Human Gene Functions【DP】

It is well known that a human gene can be considered as a sequence, consisting of four nucleotides, which are simply denoted by four letters, A, C, G, and T. Biologists have been interested in identifying human genes and determining their functions, because these can be used to diagnose human diseases and to design new drugs for them. 

A human gene can be identified through a series of time-consuming biological experiments, often with the help of computer programs. Once a sequence of a gene is obtained, the next job is to determine its function. 
One of the methods for biologists to use in determining the function of a new gene sequence that they have just identified is to search a database with the new gene as a query. The database to be searched stores many gene sequences and their functions – many researchers have been submitting their genes and functions to the database and the database is freely accessible through the Internet. 

A database search will return a list of gene sequences from the database that are similar to the query gene. 
Biologists assume that sequence similarity often implies functional similarity. So, the function of the new gene might be one of the functions that the genes from the list have. To exactly determine which one is the right one another series of biological experiments will be needed. 

Your job is to make a program that compares two genes and determines their similarity as explained below. Your program may be used as a part of the database search if you can provide an efficient one. 
Given two genes AGTGATG and GTTAG, how similar are they? One of the methods to measure the similarity 
of two genes is called alignment. In an alignment, spaces are inserted, if necessary, in appropriate positions of 
the genes to make them equally long and score the resulting genes according to a scoring matrix. 

For example, one space is inserted into AGTGATG to result in AGTGAT-G, and three spaces are inserted into GTTAG to result in –GT--TAG. A space is denoted by a minus sign (-). The two genes are now of equal 
length. These two strings are aligned: 

AGTGAT-G 
-GT--TAG 

In this alignment, there are four matches, namely, G in the second position, T in the third, T in the sixth, and G in the eighth. Each pair of aligned characters is assigned a score according to the following scoring matrix. 

denotes that a space-space match is not allowed. The score of the alignment above is (-3)+5+5+(-2)+(-3)+5+(-3)+5=9. 

Of course, many other alignments are possible. One is shown below (a different number of spaces are inserted into different positions): 

AGTGATG 
-GTTA-G 

This alignment gives a score of (-3)+5+5+(-2)+5+(-1) +5=14. So, this one is better than the previous one. As a matter of fact, this one is optimal since no other alignment can have a higher score. So, it is said that the 
similarity of the two genes is 14.

Input

The input consists of T test cases. The number of test cases ) (T is given in the first line of the input file. Each test case consists of two lines: each line contains an integer, the length of a gene, followed by a gene sequence. The length of each gene sequence is at least one and does not exceed 100.

Output

The output should print the similarity of each test case, one per line.

Sample Input

2 
7 AGTGATG 
5 GTTAG 
7 AGCTATT 
9 AGCTTTAAA 

Sample Output

14
21 

 

思路：

首先，打个表，记录点与点之间的匹配价值，之后就是推dp的状态转移方程：

•   dp[i][j]表示i匹配j的时候的情况的最大值；

• dp[i][j]可以从dp[i-1][j-1]和(i, j)直接匹配得来，即+dir[i][j]；

• dp[i][j]也可以由dp[i][j-1]（意味着此时的j得匹配空格了）+dir['_'][j]；

• dp[i][j]最后还能由dp[i-1][j]得到，与前面那个刚好相反， +dir[i]['_']。

 

完整代码：

（要用G++交哦，我这里的宏文件写了pi 和 e用了G++认可的写法、C++会CE）

#include <iostream>
#include <cstdio>
#include <cmath>
#include <string>
#include <cstring>
#include <algorithm>
#include <limits>
#include <vector>
#include <stack>
#include <queue>
#include <set>
#include <map>
#define lowbit(x) ( x&(-x) )
#define pi acos(-1.0)
#define e exp(1.0)
using namespace std;
typedef long long ll;
const int INF=1e9+7;
const int maxN=105;
int N, M, dp[maxN][maxN];       //第i位匹配下面的第j位
int dir[5][5]=
{
    5, -1, -2, -1, -3,
    -1, 5, -3, -2, -4,
    -2, -3, 5, -2, -2,
    -1, -2, -2, 5, -1,
    -3, -4, -2, -1, 0
};                      //得分表
int getID(char x)
{
    if(x=='A') return 0;
    else if(x=='C') return 1;
    else if(x=='G') return 2;
    else if(x=='T') return 3;
    else return 4;
}
char s1[maxN], s2[maxN];
int main()
{
    int T;
    scanf("%d", &T);
    while(T--)
    {
        scanf("%d", &N);    scanf("%s", s1+1);
        scanf("%d", &M);    scanf("%s", s2+1);
        if(N<M) { swap(N, M); swap(s1, s2); }
        memset(dp, 0, sizeof(dp));
        dp[1][0]=dir[getID(s1[1])][getID('_')];
        dp[0][1]=dir[getID('_')][getID(s2[1])];
        for(int i=2; i<=N; i++)
        {
            dp[i][0]=dp[i-1][0]+dir[getID(s1[i])][getID('_')];
        }
        for(int i=2; i<=M; i++)
        {
            dp[0][i]=dp[0][i-1]+dir[getID('_')][getID(s2[i])];
        }
        for(int i=1; i<=N; i++)
        {
            for(int j=1; j<=M; j++)
            {
                dp[i][j] = max(dp[i-1][j-1] + dir[getID(s1[i])][getID(s2[j])], max(dp[i][j-1] + dir[getID('_')][getID(s2[j])], dp[i-1][j] + dir[getID(s1[i])][getID('_')]));
            }
        }
        printf("%d\n", dp[N][M]);
    }
    return 0;
}