# Find frequency of each element in a limited range array in less than O(n) time

Given a sorted array of positive integers, the count number of occurrences for each element in the array. Assume all elements in the array are less than some constant M.
Do this without traversing the complete array. i.e. expected time complexity is less than O(n).
Examples:

```Input: arr[] = [1, 1, 1, 2, 3, 3, 5,
5, 8, 8, 8, 9, 9, 10]
Output:
Element 1 occurs 3 times
Element 2 occurs 1 times
Element 3 occurs 2 times
Element 5 occurs 2 times
Element 8 occurs 3 times
Element 9 occurs 2 times
Element 10 occurs 1 times

Input: arr[] = [2, 2, 6, 6, 7, 7, 7, 11]
Output:
Element 2 occurs 2 times
Element 6 occurs 2 times
Element 7 occurs 3 times
Element 11 occurs 1 times
```

Method 1: This method uses the technique of Linear Search to solve the following.

• Approach: The idea is to traverse the input array and for each distinct element of the array, store its frequency in a HashMap, and finally print the HashMap.
• Algorithm:
1. Create a HashMap to map the frequency to element, i.e to store element-frequency pair.
2. Traverse the array from start to end.
3. For each element in the array update the frequency, i.e hm[array[i]]++
4. Traverse the HashMap and print the element frequency pair.
• Implementation:

## C++

 `// C++ program to count number of occurrences of` `// each element in the array #include ` `#include ` `using` `namespace` `std;`   `// It prints number of` `// occurrences of each element in the array.` `void` `findFrequency(``int` `arr[], ``int` `n)` `{` `    ``// HashMap to store frequencies` `    ``unordered_map<``int``, ``int``> mp;`   `    ``// traverse the array` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``// update the frequency` `        ``mp[arr[i]]++;` `    ``}`   `    ``// traverse the hashmap` `    ``for` `(``auto` `i : mp) {` `        ``cout << ``"Element "` `<< i.first << ``" occurs "` `             ``<< i.second << ``" times"` `<< endl;` `    ``}` `}`   `// Driver function` `int` `main()` `{` `    ``int` `arr[] = { 1, 1, 1, 2, 3, 3, 5, 5,` `                  ``8, 8, 8, 9, 9, 10 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);`   `    ``findFrequency(arr, n);`   `    ``return` `0;` `}`

## Python3

 `# Python program to count number of occurrences of` `# each element in the array #include `   `# It prints number of` `# occurrences of each element in the array.` `def` `findFrequency(arr, n):` `    `  `    ``# HashMap to store frequencies` `    ``mp ``=` `{}`   `    ``# traverse the array` `    ``for` `i ``in` `range``(n):`   `        ``# update the frequency` `        ``if` `arr[i] ``not` `in` `mp:` `            ``mp[arr[i]] ``=` `0` `        ``mp[arr[i]] ``+``=` `1` `        `  `    ``# traverse the hashmap` `    ``for` `i ``in` `mp:` `        ``print``(``"Element"``, i, ``"occurs"``, mp[i], ``"times"``)` `    `  `# Driver function` `arr ``=` `[``1``, ``1``, ``1``, ``2``, ``3``, ``3``, ``5``, ``5``,``8``, ``8``, ``8``, ``9``, ``9``, ``10``]` `n ``=` `len``(arr)`   `findFrequency(arr, n)` `    `  `# This code is contributed by shubhamsingh10`

