Queries for count of even digit sum elements in given range using MO's Algorithm

Last Updated : 12 Jul, 2025

Given an array arr[] of N elements, the task is to answer Q queries each having two integers L and R. For each query, the task is to find the number of elements in the subarray arr[L…R] whose digit sum is even.

Examples:

Input:arr[] = {7, 3, 19, 13, 5, 4}
query = { {1, 5}, {0, 1} }
Output: 3
Explanation:
Query 1: Elements 19, 13 and 4 have even digit sum in the subarray: {3, 9, 13, 5, 4}.
Query 2: No elements have even sum in the subarray: {7, 3}

Input:arr[] = {0, 1, 2, 3, 4, 5, 6, 7}
query = { 3, 5 }
Output:1

We have already discussed this approach: Queries for the count of even digit sum elements in the given range using Segment Tree

Approach: (Using MO's Algorithm)
The idea is to pre-process all queries so that result of one query can be used in the next query.

Sort all queries in a way that queries with L values from 0 to ?n – 1 are put together, followed by queries from ?n to 2×?n – 1, and so on. All queries within a block are sorted in increasing order of R values.
Process all queries one by one and increase the count of even digit sum elements and store the result in the structure.
- Let count_even store the count of even digits sum elements in the previous query.
- Remove extra elements of previous query and add new elements for the current query. For example, if previous query was [0, 8] and the current query is [3, 9], then remove the elements arr[0], arr[1] and arr[2] and add arr[9].
In order to display the results, sort the queries in the order they were provided.

Helper functions:
Adding elements()

If the current element has even digit sum then increase the count of count_even.
Removing elements()

If the current element has even digit sum then decrease the count of count_even.

Below code is the implementation of the above approach:

C++

// C++ program to count of even
// digit sum elements in the given
// range using MO's algorithm

#include <bits/stdc++.h>
using namespace std;

#define MAX 100000

// 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 {

    // Starting index
    int L;

    // Ending index
    int R;

    // Index of query
    int index;

    // Count of even
    // even digit sum
    int even;
};

// To store the count of
// even digit sum
int count_even;

// 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 used to sort all queries
// in order of their index value so that
// results of queries can be printed
// in same order as of input
bool compare1(Query x, Query y)
{
    return x.index < y.index;
}

// Function to find the digit sum
// for a number
int digitSum(int num)
{
    int sum = 0;
    while (num) {
        sum += (num % 10);
        num /= 10;
    }

    return sum;
}

// Function to Add elements
// of current range
void add(int currL, int a[])
{
    // If digit sum of a[currL]
    // is even then increment
    if (digitSum(a[currL]) % 2 == 0)
        count_even++;
}

// Function to remove elements
// of previous range
void remove(int currR, int a[])
{

    // If digit sum of a[currL]
    // is even then decrement
    if (digitSum(a[currR]) % 2 == 0)
        count_even--;
}

// Function to generate
// the result of queries
void queryResults(int a[], int n,
                  Query q[], int m)
{

    // Initialize number of
    // even digit sum to 0
    count_even = 0;

    // 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++;
        }

        q[i].even = count_even;
    }
}
// Function to display the results of
// queries in their initial order
void printResults(Query q[], int m)
{
    sort(q, q + m, compare1);
    for (int i = 0; i < m; i++) {
        cout << q[i].even << endl;
    }
}

// Driver Code
int main()
{

    int arr[] = { 5, 2, 3, 1, 4, 8, 10, 12 };
    int n = sizeof(arr) / sizeof(arr[0]);

    Query q[] = { { 1, 3, 0, 0 },
                  { 0, 4, 1, 0 },
                  { 4, 7, 2, 0 } };

    int m = sizeof(q) / sizeof(q[0]);

    queryResults(arr, n, q, m);

    printResults(q, m);

    return 0;
}

Java

// Java program to count of even
// digit sum elements in the given
// range using MO's algorithm
import java.util.Arrays;
import java.util.Collections;
import java.util.Comparator;

class GFG{

static int MAX = 100000;

// Variable to represent block size.
// This is made global so compare()
// of sort can use it.
static int block;

// Structure to represent a query range
static class Query 
{
    
    // Starting index
    int L;

    // Ending index
    int R;

    // Index of query
    int index;

    // Count of even
    // even digit sum
    int even;

    public Query(int l, int r, 
                 int index, int even) 
    {
        this.L = l;
        this.R = r;
        this.index = index;
        this.even = even;
    }
};

// To store the count of
// even digit sum
static int count_even;

// Function to find the digit sum
// for a number
static int digitSum(int num) 
{
    int sum = 0;
    while (num > 0) 
    {
        sum += (num % 10);
        num /= 10;
    }
    return sum;
}

// Function to Add elements
// of current range
static void add(int currL, int a[])
{
    
    // If digit sum of a[currL]
    // is even then increment
    if (digitSum(a[currL]) % 2 == 0)
        count_even++;
}

// Function to remove elements
// of previous range
static void remove(int currR, int a[]) 
{
    
