Ural 1158. Censored! 有限状态自动机+DP+大整数

Ural1158 看上去很困难的一道题。

原文地址 http://blog.csdn.net/prolightsfxjh/article/details/54729646

题意：给出n个不同的字符，用这n个字符构成长度为m的字符串，要求每个串的子串都不出现给定的p个串中的任一个，求满足要求的字符串的个数。

 

AC自动机+dp

因为构成的最终串是由一个字符一个字符添加到字符串尾部构成的，那么如果一个串的后缀如果恰好是某个给定串的前缀时，这个串就可能最终成为非法串。

用k个给定串建立AC自动机，然后从根节点开始递推，

dpij表示递推到第j个字符当前在自动机上的i号节点时的方案数，如果下一个节点是k，且不是危险节点，则把dpij加到dp[k][i+1]里，

跑一遍，然后答案就是所有非危险节点的方案数的和(其实危险节点上都是0)。因为危险节点是给定串的终点或者其后缀节点是危险节点的点，遍历到危险节点的点上的方案必定是包含了给定串的方案，故不能记录这些。

此外这里dpij会很多，故要用高精度整数。笔者自己收藏的大整数类的版，可行长太长了，所以MLE了一发，调小了才过。⊙﹏⊙‖∣

复杂度 O(n^3)

 

模版串长度很小，所以Trie上节点很少，递推成本也就比较低。

 

其实这道题就是三个部分， 实现了AC自动机和大整数， 就是一道DP题了，分开来看 理解了KMP算法和Trie树 AC自动机很简单， 大整数朴素的O（n^2）实现也很简单，DP方程也很简单，但是组合起来，就很困难了。

诶 东北欧赛区的题，总是这么~

代码如下 洋洋洒洒12k 当然不是我写的， 不过这份代码非常清晰易懂，反正智障的我看的挺明白。

 

#include <iostream>  
#include <cstdio>  
#include <cstring>  
#include <queue>  
#include <map>  
using namespace std;  
const int CHAR_SIZE = 51;  
const int MAX_SIZE = 105;  
map<char, int> mp;  
  
struct AC_Machine{  
    int ch[MAX_SIZE][CHAR_SIZE], danger[MAX_SIZE], fail[MAX_SIZE];  
    int sz;  
  
    void init(){  
        sz = 1;  
        memset(ch[0], 0, sizeof ch[0]);  
        memset(danger, 0, sizeof danger);  
    }  
  
    void _insert(char *s){  
        int n = strlen(s);  
        int u = 0, c;  
        for(int i = 0; i < n; i++){  
            c = mp[s[i]];  
            if(!ch[u][c]){  
                memset(ch[sz], 0, sizeof ch[sz]);  
                danger[sz] = 0;  
                ch[u][c] = sz++;  
            }  
            u = ch[u][c];  
        }  
        danger[u] = 1;  
    }  
  
    void _build(){  
        queue<int> Q;  
        fail[0] = 0;  
        for(int c = 0, u; c < CHAR_SIZE; c++){  
            u = ch[0][c];  
            if(u){Q.push(u); fail[u] = 0;}  
        }  
        int r;  
        while(!Q.empty()){  
            r = Q.front();  
            Q.pop();  
            danger[r] |= danger[fail[r]];  
            for(int c = 0, u; c < CHAR_SIZE; c++){  
                u = ch[r][c];  
                if(!u){ch[r][c] = ch[fail[r]][c]; continue; }  
                fail[u] = ch[fail[r]][c];  
                Q.push(u);  
            }  
        }  
    }  
}ac;  
  
char s[MAX_SIZE];  
  
#include <string>  
#include <iostream>  
#include <iosfwd>  
#include <cmath>  
#include <cstring>  
#include <stdlib.h>  
#include <stdio.h>  
#include <cstring>  
#define MAX_L 205 //最大长度，可以修改  
using namespace std;  
  
class bign  
{  
public:  
    int len, s[MAX_L];//数的长度，记录数组  
//构造函数  
    bign();  
    bign(const char*);  
    bign(int);  
    bool sign;//符号 1正数 0负数  
    string toStr() const;//转化为字符串，主要是便于输出  
    friend istream& operator>>(istream &,bign &);//重载输入流  
    friend ostream& operator<<(ostream &,bign &);//重载输出流  
//重载复制  
    bign operator=(const char*);  
    bign operator=(int);  
    bign operator=(const string);  
//重载各种比较  
    bool operator>(const bign &) const;  
    bool operator>=(const bign &) const;  
    bool operator<(const bign &) const;  
    bool operator<=(const bign &) const;  
    bool operator==(const bign &) const;  
    bool operator!=(const bign &) const;  
//重载四则运算  
    bign operator+(const bign &) const;  
    bign operator++();  
    bign operator++(int);  
    bign operator+=(const bign&);  
    bign operator-(const bign &) const;  
    bign operator--();  
    bign operator--(int);  
    bign operator-=(const bign&);  
    bign operator*(const bign &)const;  
    bign operator*(const int num)const;  
    bign operator*=(const bign&);  
    bign operator/(const bign&)const;  
    bign operator/=(const bign&);  
//四则运算的衍生运算  
    bign operator%(const bign&)const;//取模（余数）  
    bign factorial()const;//阶乘  
    bign Sqrt()const;//整数开根（向下取整）  
    bign pow(const bign&)const;//次方  
//一些乱乱的函数  
    void clean();  
    ~bign();  
};  
#define max(a,b) a>b ? a : b  
#define min(a,b) a<b ? a : b  
  