## C#

 `// C# program to count number ` `// of occurrences of each ` `// element in the array ` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG{`   `// It prints number of` `// occurrences of each ` `// element in the array.` `static` `void` `findFrequency(``int` `[] arr, ` `                          ``int` `n)` `{` `  ``// HashMap to store frequencies` `  ``Dictionary <``int``,` `              ``int` `> mp = ` `              ``new` `Dictionary<``int``,` `                             ``int``>();`   `  ``// traverse the array` `  ``for` `(``int` `i = 0; i < n; i++) ` `  ``{` `    ``// update the frequency` `    ``if` `(!mp.ContainsKey(arr[i]))` `      ``mp[arr[i]] = 0;`   `    ``mp[arr[i]]++;` `  ``}`   `  ``// traverse the hashmap` `  ``foreach``(KeyValuePair<``int``, ` `                       ``int``> kvp ``in` `mp)` `    ``Console.WriteLine(``"Element "` `+ kvp.Key + ` `                      ``" occurs "` `+ kvp.Value +` `                      ``" times"``);` `}`   `// Driver function` `public` `static` `void` `Main()` `{` `  ``int` `[] arr = {1, 1, 1, 2, ` `                ``3, 3, 5, 5,` `                ``8, 8, 8, 9, ` `                ``9, 10};` `  ``int` `n = arr.Length;` `  ``findFrequency(arr, n);` `}` `}`   `// This code is contributed by Chitranayal`

• Output:

```Element 1 occurs 3 times
Element 2 occurs 1 times
Element 3 occurs 2 times
Element 5 occurs 2 times
Element 8 occurs 3 times
Element 9 occurs 2 times
Element 10 occurs 1 times
```
• Complexity Analysis:
• Time Complexity:O(n), only one traversal of the array is needed.
• Space Complexity:O(n), to store the elements in the HashMap O(n) extra space is needed.

Method 2: This method uses the technique of Binary Search to arrive at the solution.

• Approach: The problem can be solved in less than O(n) time if all its elements are sorted, i.e. if similar elements exist in the array then the elements are in a contiguous subarray or it can be said that if ends of a subarray are same then all the elements inside the subarray are equal. So the count of that element is the size of the subarray and all the elements of that subarray need not be counted.

• Algorithm:
1. Create a HashMap (hm) to store the frequency of elements.
2. Create a recursive function that accepts an array and size.
3. Check if the first element of the array is equal to the last element. If equal then all the elements are the same and update the frequency by hm[array+=size
4. Else divide the array into two equal halves and call the function recursively for both the halves.
5. Traverse the hashmap and print the element frequency pair.
• Implementation:

## C++

 `// C++ program to count number of occurrences of` `// each element in the array in less than O(n) time` `#include ` `#include ` `using` `namespace` `std;`   `// A recursive function to count number of occurrences` `// for each element in the array without traversing` `// the whole array` `void` `findFrequencyUtil(``int` `arr[], ``int` `low, ``int` `high,` `                       ``vector<``int``>& freq)` `{` `    ``// If element at index low is equal to element` `    ``// at index high in the array` `    ``if` `(arr[low] == arr[high]) {` `        ``// increment the frequency of the element` `        ``// by count of elements between high and low` `        ``freq[arr[low]] += high - low + 1;` `    ``}` `    ``else` `{` `        ``// Find mid and recurse for left and right` `        ``// subarray` `        ``int` `mid = (low + high) / 2;` `        ``findFrequencyUtil(arr, low, mid, freq);` `        ``findFrequencyUtil(arr, mid + 1, high, freq);` `    ``}` `}`   `// A wrapper over recursive function` `// findFrequencyUtil(). It print number of` `// occurrences of each element in the array.` `void` `findFrequency(``int` `arr[], ``int` `n)` `{` `    ``// create a empty vector to store frequencies` `    ``// and initialize it by 0. Size of vector is` `    ``// maximum value (which is last value in sorted` `    ``// array) plus 1.` `    ``vector<``int``> freq(arr[n - 1] + 1, 0);`   `    ``// Fill the vector with frequency` `    ``findFrequencyUtil(arr, 0, n - 1, freq);`   `    ``// Print the frequencies` `    ``for` `(``int` `i = 0; i <= arr[n - 1]; i++)` `        ``if` `(freq[i] != 0)` `            ``cout << ``"Element "` `<< i << ``" occurs "` `                 ``<< freq[i] << ``" times"` `<< endl;` `}`   `// Driver function` `int` `main()` `{` `    ``int` `arr[] = { 1, 1, 1, 2, 3, 3, 5, 5,` `                  ``8, 8, 8, 9, 9, 10 };` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr);`   `    ``findFrequency(arr, n);`   `    ``return` `0;` `}`

