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

Last Updated : 27 Apr, 2023

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[Lâ€¦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.

Prerequisites: MO’s algorithm, SQRT Decomposition

Approach: The idea is to use MOâ€™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 multiple chunks where each chunk contains the values of starting range in (0 to âˆšN – 1), (âˆšN to 2xâˆšN – 1), and so on. Sort the queries within a chunk in increasing 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[]=[0â€¦.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
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(âˆšN) 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  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 rangestruct Query {    int L;    int R;}; // Frequency array// to keep count of elementsint frequency[MAX]; // Array which contains the frequency// of a particular blockint blocks[SQRSIZE]; // Block sizeint 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 indint getblocknumber(int ind){    return (ind) / blk_sz;} // Function to get the answer// of range [0, k] which uses the// sqrt decomposition techniqueint 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    // right 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 codeint 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;}

## Java

 // Java implementation to find the // values in the range A to B // in a subarray of L to Rimport java.util.*;import java.lang.Math; public class GFG {   public static int MAX=100001;  public static int SQRSIZE=400;   // Variable to represent block size.  // This is made global so compare()  // of sort can use it.  public static int query_blk_sz;   // Frequency array  // to keep count of elements  public static int[] frequency = new int[MAX];   // Array which contains the frequency  // of a particular block  public static int[] blocks = new int[SQRSIZE];   // Block size  public static 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.   static Comparator<int[]> arrayComparator = new Comparator<int[]>() {    @Override    public int compare(int[] x, int[] y) {      if (x[0] / query_blk_sz != y[0] / query_blk_sz)        return Integer.compare(x[0] / query_blk_sz, y[0] / query_blk_sz);       return Integer.compare(x[1],y[1]);    }  };   // Function used to get the block  // number of current a[i] i.e ind  public static int getblocknumber(int ind)  {    return (ind) / blk_sz;  }   // Function to get the answer  // of range [0, k] which uses the  // sqrt decomposition technique  public static 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    // right 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;  }   public static 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]++;  }  public static 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]--;  }  public static void queryResults(int a[], int n,                                  int[][] q, int m, int A, int B)  {     // Initialize the block size    // for queries    query_blk_sz = (int)Math.sqrt(m);     // Sort all queries so that queries    // of same blocks are arranged    // together.    Arrays.sort(q,arrayComparator);     // 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][0], R = q[i][1];       // 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++;      }      System.out.println(getans(A, B));    }  }   // Driver code  public static void main(String[] args) {    int arr[] = { 2, 0, 3, 1, 4, 2, 5, 11 };    int N = arr.length;     int A = 1, B = 5;    blk_sz = (int) Math.sqrt(N);    int[][] Q = { { 0, 2 }, { 0, 3 }, { 5, 7 } };     int M = Q.length;     // Answer the queries    queryResults(arr, N, Q, M, A, B);  }} // This code is contributed// by Shubham Singh

