ZOJ 3940 Modulo Query（YY+二分）

Modulo Query

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Time Limit: 2 Seconds      Memory Limit: 65536 KB

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One day, Peter came across a function which looks like:

• F(1, X) = X mod A₁.
• F(i, X) = F(i - 1, X) mod A_i, 2 ≤ i ≤ N.

Where A is an integer array of length N, X is a non-negative integer no greater than M.

Peter wants to know the number of solutions for equation F(N, X) = Y, where Y is a given number.

Input

There are multiple test cases. The first line of input contains an integer T, indicating the number of test cases. For each test case:

The first line contains two integers N and M (2 ≤ N ≤ 10⁵, 0 ≤ M ≤ 10⁹).

The second line contains N integers: A₁, A₂, ..., A_N (1 ≤ A_i ≤ 10⁹).

The third line contains an integer Q (1 ≤ Q ≤ 10⁵) - the number of queries. Each of the following Q lines contains an integer Y_i (0 ≤ Y_i ≤ 10⁹), which means Peter wants to know the number of solutions for equation F(N, X) = Y_i.

Output

For each test cases, output an integer S = (1 ⋅ Z₁ + 2 ⋅ Z₂ + ... + Q ⋅ Z_Q) mod (10⁹ + 7), where Z_i is the answer for the i-th query.

Sample Input

1
3 5
3 2 4
5
0
1
2
3
4

Sample Output

8

Hint

The answer for each query is: 4, 2, 0, 0, 0.

题目链接：ZOJ 3940

题意：给出N个数Ai和M，又给Q个询问，每一个询问都是求[0,M]中求是否存在X使得X%A1%A2%A3%......%An=Yi，输出符合有几个这种整数X。

可以发现任何情况下对连续的数取模除非当前值比上一个取模值小，否则直接跳过即可，当然最重要的不是这里，而是如何把题目转换一下，每一次问[0,M]中符合题意的X个数，那么我们可以把[0,M]区间分割为无数个被取模后的小区间，若计这些小区间的贡献均为1，则覆盖在点Yi的情况就是询问Yi的答案，怎么分割呢，当然是用题目给的A数组分割，顺序地输入数组，这里就可以用到上面讲到的取模的技巧来减少分割的次数，分割之后做一遍前缀或后缀和（比如小区间0-1与0-2，0-2显然是包括0-1的，因此是前缀或后缀和关系，两种不同的和只会影响统计时候的加减法问题不会影响答案）。然后询问的时候二分到第一个大于Yi的区间，假设你输入的是3，二分出来的位置是pos，pos对应的子区间为0-4，答案就是从pos~end的贡献和。因为在pos之前只会产生小于3的数，不可能出现3的情况，只有从至少%4开始，才会出现3

代码：

#include <bits/stdc++.h>
using namespace std;
#define INF 0x3f3f3f3f
typedef pair<int, int> pii;
typedef long long LL;
const int N = 1e5 + 7;
const LL MOD = 1e9 + 7;
int A[N];
map<int, int>pos;
pii prefix[N << 2];

inline int getpos(int l, int r, int key)
{
    int ans = -1;
    while (l <= r)
    {
        int mid = (l + r) >> 1;
        if (prefix[mid].first > key)
        {
            ans = mid;
            r = mid - 1;
        }
        else
            l = mid + 1;
    }
    return ans;
}
int main(void)
{
    int tcase, n, m, i, q;
    scanf("%d", &tcase);
    while (tcase--)
    {
        int Min = INF;
        pos.clear();
        scanf("%d%d", &n, &m);
        pos[m + 1] = 1;
        prefix[0] = {0, 0};
        for (i = 1; i <= n; ++i)
        {
            scanf("%d", &A[i]);
            if (A[i] < Min)
                Min = A[i];
            while (1)
            {
                auto it = pos.upper_bound(A[i]);
                if (it == pos.end())
                    break;
                int olde = it->first;
                int oldcnt = it->second;
                pos[A[i]] += olde / A[i] * oldcnt;
                if (olde % A[i] != 0)
                    pos[olde % A[i]] += oldcnt;
                pos.erase(it);
            }
        }
        auto it = pos.begin();
        int sz = 0;
        while (it != pos.end())
        {
            ++sz;
            prefix[sz].second = prefix[sz - 1].second + it->second;
            prefix[sz].first = it->first;
            ++it;
        }
        scanf("%d", &q);
        LL ans = 0LL;
        for (i = 1; i <= q; ++i)
        {
            LL curans;
            int x;
            scanf("%d", &x);
            if (x >= Min)
                curans = 0LL;
            else
            {
                int l = getpos(1, sz, x);
                if (~l)
                    curans = prefix[sz].second - prefix[l - 1].second;
                else
                    curans = 0LL;
            }
            if (curans)
            {
                ans = ans + (LL)i * curans % MOD;
                if (ans > MOD)
                    ans %= MOD;
            }
        }
        printf("%lld\n", ans);
    }
    return 0;
}