## Java

 `// Java program to count number of occurrences of` `// each element in the array in less than O(n) time` `import` `java.util.*;`   `class` `GFG {`   `    ``// A recursive function to count number of occurrences` `    ``// for each element in the array without traversing` `    ``// the whole array` `    ``static` `void` `findFrequencyUtil(``int` `arr[], ``int` `low,` `                                  ``int` `high, ``int``[] freq)` `    ``{` `        ``// If element at index low is equal to element` `        ``// at index high in the array` `        ``if` `(arr[low] == arr[high]) {` `            ``// increment the frequency of the element` `            ``// by count of elements between high and low` `            ``freq[arr[low]] += high - low + ``1``;` `        ``}` `        ``else` `{` `            ``// Find mid and recurse for left and right` `            ``// subarray` `            ``int` `mid = (low + high) / ``2``;` `            ``findFrequencyUtil(arr, low, mid, freq);` `            ``findFrequencyUtil(arr, mid + ``1``, high, freq);` `        ``}` `    ``}`   `    ``// A wrapper over recursive function` `    ``// findFrequencyUtil(). It print number of` `    ``// occurrences of each element in the array.` `    ``static` `void` `findFrequency(``int` `arr[], ``int` `n)` `    ``{` `        ``// create a empty vector to store frequencies` `        ``// and initialize it by 0. Size of vector is` `        ``// maximum value (which is last value in sorted` `        ``// array) plus 1.` `        ``int``[] freq = ``new` `int``[arr[n - ``1``] + ``1``];`   `        ``// Fill the vector with frequency` `        ``findFrequencyUtil(arr, ``0``, n - ``1``, freq);`   `        ``// Print the frequencies` `        ``for` `(``int` `i = ``0``; i <= arr[n - ``1``]; i++)` `            ``if` `(freq[i] != ``0``)` `                ``System.out.println(``"Element "` `+ i + ``" occurs "` `+ freq[i] + ``" times"``);` `    ``}`   `    ``// Driver Code` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `arr[] = { ``1``, ``1``, ``1``, ``2``, ``3``, ``3``, ``5``,` `                      ``5``, ``8``, ``8``, ``8``, ``9``, ``9``, ``10` `};` `        ``int` `n = arr.length;`   `        ``findFrequency(arr, n);` `    ``}` `}`   `// This code is contributed by 29AjayKumar`

## Python3

 `# Python 3 program to count number of occurrences of` `# each element in the array in less than O(n) time`   `# A recursive function to count number of occurrences ` `# for each element in the array without traversing ` `# the whole array` `def` `findFrequencyUtil(arr, low, high, freq):` `    `  `    ``# If element at index low is equal to element ` `    ``# at index high in the array` `    ``if` `(arr[low] ``=``=` `arr[high]):` `        `  `        ``# increment the frequency of the element` `        ``# by count of elements between high and low` `        ``freq[arr[low]] ``+``=` `high ``-` `low ``+` `1` `    `  `    ``else``:` `        `  `        ``# Find mid and recurse for left  ` `        ``# and right subarray` `        ``mid ``=` `int``((low ``+` `high) ``/` `2``)` `        ``findFrequencyUtil(arr, low, mid, freq)` `        ``findFrequencyUtil(arr, mid ``+` `1``, high, freq)` `    `  `# A wrapper over recursive function ` `# findFrequencyUtil(). It print number of ` `# occurrences of each element in the array.` `def` `findFrequency(arr, n):` `    `  `    ``# create a empty vector to store frequencies` `    ``# and initialize it by 0. Size of vector is ` `    ``# maximum value (which is last value in sorted` `    ``# array) plus 1.` `    ``freq ``=` `[``0` `for` `i ``in` `range``(n ``-` `1` `+` `1``)]` `    `  `    ``# Fill the vector with frequency` `    ``findFrequencyUtil(arr, ``0``, n ``-` `1``, freq)`   `    ``# Print the frequencies` `    ``for` `i ``in` `range``(``0``, arr[n ``-` `1``] ``+` `1``, ``1``):` `        ``if` `(freq[i] !``=` `0``):` `            ``print``(``"Element"``, i, ``"occurs"``, ` `                        ``freq[i], ``"times"``)`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    ``arr ``=` `[``1``, ``1``, ``1``, ``2``, ``3``, ``3``, ``5``,` `           ``5``, ``8``, ``8``, ``8``, ``9``, ``9``, ``10``]` `    ``n ``=` `len``(arr)` `    ``findFrequency(arr, n)` `    `  `# This code is contributed by` `# Surendra_Gangwar`