    // If digit sum of a[currL]
    // is even then decrement
    if (digitSum(a[currR]) % 2 == 0)
        count_even--;
}

// Function to generate
// the result of queries
static void queryResults(int a[], int n, 
                       Query q[], int m)
{
    
    // Initialize number of
    // even digit sum to 0
    count_even = 0;

    // Find block size
    block = (int) Math.sqrt(n);

    // Sort all queries so that queries of
    // same blocks are arranged together.
    Collections.sort(Arrays.asList(q), 
                     new Comparator<Query>()
    {
        
        // 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.
        public int 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;
        }
    });

    // 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++;
        }
        q[i].even = count_even;
    }
}

// Function to display the results of
// queries in their initial order
static void printResults(Query q[], int m) 
{
    Collections.sort(Arrays.asList(q), 
             new Comparator<Query>()
    {
        
        // Function used to sort all queries
        // in order of their index value so that
        // results of queries can be printed
        // in same order as of input
        public int compare(Query x, Query y) 
        {
            return x.index - y.index;
        }
    });
    for(int i = 0; i < m; i++)
    {
        System.out.println(q[i].even);
    }
}

// Driver Code
public static void main(String[] args)
{
    int arr[] = { 5, 2, 3, 1, 4, 8, 10, 12 };
    int n = arr.length;

    Query q[] = { new Query(1, 3, 0, 0), 
                  new Query(0, 4, 1, 0), 
                  new Query(4, 7, 2, 0) };

    int m = q.length;

    queryResults(arr, n, q, m);

    printResults(q, m);
}
}

// This code is contributed by sanjeev2552

Python3

# Python program to count of even
# digit sum elements in the given
# range using MO's algorithm
import math
import functools

MAX = 100000

arr = [5, 2, 3, 1, 4, 8, 10, 12]
n = len(arr)

# Variable to represent block size.
# This is made here so compare()
# of sort can use it.
block = int(math.sqrt(n))

# Class to represent a query range


class Query:
    def __init__(self, l, r, index, even):
        self.L = l  # Starting index
        self.R = r  # Ending index
        self.index = index  # Index of query
        self.even = even  # Count of even digit sum

# 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):
  
    # Different blocks, sort by block.
    if x.L / block != y.L / block:
        return x.L / block < y.L / block
    else:
        # Same block, sort by R value
        return x.R < y.R

# Function to find the digit sum
# for a number
def digitSum(num):
    sum = 0
    while num > 0:
        sum += (num % 10)
        num //= 10
    return sum

# Function to Add elements
# of current range
def add(currL, a, count_even):
  
    # If digit sum of a[currL]
    # is even then increment
    if digitSum(a[currL]) % 2 == 0:
        count_even += 1

    return count_even

# Function to remove elements
# of previous range
def remove(currR, a, count_even):
    # If digit sum of a[currL]
    # is even then decrement
    if digitSum(a[currR]) % 2 == 0:
        count_even -= 1
    return count_even

# Function to generate
# the result of queries
def queryResults(a, n, q, m):

    # Initialize number of
    # even digit sum to 0
    count_even = 0

    # Find block size
    block = int(math.sqrt(n))

    # Sort all queries so that queries of
    # same blocks are arranged together.
    q.sort(key=functools.cmp_to_key(compare))

    # Initialize current L, current R and
    # current result
    currL = 0
    currR = 0

    for i in range(m):
        # L and R values of current range
        L = q[i].L
        R = q[i].R

        # Add Elements of current range
        while currR <= R:
            count_even = add(currR, a, count_even)
            currR += 1

        while currL > L:
            count_even = add(currL - 1, a, count_even)
            currL -= 1

        # Remove element of previous range
        while currR > R + 1:
            count_even = remove(currR - 1, a, count_even)
            currR -= 1

        while currL < L:
            count_even = remove(currL, a, count_even)
            currL += 1

        q[i].even = count_even

# Function to display the results of
# queries in their initial order
def printResults(q, m):
    q.sort(key=lambda x: x.index)
    for i in range(m):
        print(q[i].even)

# Driver Code
q = [Query(1, 3, 0, 0),
     Query(0, 4, 1, 0),
     Query(4, 7, 2, 0)]
m = len(q)

queryResults(arr, n, q, m)
printResults(q, m)

# This code is contributed by Tapesh (tapeshdua420)

//  program to count of even
// digit sum elements in the given
// range using MO's algorithm
using System;
using System.Linq;
using System.Collections.Generic;

// Class to represent a query range
public class Query
{
  public int L { get; set; }
  public int R { get; set; }
  public int Index { get; set; }
  public int Even { get; set; }

  public Query(int l, int r, int index, int even)
  {
    L = l;
    R = r;
    Index = index;
    Even = even;
  }
}

public class GFG
{
  static int MAX = 100000;
  static int[] arr = new int[] { 5, 2, 3, 1, 4, 8, 10, 12 };
  static int n = arr.Length;


  // Variable to represent block size.
  // This is made here so compare()
  // of sort can use it
  static int block = (int)Math.Sqrt(n);


