# Range Queries for count of Armstrong numbers in subarray using MO’s algorithm

Given an array arr[] consisting of N elements and Q queries represented by L and R denoting a range, the task is to print the number of Armstrong numbers in the subarray [L, R].

Examples:

Input: arr[] = {18, 153, 8, 9, 14, 5}
Query 1: query(L=0, R=5)
Query 2: query(L=3, R=5)
Output: 4
2
Explaination:
18 => 1*1 + 8*8 != 18
153 => 1*1*1 + 5*5*5 + 3*3*3 = 153
8 => 8 = 8
9 => 9 = 9
14 => 1*1 + 4*4 != 14
Query 1: The subarray[0…5] has 4 Armstrong numbers viz. {153, 8, 9, 5}
Query 2: The subarray[3…5] has 2 Armstrong numbers viz. {9, 5}

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.

1. 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.
2. Process all queries one by one and increase the count of Armstrong numbers and store the result in the structure.
• Let ‘count_Armstrong‘ store the count of Armstrong numbers 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].
3. In order to display the results, sort the queries in the order they were provided.

• If the current element is a Armstrong number then increment count_Armstrong.
• Removing elements

• If the current element is a Armstrong number then decrement count_Armstrong.
• Below is the implementation of the above approach:

## C++

 `// C++ implementation to count the ` `// number of Armstrong numbers ` `// in subarray using MO’s algorithm ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// 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, index; ` ` `  `    ``// Count of Armstrong ` `    ``// numbers ` `    ``int` `armstrong; ` `}; ` ` `  `// To store the count of ` `// Armstrong numbers ` `int` `count_Armstrong; ` ` `  `// 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 that return true ` `// if num is armstrong ` `// else return false ` `bool` `isArmstrong(``int` `x) ` `{ ` `    ``int` `n = to_string(x).size(); ` `    ``int` `sum1 = 0; ` `    ``int` `temp = x; ` `    ``while` `(temp > 0) { ` `        ``int` `digit = temp % 10; ` `        ``sum1 += ``pow``(digit, n); ` `        ``temp /= 10; ` `    ``} ` `    ``if` `(sum1 == x) ` `        ``return` `true``; ` `    ``return` `false``; ` `} ` ` `  `// Function to Add elements ` `// of current range ` `void` `add(``int` `currL, ``int` `a[]) ` `{ ` `    ``// If a[currL] is a Armstrong number ` `    ``// then increment count_Armstrong ` `    ``if` `(isArmstrong(a[currL])) ` `        ``count_Armstrong++; ` `} ` ` `  `// Function to remove elements ` `// of previous range ` `void` `remove``(``int` `currR, ``int` `a[]) ` `{ ` `    ``// If a[currL] is a Armstrong number ` `    ``// then decrement count_Armstrong ` `    ``if` `(isArmstrong(a[currR])) ` `        ``count_Armstrong--; ` `} ` ` `  `// Function to generate the result of queries ` `void` `queryResults(``int` `a[], ``int` `n, Query q[], ` `                  ``int` `m) ` `{ ` ` `  `    ``// Initialize count_Armstrong to 0 ` `    ``count_Armstrong = 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].armstrong = count_Armstrong; ` `    ``} ` `} ` ` `  `// 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].armstrong << endl; ` `    ``} ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` ` `  `    ``int` `arr[] = { 18, 153, 8, 9, 14, 5 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]); ` ` `  `    ``Query q[] = { { 0, 5, 0, 0 }, ` `                  ``{ 3, 5, 1, 0 } }; ` ` `  `    ``int` `m = ``sizeof``(q) / ``sizeof``(q[0]); ` ` `  `    ``queryResults(arr, n, q, m); ` ` `  `    ``printResults(q, m); ` ` `  `    ``return` `0; ` `} `

Output:

```4
2
```

Time Complexity: O(Q * √N)

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