiple test cases in the input. The first line of the input file is an integer T demonstrating the number of test cases. (0< T<= 200).
　　For each test case, there is only one line containing 4 integers, N, M, P and Q ,meaning that the grid has N rows and M columns, and you have P heavy towers and Q light towers. You do not have to put all the towers in the grid. (1 <= N, M <= 200, 0 <= P, Q <= 200)

 

Output

　　For each test case, output the number of different formations mod (10⁹ + 7) in a single line.

 

Sample Input

3
2 2 0 1
2 2 2 0
3 3 2 1

 

Sample Output

4
10
144

 

Source

2013 Asia Hangzhou Regional Contest

思路来源于：点击打开链接

题意：

有两种塔，重塔，轻塔。

每种塔，能攻击他所在的一行和他所在的一列， 轻塔不 能被攻击。而重塔能够被至多一个塔攻击，也就是说重塔仅仅能被重塔攻击。

在一个n*m 的矩阵中，最少放一个塔，可放多个。
问。给定p个重塔，q个轻塔。问有多少种放法。

思路：

1、 一行中有两个重塔，
2、 一列中有两个重塔
3、 在该行及在该行塔所在的列仅仅有一个塔，重塔或者轻塔。

对以上三种情况挨个处理：
1、 设有i行有两个重塔，j列有两个重塔，则一共占 i+2*j 行， j+2*i列，共用2*（i+j）个重塔，，由于一行或一列两个塔
2、 对剩余的塔，进行枚举，0<-->剩余的塔。枚举这些塔中重塔的个数进行枚举

对于1： 在行中有两个重塔 c(n,i)*c(m,2*i)*((2*i)!/2^i)   意思 是在n行中选i行，在m列中选2*i列，  对于选出来的2*i列，分成i组。须要进行全排列，可是组内不须要进行全排列。所以为(2*i)!/2^i 在列中有两个重塔，c(m-2*i,j)*c(n-i,2*j)*((2*j)!/2^j)  原理同上

对于2:设有k个塔， 在剩余的n-(i+2*j) 行  m-(2*i+j) 列中 选 k个 点 （现有的行列组合*k!），k最大为  p-2*(i+j)+q，对于k个塔，则重塔最多有b =  min （k， p-2*(i+j) ） 个，  最少有a =  max（0，k-q） 个 。k个塔，最少 a 。最多b  则为重塔的取法有（c[k][0]+c[k][1]...+c[k][b]）- (c[k][0]+c[k][1]+...+c[k][a-1])；
最后将不放的情况减掉就可以，也就是减1；

代码：

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
#include <string>
#include <map>
#include <stack>
#include <vector>
#include <set>
#include <queue>
#define maxn 405
#define MAXN 200005
#define INF 0x3f3f3f3f
#define mod 1000000007
#define eps 1e-6
const double pi=acos(-1.0);
typedef long long ll;
using namespace std;

ll n,m,ans,cnt,p,q;
ll c[maxn][maxn],sum[maxn][maxn],fac[maxn],odd[maxn];

void presolve()
{
    ll i,j;
    for(i=0;i<=200;i++) c[i][0]=sum[i][0]=1;
    for(i=1;i<=200;i++)
    {
        for(j=1;j<=i;j++)
        {
            c[i][j]=c[i-1][j]+c[i-1][j-1];
            c[i][j]%=mod;
        }
    }
    for(i=0;i<=200;i++)
    {
        for(j=1;j<=200;j++)
        {
            sum[i][j]=sum[i][j-1]+c[i][j];
            sum[i][j]%=mod;
        }
    }
    fac[0]=odd[1]=1;
    for(i=1;i<=400;i++)
    {
        fac[i]=(fac[i-1]*i)%mod;
        if(i>1&&(i&1)) odd[i]=(odd[i-2]*i)%mod;
    }
}
void solve()
{
    ll i,j,t,k;
    ans=0;
    for(i=0; i<=n; i++)
    {
        for(j=0; j<=m; j++)
        {
            if(i+2*j>n||(j+2*i)>m||2*(i+j)>p) break ;
            if(2*i-1>0) t=odd[2*i-1];
            else t=1;
            ll res=(((((c[n][i]*c[m][2*i])%mod)*t)%mod)*fac[i])%mod;
            if(2*j-1>0) t=odd[2*j-1];
            else t=1;
            res=((((res*c[m-2*i][j])%mod*c[n-i][2*j])%mod*t)%mod*fac[j])%mod;

            ll tp=p-2*(i+j),tq=q,tn=n-(i+2*j),tm=m-(j+2*i);
            for(k=0;k<=tp+tq;k++)
            {
                if(k>tn||k>tm) break ;
                ll minp=max(0LL,k-tq),maxp=min(k,tp);
                if(minp==0) t=0;
                else t=sum[k][minp-1];
                ll tmp=((((c[tn][k]*c[tm][k])%mod*fac[k])%mod)*(sum[k][maxp]-t))%mod;
                ans=(ans+tmp*res)%mod;
            }
        }
    }
    ans=(ans-1+mod)%mod;
    printf("%lld\\n",ans);
}
int main()
{
    ll i,j,t;
    presolve();
    scanf("%lld",&t);
    while(t--)
    {
        scanf("%lld%lld%lld%lld",&n,&m,&p,&q);
        solve();
    }
    return 0;
}
/*
3
2 2 0 1
2 2 2 0
3 3 2 1
*/