Problem Description
对于一个串S,当它同时满足如下条件时,它就是一个01偏串: 1、只由0和1两种符组成; 2、在S的每一个前缀中,0的个数不超过1的个数; 3、S中0的个数和1的个数相等。
现在给定01偏串S,请计算一下S在所有长度为n的01偏串中作为子串出现的次数的总和。 由于结果比较大,结果对1e9+7取余后输出。
样例解释: 在第二个样例中,长度为4的偏串共两个1010,1100。10在1010中出现了两次,在1100中出现了1次。所以答案是3。
Input
第一行给出一个整数T(1<=T<=40),表示测试数据的数目。 每一组测试包含一个整数n和字符串S,中间用空格分开。(1<=|S|<=100000,1<=n<=1000000000)
输入保证S是一个01偏串。
Output
对于每一组数据,输出一个整数占一行,表示答案。
Sample Input
2 2 10 4 10
Sample Output
1 3
解法:
1 我们求等于0的情况 n为奇数或者字符长度>n
2 我们暴力的算一算发现,这个和字符串长度有关,和具体内容无关
3 通过查询发现这是个ans=(n-len)/2
![](https://images2017.cnblogs.com/blog/843428/201708/843428-20170806230207959-1810509273.png)
但数据太大,1s算不完,只能化简看看啦
4 x=(n-len)/2+1
有
![](https://images2017.cnblogs.com/blog/843428/201708/843428-20170806231133803-391219992.png)
我们有个叫卡特兰数的东西带进去
5 我们只需要算f(x),他有一个
但从1算到1e9还是大了,我们就分段计算,每1e5个数字保存一次,然后每一次计算找到它最近的起点就行
1 #include<bits/stdc++.h> 2 using namespace std; 3 typedef long long ll; 4 //计算n的k次方对M取模,二分法 5 ll Pow(ll n, ll k, ll p){ 6 ll ans=1; 7 while(k){ 8 if(k&1) 9 ans=(ans*n)%p; 10 n = (n*n)%p; 11 k>>=1; //k=k>>1 k=k/2; 12 } 13 return ans; 14 } 15 ll C(ll n,ll m,ll p){ 16 if(m==0) return 1; 17 if(m>n-m) m=n-m; 18 ll up=1,down=1; 19 for(int i=1;i<=m;i++){ 20 up = (up*(n-i+1))%p; 21 down=(down*i)%p; 22 } 23 return up*Pow(down,p-2,p)%p; 24 } 25 ll lucas(ll n,ll m,ll p){ 26 if(m==0) return 1; 27 return C(n%p,m%p,p)*lucas(n/p,m/p,p); 28 } 29 #define mod 1000000007 30 int 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31 ll Pow(ll a, ll b) 32 { 33 ll sum = 1; 34 while(b) 35 { 36 if(b%2) 37 sum = sum*a%mod; 38 a = a*a%mod; 39 b /= 2; 40 } 41 return sum; 42 } 43 int main(){ 44 int t=1; 45 char s[123456]; 46 ll m,n,ans; 47 while(~scanf("%d",&t)){ 48 while(t--){ 49 scanf("%lld%s",&n,s); 50 m=strlen(s); 51 //cout<<m<<endl; 52 if(m>n||n&1){ 53 printf("0 "); 54 }else{ 55 ans=(n-m)/2; 56 if(ans<=4999999){ 57 printf("%lld ",lucas(2*ans+1,ans,1e9+7)); 58 }else{ 59 int i=1; 60 ans+=1; 61 for(i=1;i<=5000;i++){ 62 if(i*100000>ans){ 63 break; 64 } 65 } 66 i-=1; 67 // cout<<i<<endl; 68 ll now=ct[i]; 69 for(int j=i*100000+1;j<=ans;j++){ 70 // cout<<"B"<<endl; 71 now = 2*(2*j-1)*now%mod*Pow(j+1, mod-2)%mod; 72 } 73 ll x=now%mod*(((ans+1)%mod*500000004)%mod)%mod; 74 cout<<x<<endl; 75 } 76 } 77 } 78 } 79 return 0; 80 }