Queries for elements having values within the range A to B using MO’s Algorithm

Prerequisites: MO’s algorithm, SQRT Decomposition

Given an array arr[] of N elements and two integers A to B, the task is to answer Q queries each having two integers L and R. For each query, find the number of elements in the subarray arr[L, R] which lies within the range A to B (inclusive).

Examples:

Input: arr[] = {3, 4, 6, 2, 7, 1}, A = 1, B = 6, query = {0, 4}
Output: 4
Explanation:
All 3, 4, 6, 2 lies within 1 to 6 in the subarray {3, 4, 6, 2}
Therefore, the count of such elements is 4.

Input: arr[] = {0, 1, 2, 3, 4, 5, 6, 7}, A = 1, B = 5, query = {3, 5}
Output: 3
Explanation:
All the elements 3, 4 and 5 lies within the range 1 to 5 in the subarray {3, 4, 5}.
Therefore, the count of such elements is 3.



Approach: The idea is to use MO’s algorithm to pre-process all queries so that result of one query can be used in the next query. Below is the illustration of the steps:

  1. Group the queries into mutiple chunks where each chunk contains the values of starting range in (0 to √N – 1), (√N to 2x√N – 1) and so on. Sort the queries within a chunk in incresing order of R.
  2. Process all queries one by one in a way that every query uses result computed in the previous query.
  3. Maintain the frequency array that will count the frequency of arr[i] as they appear in the range [L, R].

For example:

arr[] = [3, 4, 6, 2, 7, 1], L = 0, R = 4 and A = 1, B = 6

Initially frequency array is initialized to 0 i.e freq[]=[0….0]
Step 1: Add arr[0] and increment its frequency as freq[arr[0]]++
i.e freq[3]++ and freq[]=[0, 0, 0, 1, 0, 0, 0, 0]

Step 2: Add arr[1] and increment freq[arr[1]]++
i.e freq[4]++ and freq[]=[0, 0, 0, 1, 1, 0, 0, 0]

Step 3: Add arr[2] and increment freq[arr[2]]++
i.e freq[6]++ and freq[]=[0, 0, 0, 1, 1, 0, 1, 0]

Step 4: Add arr[3] and increment freq[arr[3]]++
i.e freq[2]++ and freq[]=[0, 0, 1, 1, 1, 0, 1, 0]

Step 5: Add arr[4] and increment freq[arr[4]]++
i.e freq[7]++ and freq[]=[0, 0, 1, 1, 1, 0, 1, 1]



Step 6: Now we need to find the numbers of elements between A and B.

Step 7: The answer is equal to  \sum_{i=A}^B freq[i]

To calculate the sum in Step 7, we cannot do iteration because that would lead to O(N) time complexity per query. So we will use square root decomposition technique to find the sum, whose time complexity is O(√N) per query.

Below is the implementation of the above approach:

C++

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// C++ implementation to find the
// values in the range A to B
// in a subarray of L to R
  
#include <bits/stdc++.h>
using namespace std;
  
#define MAX 100001
#define SQRSIZE 400
  
// Variable to represent block size.
// This is made global so compare()
// of sort can use it.
int query_blk_sz;
  
// Structure to represent a query range
struct Query {
    int L;
    int R;
};
  
// Frequency array
// to keep count of elements
int frequency[MAX];
  
// Array which contains the frequency
// of a particular block
int blocks[SQRSIZE];
  
// Block size
int blk_sz;
  
// 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)
{
    if (x.L / query_blk_sz
        != y.L / query_blk_sz)
        return (x.L / query_blk_sz
                < y.L / query_blk_sz);
  
    return x.R < y.R;
}
  
// Function used to get the block
// number of current a[i] i.e ind
int getblocknumber(int ind)
{
    return (ind) / blk_sz;
}
  
// Function to get the answer
// of range [0, k] which uses the
// sqrt decompostion technique
int getans(int A, int B)
{
    int ans = 0;
    int left_blk, right_blk;
    left_blk = getblocknumber(A);
    right_blk = getblocknumber(B);
  
    // If left block is equal to rigth block
    // then we can traverse that block
    if (left_blk == right_blk) {
        for (int i = A; i <= B; i++)
            ans += frequency[i];
    }
    else {
        // Traversing first block in
        // range
        for (int i = A;
             i < (left_blk + 1) * blk_sz;
             i++)
            ans += frequency[i];
  
        // Traversing completely overlapped
        // blocks in range
        for (int i = left_blk + 1;
             i < right_blk; i++)
            ans += blocks[i];
  
        // Traversing last block in range
        for (int i = right_blk * blk_sz;
             i <= B; i++)
            ans += frequency[i];
    }
    return ans;
}
  
void add(int ind, int a[])
{
    // Increment the frequency of a[ind]
    // in the frequency array
    frequency[a[ind]]++;
  
    // Get the block number of a[ind]
    // to update the result in blocks
    int block_num = getblocknumber(a[ind]);
  
    blocks[block_num]++;
}
  
void remove(int ind, int a[])
{
    // Decrement the frequency of
    // a[ind] in the frequency array
    frequency[a[ind]]--;
  
    // Get the block number of a[ind]
    // to update the result in blocks
    int block_num = getblocknumber(a[ind]);
  
    blocks[block_num]--;
}
void queryResults(int a[], int n,
                  Query q[], int m,
                  int A, int B)
{
  
    // Initialize the block size
    // for queries
    query_blk_sz = sqrt(m);
  
    // 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 the
        // 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++;
        }
        printf("%d\n", getans(A, B));
    }
}
  
// Driver code
int main()
{
  
    int arr[] = { 3, 4, 6, 2, 7, 1 };
    int N = sizeof(arr) / sizeof(arr[0]);
  
    int A = 1, B = 6;
    blk_sz = sqrt(N);
    Query Q[] = { { 0, 4 } };
  
    int M = sizeof(Q) / sizeof(Q[0]);
  
    // Answer the queries
    queryResults(arr, N, Q, M, A, B);
  
    return 0;
}

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Output:

4

Time Complexity: O(Q*√N)
Space Complexity: O(N)

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