## Python3

 # Python implementation to find the# values in the range A to B# in a subarray of L to Rimport mathfrom functools import cmp_to_keyMAX = 100001SQRSIZE = 400 # Variable to represent block size.# This is made global so compare()# of sort can use it.query_blk_sz = 1 # Frequency array# to keep count of elementsfrequency = [0] * MAX # Array which contains the frequency# of a particular blockblocks = [0]*SQRSIZE # Block sizeblk_sz = 0 # 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.  def compare(x, y):    if ((x[0] // query_blk_sz) != (y[0] // query_blk_sz)):        return x[0] // query_blk_sz < y[0] // query_blk_sz     return x[1] < y[1]  # Function used to get the block# number of current a[i] i.e inddef getblocknumber(ind):    return (ind) // blk_sz # Function to get the answer# of range [0, k] which uses the# sqrt decomposition technique  def getans(A, B):    ans = 0     left_blk = getblocknumber(A)    right_blk = getblocknumber(B)     # If left block is equal to    # right block then we can traverse    # that block    if (left_blk == right_blk):        for i in range(A, B+1):            ans += frequency[i]     else:        # Traversing first block in        # range        i = A        while(i < (left_blk+1)*blk_sz):            ans += frequency[i]            i += 1         # Traversing completely overlapped        # blocks in range        i = (int)(left_blk + 1)        while(i < right_blk):            ans += blocks[i]            i += 1         # Traversing last block in range        i = (int)(right_blk * blk_sz)        while(i <= B):            ans += frequency[i]            i += 1     return ans  def add(ind, a):     # Increment the frequency of a[ind]    # in the frequency array    frequency[a[ind]] += 1     # Get the block number of a[ind]    # to update the result in blocks    block_num = getblocknumber(a[ind])     blocks[(int)(block_num)] += 1  def remove(ind, a):     # Decrement the frequency of    # a[ind] in the frequency array    frequency[a[ind]] -= 1     # Get the block number of a[ind]    # to update the result in blocks    block_num = getblocknumber(a[ind])     blocks[(int)(block_num)] -= 1  def queryResults(a, n, q, m, A, B):     # Initialize the block size    # for queries    query_blk_sz = (int)(math.sqrt(m))     # Sort all queries so that queries    # of same blocks are arranged    # together.    q.sort(key=cmp_to_key(compare))     # Initialize current L,    # current R and current result    currL = 0    currR = 0     for i in range(0, m):         # L and R values of the        # current range         L = q[i][0]        R = q[i][1]         # Add Elements of current        # range        while (currR <= R):            add(currR, a)            currR += 1         while (currL > L):            add(currL - 1, a)            currL -= 1         # Remove element of previous        # range        while (currR > R + 1):            remove(currR - 1, a)            currR -= 1         while (currL < L):            remove(currL, a)            currL += 1         print(getans(A, B))  # Driver codearr = [2, 0, 3, 1, 4, 2, 5, 11]N = len(arr) A = 1B = 5blk_sz = (int)(math.sqrt(N)) Q = [[0, 2], [0, 3], [5, 7]] M = len(Q) # Answer the queriesqueryResults(arr, N, Q, M, A, B) # This code is contributed by rj113to.

## C#

 // C# implementation to find the // values in the range A to B // in a subarray of L to Rusing System;using System.Collections.Generic; public class GFG {    public static int MAX = 100001;    public static int SQRSIZE = 400;         // Variable to represent block size.    // This is made global so compare()    // of sort can use it.    public static int query_blk_sz;         // Frequency array    // to keep count of elements    public static int[] frequency = new int[MAX];              // Array which contains the frequency    // of a particular block    public static int[] blocks = new int[SQRSIZE];         // Block size    public static 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.    static Comparer<int[]> arrayComparator = Comparer<int[]>.Create((x, y) => {        if (x[0] / query_blk_sz != y[0] / query_blk_sz)            return x[0] / query_blk_sz.CompareTo(y[0] / query_blk_sz);        return x[1].CompareTo(y[1]);    });         // Function used to get the block    // number of current a[i] i.e ind    public static int getblocknumber(int ind)    {        return (ind) / blk_sz;    }              // Function to get the answer    // of range [0, k] which uses the    // sqrt decomposition technique    public static 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        // right 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;    }         public static 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]++;    }         public static 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]--;    }         public static void queryResults(int[] a, int n, int[][] q, int m, int A, int B)    {        // Initialize the block size        // for queries        query_blk_sz = (int)Math.Sqrt(m);                 // Sort all queries so that queries        // of same blocks are arranged        // together.        Array.Sort(q, arrayComparator);        int currL = 0, currR = 0;                 // Initialize current L,        // current R and current result        for (int i = 0; i < m; i++) {                                      // L and R values of the            // current range            int L = q[i][0], R = q[i][1];                         // 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++;            }            Console.WriteLine(getans(A, B));        }    }         // Driver code    public static void Main(string[] args) {        int[] arr = { 2, 0, 3, 1, 4, 2, 5, 11 };        int N = arr.Length;        int A = 1, B = 5;        blk_sz = (int)Math.Sqrt(N);        int[][] Q = { new int[] { 0, 2 }, new int[] { 0, 3 }, new int[] { 5, 7 } };        int M = Q.Length;                 // Answer the queries        queryResults(arr, N, Q, M, A, B);    }}     // This code is contributed by shivhack999