## C#

 `// C# program to count number of occurrences of` `// each element in the array in less than O(n) time` `using` `System;`   `class` `GFG {`   `    ``// A recursive function to count number of occurrences` `    ``// for each element in the array without traversing` `    ``// the whole array` `    ``static` `void` `findFrequencyUtil(``int``[] arr, ``int` `low,` `                                  ``int` `high, ``int``[] freq)` `    ``{` `        ``// If element at index low is equal to element` `        ``// at index high in the array` `        ``if` `(arr[low] == arr[high]) {` `            ``// increment the frequency of the element` `            ``// by count of elements between high and low` `            ``freq[arr[low]] += high - low + 1;` `        ``}` `        ``else` `{` `            ``// Find mid and recurse for left and right` `            ``// subarray` `            ``int` `mid = (low + high) / 2;` `            ``findFrequencyUtil(arr, low, mid, freq);` `            ``findFrequencyUtil(arr, mid + 1, high, freq);` `        ``}` `    ``}`   `    ``// A wrapper over recursive function` `    ``// findFrequencyUtil(). It print number of` `    ``// occurrences of each element in the array.` `    ``static` `void` `findFrequency(``int``[] arr, ``int` `n)` `    ``{` `        ``// create a empty vector to store frequencies` `        ``// and initialize it by 0. Size of vector is` `        ``// maximum value (which is last value in sorted` `        ``// array) plus 1.` `        ``int``[] freq = ``new` `int``[arr[n - 1] + 1];`   `        ``// Fill the vector with frequency` `        ``findFrequencyUtil(arr, 0, n - 1, freq);`   `        ``// Print the frequencies` `        ``for` `(``int` `i = 0; i <= arr[n - 1]; i++)` `            ``if` `(freq[i] != 0)` `                ``Console.WriteLine(``"Element "` `+ i + ``" occurs "` `+ freq[i] + ``" times"``);` `    ``}`   `    ``// Driver Code` `    ``public` `static` `void` `Main(String[] args)` `    ``{` `        ``int``[] arr = { 1, 1, 1, 2, 3, 3, 5,` `                      ``5, 8, 8, 8, 9, 9, 10 };` `        ``int` `n = arr.Length;`   `        ``findFrequency(arr, n);` `    ``}` `}`   `// This code is contributed by Princi Singh`

• Output:
```Element 1 occurs 3 times
Element 2 occurs 1 times
Element 3 occurs 2 times
Element 5 occurs 2 times
Element 8 occurs 3 times
Element 9 occurs 2 times
Element 10 occurs 1 times
```
• Complexity Analysis:
• Time Complexity: O(m log n).
Where m is the number of distinct elements in the array of size n. Since m <= M (a constant) (elements are in a limited range), the time complexity of this solution is O(log n).
• Space Complexity:O(n).
To store the elements in the HashMap O(n) extra space is needed.

https://youtu.be/B2hI-QPoisk