查询给定范围内数组乘积的除数计数 |设置 2（MO 的算法）

给定大小为N的数组arr和形式为[L, R] 的Q查询，任务是在给定范围内找到该数组的乘积的除数数。

先决条件： MO 算法、模乘逆、使用筛法的质因数分解

例子：

Input: arr[] = {4, 1, 9, 12, 5, 3}, Q = {{1, 3}, {3, 5}}
Output:
9
24

Input: arr[] = {5, 2, 3, 1, 4}, Q = {{2, 4}, {1, 5}}
Output:
4
16

方法：

MO 算法的思想是对所有查询进行预处理，以便一个查询的结果可以在下一个查询中使用。
让a[0…n-1]是输入数组， q[0..m-1]是一个查询数组。

1. 对所有查询进行排序，将 L 值从0到√n – 1的查询放在一起，然后将所有查询从√n到2×√n – 1 ，依此类推。块内的所有查询都按 R 值的升序排序。
2. 以每个查询都使用在前一个查询中计算出的结果的方式一个一个地处理所有查询。让 ‘result’ 为上一个查询的结果
3. 一个数 n 可以表示为n = $\prod_{i=1}^{n} a_{i}^{p_{i}}$ ，其中a _i是质因数， _{pi i}是它们的整数幂。
   因此，对于这种分解，我们有公式来找到 n 的除数总数，即：

   $\prod_{i=1}^{n} (p_{i}+1)$

4. 在Add函数，我们增加计数器数组，即counter[a[i]]=counter[a[i]]+ p _i 。让 ‘prev’ 存储 counter[a[i]] 的先前值。现在随着计数器数组的变化，结果更改为：
   • result = result / (prev + 1) (Dividing by prev+1 neutralizes the effect of previous p_i)
   • result = (result × (counter[p_i] + 1) (Now the previous result is neutralized so we multiply with the new count i.e counter[a[i]]+1)
5. 在Remove函数，我们将计数器数组递减为counter[a[i]] = counter[a[i]] – p _i 。现在随着计数器数组的变化，结果更改为：
   • result = result / (prev + 1) (Dividing by prev+1 neutralizes the effect of previous p_i)
   • result = (result × (counter[p_i] + 1) (Now the previous result is neutralized so we
     multiply with the new count i.e counter[a[i]]+1)

下面是上述方法的实现

// C++ program to Count the divisors 
// of product of an Array in range 
// L to R for Q queries
#include 
using namespace std;
  
#define MAX 1000000
#define MOD 1000000007
#define ll long long int
  
// Variable to represent block size.
// This is made global so compare()
// of sort can use it.
int block;
  
// Structure to represent a query range
struct Query {
    int L, R;
};
  
// Store the prime factor of numbers
// till MAX
vector > store[MAX + 1];
  
// Initialized to store the count 
// of prime fators
int counter[MAX + 1] = {};
  
// Result is Initialized to 1
int result = 1;
  
// Inverse array to store
// inverse of number from 1 to MAX
ll inverse[MAX + 1];
  
// Function used to sort all queries so that 
// all queries of the same block are arranged 
// together and within a block, queries are
// sorted in increasing order of R values.
bool compare(Query x, Query y)
{
    // Different blocks, sort by block.
    if (x.L / block != y.L / block)
        return x.L / block < y.L / block;
  
    // Same block, sort by R value
    return x.R < y.R;
}
  
// Function to calculate modular
// inverse and storing it in Inverse array
void modularInverse()
{
  
    inverse[0] = inverse[1] = 1;
    for (int i = 2; i <= MAX; i++)
        inverse[i] = inverse[MOD % i]
                    * (MOD - MOD / i) 
                      % MOD;
}
  
// Function to use Sieve to compute
// and store prime numbers
void sieve()
{
  
    store[1].push_back({ 1, 0 });
    for (int i = 2; i <= MAX; i++)
    {
        if (store[i].size() == 0)
        {
            store[i].push_back({ i, 1 });
              
            for (int j = 2 * i; j <= MAX; j += i)
            {
                int cnt = 0;
                int x = j;
                while (x % i == 0)
                    cnt++, x /= i;
                store[j].push_back({ i, cnt });
            }
        }
    }
}
  
// Function to Add elements
// of current range
void add(int currL, int a[])
{
    int value = a[currL];
    for (auto it = store[value].begin();
         it != store[value].end(); it++) {
        // it->first is ai
        // it->second is its integral power
        int prev = counter[it->first];
        counter[it->first] += it->second;
        result = (result * inverse[prev + 1])
                 % MOD;
          
        result = (result *
                  (counter[it->first] + 1))
                  % MOD;
    }
}
  
// Function to remove elements
// of previous range
void remove(int currR, int a[])
{
    int value = a[currR];
    for (auto it = store[value].begin(); 
         it != store[value].end(); it++) {
        // it->first is ai
        // it->second is its integral power
        int prev = counter[it->first];
        counter[it->first] -= it->second;
        result = (result * inverse[prev + 1])
                  % MOD;
        result = (result *
                  (counter[it->first] + 1)) 
                  % MOD;
    }
}
  
// Function to print the answer.
void queryResults(int a[], int n, Query q[],
                  int m)
{
    // Find block size
    block = (int)sqrt(n);
  
    // Sort all queries so that queries of 
    // same blocks are arranged together.
    sort(q, q + m, compare);
  
    // Initialize current L, current R and
    // current result
    int currL = 0, currR = 0;
  
    for (int i = 0; i < m; i++) {
        // L and R values of current range
        int L = q[i].L, R = q[i].R;
  
        // Add Elements of current range
        while (currR <= R) {
            add(currR, a);
            currR++;
        }
        while (currL > L) {
            add(currL - 1, a);
            currL--;
        }
  
        // Remove element of previous range
        while (currR > R + 1)
  
        {
            remove(currR - 1, a);
            currR--;
        }
        while (currL < L) {
            remove(currL, a);
            currL++;
        }
  
        cout << result << endl;
    }
}
  
// Driver Code
int main()
{
    // Precomputing the prime numbers 
    // using sieve
    sieve();
  
    // Precomputing modular inverse of 
    // numbers from 1 to MAX
    modularInverse();
  
    int a[] = { 5, 2, 3, 1, 4 };
    int n = sizeof(a) / sizeof(a[0]);
      
    Query q[] = { { 1, 3 }, { 0, 4 } };
      
    int m = sizeof(q) / sizeof(q[0]);
      
    // Answer the queries
    queryResults(a, n, q, m);
    return 0;
}

输出：

4
16

时间复杂度： O(Q×sqrt(N))

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