划分型DP

1.乘积最大（NOIP2000）
http://codevs.cn/problem/1017/

方程：

f[i][j] = max{f[k][j-1] + to_num(k+1~i)} (1 <= k < i)
to_num(low~high)表示将第low位到第high位转化成一个数字

代码：

#include<bits/stdc++.h>#define MAXN 50#define MAXK 10using namespace std;int n, k;char str[MAXN];long long f[MAXN][MAXK];long long to_num(int low, int high){    long long ret = 0;    for (int i = low; i <= high; i++)        ret = ret * 10 + (str[i - 1] - '0');    return ret;}int main(){    scanf("%d%d", &n, &k);    scanf("%s", str);    memset(f, 0, sizeof(f));    for (int i = 1; i <= n; i++)        f[i][0] = to_num(1, i);    for (int j = 1; j <= k; j++)        for (int i = 1; i <= n; i++)            for (int l = j; l < i; l++)                f[i][j] = max(f[i][j], f[l][j - 1] * to_num(l + 1, i));    printf("%d\n", f[n][k]);    return 0;}

2.数的划分（NOIP2001提高组）
http://codevs.cn/problem/1039/

方程：

f[i][j] = f[i - j][j] + f[i - 1][j - 1] (1 <= j <= min{i, k})

代码：

#include<bits/stdc++.h>#define MAXN 300#define MAXK 10using namespace std;int n, k;int f[MAXN][MAXK];int main(){    scanf("%d%d", &n, &k);    memset(f, 0, sizeof(f));    f[0][0] = 1;    for (int i = 1; i <= n; i++)        for (int j = 1; j <= min(i, k); j++)            f[i][j] = f[i - j][j] + f[i - 1][j - 1];    printf("%d", f[n][k]);    return 0;}

3.统计单词个数（NOIP2001提高组）
http://codevs.cn/problem/1040/

方程：

f[i][k]=max(f[i][k], f[j][k-1]+word_cnt(j+1~i)）
f[i][k]表示把前i个字母分成k个部分的最优解
word_cnt(low~high)表示从low到high位置的最多的单词数

PS：预处理很麻烦。

代码：

#include<bits/stdc++.h>#define MAX_LEN 300#define MAX_K 50#define MAX_S 10using namespace std;int n, p, k, s;char str[MAX_LEN];//字母串 char word[MAX_S][MAX_LEN];//字典 char temp[MAX_LEN];//临时字符串 int word_len[MAX_S];//word_len[i]-word[i]的长度 int strt[MAX_LEN];//temp[i]-以str[i]开头的单词长度 int word_cnt[MAX_LEN][MAX_LEN];//word_cnt[i][j]-str[i]~str[j]之间最多的单词数int f[MAX_LEN][MAX_K];//f[i][j]-到str[i]为止分成j份最多的单词数void get_substr(int start, int length){    for (int i = 0; i < MAX_LEN; i++)        temp[i] = '\0';    for (int i = 0; i < length; i++)        temp[i] = str[start + i];}void work(){    scanf("%d%d", &p, &k);    for (int i = 0; i < p; i++)        scanf("%s", &str[1 + i * 20]);    p *= 20;    scanf("%d", &s);    for (int i = 0; i < s; i++)    {        scanf("%s", word[i]);        word_len[i] = strlen(word[i]);    }    memset(strt, 0, sizeof(strt));    for (int i = 1; i <= p; i++)        for (int j = 0; j < s; j++)        {            get_substr(i, word_len[j]);            if ((strcmp(word[j], temp) == 0) && ((strt[i] == 0) || (strt[i] > word_len[j])))                strt[i] = word_len[j];        }    memset(word_cnt, 0, sizeof(word_cnt));    for (int i = 1; i <= p; i++)        for (int j = i; j <= p; j++)        {            word_cnt[i][j] = word_cnt[i][j - 1];            for (int l = i; l <= j; l++)                if (strt[l] == j - l + 1)                     word_cnt[i][j] += 1;        }    memset(f, 0, sizeof(f));    for (int j = 1; j <= k; j++)        for (int i = j; i <= p; i++)            for (int l = j - 1; l < i; l++)                f[i][j] = max(f[i][j], f[l][j - 1] + word_cnt[l + 1][i]);    printf("%d\n", f[p][k]);}int main(){    scanf("%d", &n);    for (int i = 0; i < n; i++)        work();     return 0;}