bign::bign()  
{  
    memset(s, 0, sizeof(s));  
    len = 1;  
    sign = 1;  
}  
  
bign::bign(const char *num)  
{  
    *this = num;  
}  
  
bign::bign(int num)  
{  
    *this = num;  
}  
  
string bign::toStr() const  
{  
    string res;  
    res = "";  
    for (int i = 0; i < len; i++)  
        res = (char)(s[i] + '0') + res;  
    if (res == "")  
        res = "0";  
    if (!sign&&res != "0")  
        res = "-" + res;  
    return res;  
}  
  
istream &operator>>(istream &in, bign &num)  
{  
    string str;  
    in>>str;  
    num=str;  
    return in;  
}  
  
ostream &operator<<(ostream &out, bign &num)  
{  
    out<<num.toStr();  
    return out;  
}  
  
bign bign::operator=(const char *num)  
{  
    memset(s, 0, sizeof(s));  
    char a[MAX_L] = "";  
    if (num[0] != '-')  
        strcpy(a, num);  
    else  
        for (int i = 1; i < strlen(num); i++)  
            a[i - 1] = num[i];  
    sign = !(num[0] == '-');  
    len = strlen(a);  
    for (int i = 0; i < strlen(a); i++)  
        s[i] = a[len - i - 1] - 48;  
    return *this;  
}  
  
bign bign::operator=(int num)  
{  
    char temp[MAX_L];  
    sprintf(temp, "%d", num);  
    *this = temp;  
    return *this;  
}  
  
bign bign::operator=(const string num)  
{  
    const char *tmp;  
    tmp = num.c_str();  
    *this = tmp;  
    return *this;  
}  
  
bool bign::operator<(const bign &num) const  
{  
    if (sign^num.sign)  
        return num.sign;  
    if (len != num.len)  
        return len < num.len;  
    for (int i = len - 1; i >= 0; i--)  
        if (s[i] != num.s[i])  
            return sign ? (s[i] < num.s[i]) : (!(s[i] < num.s[i]));  
    return !sign;  
}  
  
bool bign::operator>(const bign&num)const  
{  
    return num < *this;  
}  
  
bool bign::operator<=(const bign&num)const  
{  
    return !(*this>num);  
}  
  
bool bign::operator>=(const bign&num)const  
{  
    return !(*this<num);  
}  
  
bool bign::operator!=(const bign&num)const  
{  
    return *this > num || *this < num;  
}  
  
bool bign::operator==(const bign&num)const  
{  
    return !(num != *this);  
}  
  
bign bign::operator+(const bign &num) const  
{  
    if (sign^num.sign)  
    {  
        bign tmp = sign ? num : *this;  
        tmp.sign = 1;  
        return sign ? *this - tmp : num - tmp;  
    }  
    bign result;  
    result.len = 0;  
    int temp = 0;  
    for (int i = 0; temp || i < (max(len, num.len)); i++)  
    {  
        int t = s[i] + num.s[i] + temp;  
        result.s[result.len++] = t % 10;  
        temp = t / 10;  
    }  
    result.sign = sign;  
    return result;  
}  
  
bign bign::operator++()  
{  
    *this = *this + 1;  
    return *this;  
}  
  
bign bign::operator++(int)  
{  
    bign old = *this;  
    ++(*this);  
    return old;  
}  
  
bign bign::operator+=(const bign &num)  
{  
    *this = *this + num;  
    return *this;  
}  
  
bign bign::operator-(const bign &num) const  
{  
    bign b=num,a=*this;  
    if (!num.sign && !sign)  
    {  
        b.sign=1;  
        a.sign=1;  
        return b-a;  
    }  
    if (!b.sign)  
    {  
        b.sign=1;  
        return a+b;  
    }  
    if (!a.sign)  
    {  
        a.sign=1;  
        b=bign(0)-(a+b);  
        return b;  
    }  
    if (a<b)  
    {  
        bign c=(b-a);  
        c.sign=false;  
        return c;  
    }  
    bign result;  
    result.len = 0;  
    for (int i = 0, g = 0; i < a.len; i++)  
    {  
        int x = a.s[i] - g;  
        if (i < b.len) x -= b.s[i];  
        if (x >= 0) g = 0;  
        else  
        {  
            g = 1;  
            x += 10;  
        }  
        result.s[result.len++] = x;  
    }  
    result.clean();  
    return result;  
}  
  