  // Function to find the digit sum
  // for a number
  static int DigitSum(int num)
  {
    int sum = 0;
    while (num > 0)
    {
      sum += (num % 10);
      num /= 10;
    }
    return sum;
  }

  // Function to Add elements
  // of current range
  static int Add(int currL, int[] a, int count_even)
  {

    // If digit sum of a[currL]
    // is even then increment
    if (DigitSum(a[currL]) % 2 == 0)
    {
      count_even++;
    }
    return count_even;
  }


  // Function to remove elements
  // of previous range
  static int Remove(int currR, int[] a, int count_even)
  {

    // If digit sum of a[currL]
    // is even then decrement
    if (DigitSum(a[currR]) % 2 == 0)
    {
      count_even--;
    }
    return count_even;
  }


  // 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 int Compare(Query x, Query y)
  {
    // Different blocks, sort by block.
    if (x.L / block != y.L / block)
    {
      return x.L / block - y.L / block;
    }
    else
    {
      // Same block, sort by R value
      return x.R - y.R;
    }
  }


  // Function to generate
  // the result of queries
  static void QueryResults(int[] a, int n, Query[] q, int m)
  {

    // Initialize number of
    // even digit sum to 0
    int count_even = 0;

    // Sort all queries so that queries of
    // same blocks are arranged together.
    q = q.OrderBy(x => x, Comparer<Query>.Create(Compare)).ToArray();

    // 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)
      {
        count_even = Add(currR, a, count_even);
        currR++;
      }

      while (currL > L)
      {
        count_even = Add(currL - 1, a, count_even);
        currL--;
      }

      // Remove element of previous range
      while (currR > R + 1)
      {
        count_even = Remove(currR - 1, a, count_even);
        currR--;
      }

      while (currL < L)
      {
        count_even = Remove(currL, a, count_even);
        currL++;
      }

      q[i].Even = count_even;
    }
  }


  // Function to display the results of
  // queries in their initial order
  static void PrintResults(Query[] q, int m)
  {
    q = q.OrderBy(x => x.Index).ToArray();
    for (int i = 0; i < m; i++)
    {
      Console.WriteLine(q[i].Even);
    }
  }

  // Driver Code
  public static void Main()
  {
    Query[] q = new Query[] {
      new Query(1, 3, 0, 0),
      new Query(0, 4, 1, 0),
      new Query(4, 7, 2, 0)
      };
    int m = q.Length;

    QueryResults(arr, n, q, m);
    PrintResults(q, m);
  }
}

// This code is contributed by shivhach999

JavaScript

// Function to find the digit sum for a number
function digitSum(num) {
let sum = 0;
while (num) {
sum += (num % 10);
num = Math.floor(num / 10);
}
return sum;
}

// Function to Add elements of current range
function add(currL, a, count_even) {
// If digit sum of a[currL] is even then increment
if (digitSum(a[currL]) % 2 === 0) {
count_even++;
}
return count_even;
}

// Function to remove elements of previous range
function remove(currR, a, count_even) {
// If digit sum of a[currR] is even then decrement
if (digitSum(a[currR]) % 2 === 0) {
count_even--;
}
return count_even;
}

// 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) {
// Different blocks, sort by block.
if (Math.floor(x.L / block) !== Math.floor(y.L / block)) {
return Math.floor(x.L / block) < Math.floor(y.L / block);
}
// Same block, sort by R value
return x.R < y.R;
}

// Function used to sort all queries in order of their index value so that results of queries can be printed in same order as of input
function compare1(x, y) {
return x.index < y.index;
}

// Function to generate the result of queries
function queryResults(a, n, q, m) {
// Initialize number of even digit sum to 0
let count_even = 0;

// Find block size
block = Math.floor(Math.sqrt(n));

// Sort all queries so that queries of same blocks are arranged together.
q.sort(compare);

// Initialize current L, current R and current result
let currL = 0, currR = 0;

for (let i = 0; i < m; i++) {
// L and R values of current range
let L = q[i].L, R = q[i].R;

// Add Elements of current range
while (currR <= R) {
  count_even = add(currR, a, count_even);
  currR++;
}
while (currL > L) {
  count_even = add(currL - 1, a, count_even);
  currL--;
}

// Remove element of previous range
while (currR > R + 1) {
  count_even = remove(currR - 1, a, count_even);
  currR--;
}
while (currL < L) {
  count_even = remove(currL, a, count_even);
  currL++;
}

q[i].even = count_even;
}
}

// Function to display the results of queries in their initial order
function printResults(q, m) {
q.sort(compare1);
for (let i = 0; i < m; i++) {
document.write(q[i].even);
}
}

// Driver Code
const arr = [5, 2, 3, 1, 4, 8, 10, 12];
const n = arr.length;

const q = [
{ L: 1, R: 3, index: 0, even: 0 },
{ L: 0, R: 4, index: 1, even: 0 },
{ L: 4, R: 7, index: 2, even: 0 }]


    let m = q.length;

    queryResults(arr, n, q, m);

    printResults(q, m);

Output

1
2
2

Time Complexity: O(Q x ?N)
Auxiliary Space: O(1)

saurabhshadow

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