## Javascript

 // JavaScript implementation to find the// values in the range A to B// in a subarray of L to Rconst MAX = 100001;const SQRSIZE = 400; // Variable to represent block size.// This is made global so compare()// of sort can use it.let query_blk_sz = 1; // Frequency array// to keep count of elementsconst frequency = new Array(MAX).fill(0); // Array which contains the frequency// of a particular blockconst blocks = new Array(SQRSIZE).fill(0); // Block sizelet blk_sz = 0; // 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.function compare(x, y) {if (Math.floor(x[0] / query_blk_sz) !== Math.floor(y[0] / query_blk_sz)) {return Math.floor(x[0] / query_blk_sz) < Math.floor(y[0] / query_blk_sz) ? -1 : 1;}return x[1] < y[1] ? -1 : 1;} // Function used to get the block// number of current a[i] i.e indfunction getblocknumber(ind) {return Math.floor(ind / blk_sz);} // Function to get the answer// of range [0, k] which uses the// sqrt decomposition techniquefunction getans(A, B) {let ans = 0; const left_blk = getblocknumber(A);const right_blk = getblocknumber(B); // If left block is equal to// right block then we can traverse// that blockif (left_blk == right_blk) {for (let i = A; i <= B; i++) {ans += frequency[i];}} else {// Traversing first block in// rangelet i = A;while (i < (left_blk + 1) * blk_sz) {ans += frequency[i];i++;}// Traversing completely overlapped// blocks in rangefor (let i = left_blk + 1; i < right_blk; i++) {  ans += blocks[i];} // Traversing last block in rangelet j = right_blk * blk_sz;for (let i = j - blk_sz; i <= B; i++) {  ans += frequency[i];}}return ans;} function add(ind, a) {// Increment the frequency of a[ind]// in the frequency arrayfrequency[a[ind]]++; // Get the block number of a[ind]// to update the result in blocksconst block_num = getblocknumber(a[ind]); blocks[block_num]++;} function remove(ind, a) {// Decrement the frequency of// a[ind] in the frequency arrayfrequency[a[ind]]--; // Get the block number of a[ind]// to update the result in blocksconst block_num = getblocknumber(a[ind]); blocks[block_num]--;} function queryResults(a, n, q, m, A, B) {// Initialize the block size// for queriesquery_blk_sz = Math.floor(Math.sqrt(m)); // Sort all queries so that queries// of same blocks are arranged// together.q.sort(compare); // Initialize current L,// current R and current resultlet currL = 0;let currR = 0; for (let i = 0; i < m; i++) {// L and R values of the current rangeconst L = q[i][0];const R = q[i][1]; // Add elements of current rangewhile (currR <= R) {add(currR, a);currR++;} while (currL > L) {add(currL - 1, a);currL--;} // Remove element of previous rangewhile (currR > R + 1) {remove(currR - 1, a);currR--;} while (currL < L) {remove(currL, a);currL++;} console.log(getans(A, B));}} // Driver codeconst arr = [2, 0, 3, 1, 4, 2, 5, 11];const N = arr.length; const aa = 1;const bb = 5; const Q = [[0, 2], [0, 3], [5, 7]]; const M = Q.length; // Answer the queriesqueryResults(arr, N, Q, M, aa, bb);

Output:
2
3
2

Time Complexity: O(M+ sqrt(N)) where N is the size of the array and M is the number of queries.
Auxiliary Space: O(sqrt(N))

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