bign bign::operator * (const bign &num)const  
{  
    bign result;  
    result.len = len + num.len;  
  
    for (int i = 0; i < len; i++)  
        for (int j = 0; j < num.len; j++)  
            result.s[i + j] += s[i] * num.s[j];  
  
    for (int i = 0; i < result.len; i++)  
    {  
        result.s[i + 1] += result.s[i] / 10;  
        result.s[i] %= 10;  
    }  
    result.clean();  
    result.sign = !(sign^num.sign);  
    return result;  
}  
  
bign bign::operator*(const int num)const  
{  
    bign x = num;  
    bign z = *this;  
    return x*z;  
}  
bign bign::operator*=(const bign&num)  
{  
    *this = *this * num;  
    return *this;  
}  
  
bign bign::operator /(const bign&num)const  
{  
    bign ans;  
    ans.len = len - num.len + 1;  
    if (ans.len < 0)  
    {  
        ans.len = 1;  
        return ans;  
    }  
  
    bign divisor = *this, divid = num;  
    divisor.sign = divid.sign = 1;  
    int k = ans.len - 1;  
    int j = len - 1;  
    while (k >= 0)  
    {  
        while (divisor.s[j] == 0) j--;  
        if (k > j) k = j;  
        char z[MAX_L];  
        memset(z, 0, sizeof(z));  
        for (int i = j; i >= k; i--)  
            z[j - i] = divisor.s[i] + '0';  
        bign dividend = z;  
        if (dividend < divid) { k--; continue; }  
        int key = 0;  
        while (divid*key <= dividend) key++;  
        key--;  
        ans.s[k] = key;  
        bign temp = divid*key;  
        for (int i = 0; i < k; i++)  
            temp = temp * 10;  
        divisor = divisor - temp;  
        k--;  
    }  
    ans.clean();  
    ans.sign = !(sign^num.sign);  
    return ans;  
}  
  
bign bign::operator/=(const bign&num)  
{  
    *this = *this / num;  
    return *this;  
}  
  
bign bign::operator%(const bign& num)const  
{  
    bign a = *this, b = num;  
    a.sign = b.sign = 1;  
    bign result, temp = a / b*b;  
    result = a - temp;  
    result.sign = sign;  
    return result;  
}  
  
bign bign::pow(const bign& num)const  
{  
    bign result = 1;  
    for (bign i = 0; i < num; i++)  
        result = result*(*this);  
    return result;  
}  
  
bign bign::factorial()const  
{  
    bign result = 1;  
    for (bign i = 1; i <= *this; i++)  
        result *= i;  
    return result;  
}  
  
void bign::clean()  
{  
    if (len == 0) len++;  
    while (len > 1 && s[len - 1] == '')  
        len--;  
}  
  
bign bign::Sqrt()const  
{  
    if(*this<0)return -1;  
    if(*this<=1)return *this;  
    bign l=0,r=*this,mid;  
    while(r-l>1)  
    {  
        mid=(l+r)/2;  
        if(mid*mid>*this)  
            r=mid;  
        else  
            l=mid;  
    }  
    return l;  
}  
  
bign::~bign()  
{  
}  
  
bign dp[MAX_SIZE][CHAR_MAX];  
  
int main()  
{  freopen("t.txt", "r", stdin);  
    #ifdef LOCAL  
    
    //freopen("1.out", "w", stdout);  
    int T = 1;  
    while(T--){  
    #endif // LOCAL  
    //ios::sync_with_stdio(false); cin.tie(0);  
  
    int n, m, p;  
    scanf("%d%d%d", &n, &m, &p);  
    scanf("%s", s);  
    for(int i = 0; i < n; i++){mp[s[i]] = i;}  
    ac.init();  
    while(p--){  
        scanf("%s", s);  
        ac._insert(s);  
    }  
    int i, j, k;  
    for(i = 0; i < ac.sz; i++){  
        for(j = 0; j <= m; j++){  
            dp[i][j] = 0;  
        }  
    }  
    ac._build();  
    dp[0][0] = 1;  
    for(i = 1; i <= m; i++){  
        for(j = 0; j < ac.sz; j++){  
            for(k = 0; k < n; k++){  
                if(!ac.danger[ac.ch[j][k]]){  
                    dp[ac.ch[j][k]][i] += dp[j][i-1];  
                }  
            }  
        }  
    }  
    bign ans = 0;  
    for(i = 0; i < ac.sz; i++){ans += dp[i][m];}  
    cout << ans << endl;  
  
    #ifdef LOCAL  
    cout << endl;  
    }  
    #endif // LOCAL  
    return 0;  
}