Range Queries to count elements lying in a given Range : MO's Algorithm

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[] = {7, 3, 9, 13, 5, 4}, A = 4, B = 7
query = {1, 5}
Output: 2
Explanation:
Only 5 and 4 lies within 4 to 7
in the subarray {3, 9, 13, 5, 4}
Therefore, the count of such elements is 2.

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.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Prerequisites: MO’s algorithm, SQRT Decomposition

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:

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.
Process all queries one by one in a way that every query uses result computed in the previous query.
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 sqrt decomposition technique to find the sum whose time complexity is O(√N) per query.

Below is the implementation of the above approach:

C++

// 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[] = { 2, 0, 3, 1, 4, 2, 5, 11 }; 
    int N = sizeof(arr) / sizeof(arr[0]); 
  
    int A = 1, B = 5; 
    blk_sz = sqrt(N); 
    Query Q[] = { { 0, 2 }, { 0, 3 }, { 5, 7 } }; 
  
    int M = sizeof(Q) / sizeof(Q[0]); 
  
    // Answer the queries 
    queryResults(arr, N, Q, M, A, B); 
    return 0; 
} 

Output:

2
3
2

My Personal Notes

Range Queries to count elements lying in a given Range : MO’s Algorithm

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

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