D. Fafa and Ancient Alphabet【求概率+分数取模】

Ancient Egyptians are known to have used a large set of symbols to write on the walls of the temples. Fafa and Fifa went to one of the temples and found two non-empty words S1 and S2 of equal lengths on the wall of temple written one below the other. Since this temple is very ancient, some symbols from the words were erased. The symbols in the set have equal probability for being in the position of any erased symbol.

Fifa challenged Fafa to calculate the probability that S1 is lexicographically greater than S2. Can you help Fafa with this task?

You know that , i. e. there were m distinct characters in Egyptians' alphabet, in this problem these characters are denoted by integers from 1 to m in alphabet order. A word x is lexicographically greater than a word y of the same length, if the words are same up to some position, and then the word x has a larger character, than the word y.

We can prove that the probability equals to some fraction , where P and Q are coprime integers, and . Print as the answer the value , i. e. such a non-negative integer less than 109 + 7, such that , where means that a and b give the same remainders when divided by m.

Input

The first line contains two integers n and m (1 ≤ n, m ≤ 105) — the length of each of the two words and the size of the alphabet , respectively.

The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ m) — the symbols of S1. If ai = 0, then the symbol at position i was erased.

The third line contains n integers representing S2 with the same format as S1.

Output

Print the value , where P and Q are coprime and is the answer to the problem.

         Examples       
           input                       Copy          
1 2
0
1
           output         
500000004
           input                       Copy          
1 2
1
0
           output         
0
           input                       Copy          
7 26
0 15 12 9 13 0 14
11 1 0 13 15 12 0
           output         
230769233

Note

In the first sample, the first word can be converted into (1) or (2). The second option is the only one that will make it lexicographically larger than the second word. So, the answer to the problem will be , that is 500000004, because .

In the second example, there is no replacement for the zero in the second word that will make the first one lexicographically larger. So, the answer to the problem is , that is 0.

题意：两个数字串s1,s2，每个串含n个数字。如果数字为0，那么该数字可以变为1~m中的任意一个。问P（s1字典序>s2字典序）？并对P取模qe9+7;

思路：重要一点，只有前面的数字都相等，才讨论当前的数字情况有意义。ans=∑P*(Q-1)%MOD*qm(m,1e9+5)。

再掌握对分数取模运算以及计算逆元的方法既可以。由费马小定理，有a^(p-1)==1(mod p) → a^(p-2)mod p==inv(a) ，i.e. a对p的逆元= a^(p-2)%p;

复杂度为O(n*sqrt(qe9+7))

AC代码：

#include<bits/stdc++.h>
#define FAST ios::sync_with_stdio(false);cin.tie(0);
using namespace std;
typedef long long ll;
const int MAX_N=1e5+50;
const int MOD=1e9+7;

int a[MAX_N];
int b[MAX_N];

ll qm(ll a,ll b){
    ll ans=1;
    while(b){
        if(b&1) ans=(ans*a)%MOD;
        a=(a*a)%MOD;
        b>>=1;
    }
    return ans%MOD;
}

void solve(){
    ll n,m,p=1;//p：前i-1个都相同的概率，因为只有前面都相同的情况下，比较当前的才有意义
    ll ans=0;
    cin >> n>>m;
    for(int i=1;i<=n;++i)   cin >> a[i];
    for(int i=1;i<=n;++i)   cin >> b[i];
    for(int i=1;i<=n;i++){
        if(a[i]&&b[i]){
            if(a[i]>b[i])   {ans+=p,ans%=MOD;break;}
            else if(a[i]<b[i])  break;
        }
        else if(!a[i]&&b[i]){
            ans+=p*(m-b[i])%MOD*qm(m,MOD-2); //取模的合理性在于(a+b)%mod=(a+b%mod)%mod，仅仅可以一个取模
            ans%=MOD;
            p*=qm(m,MOD-2);
            p%=MOD;
        }
        else if(a[i]&&!b[i]){
            ans+=p*(a[i]-1)%MOD*qm(m,MOD-2);
            ans%=MOD;
            p*=qm(m,MOD-2);
            p%=MOD;
        }
        else if(!a[i]&&!b[i]){
            ans+=p*(m-1)%MOD*qm(2*m,MOD-2); 这里是化简后的结果
            ans%=MOD;
            p*=qm(m,MOD-2);
            p%=MOD;
        }
    }
    cout << ans%MOD <<endl;
}

int main(void){
    FAST;
    solve();
    return 0;
}