Subarrays with distinct elements

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Given an array, the task is to calculate the sum of lengths of contiguous subarrays having all elements distinct.

Examples: 

Input :  arr[] = {1, 2, 3}
Output : 10
{1, 2, 3} is a subarray of length 3 with 
distinct elements. Total length of length
three = 3.
{1, 2}, {2, 3} are 2 subarray of length 2 
with distinct elements. Total length of 
lengths two = 2 + 2 = 4
{1}, {2}, {3} are 3 subarrays of length 1
with distinct element. Total lengths of 
length one = 1 + 1 + 1 = 3
Sum of lengths = 3 + 4 + 3 = 10

Input :  arr[] = {1, 2, 1}
Output : 7

Input :  arr[] = {1, 2, 3, 4}
Output : 20

A simple solution is to consider all subarrays and for every subarray check if it has distinct elements or not using hashing. And add lengths of all subarrays having distinct elements. If we use hashing to find distinct elements, then this approach takes O(n²) time under the assumption that hashing search and insert operations take O(1) time.

An efficient solution is based on the fact that if we know all elements in a subarray arr[i..j] are distinct, sum of all lengths of distinct element subarrays in this subarray is ((j-i+1)*(j-i+2))/2. How? the possible lengths of subarrays are 1, 2, 3,......, j - i +1. So, the sum will be ((j - i +1)*(j - i +2))/2.

We first find the largest subarray (with distinct elements) starting from first element. We count sum of lengths in this subarray using above formula. For finding next subarray of the distinct element, we increment starting point, i and ending point, j unless (i+1, j) are distinct. If not possible, then we increment i again and move forward the same way.

Below is the implementation of this approach:

// C++ program to calculate sum of lengths of subarrays
// of distinct elements.
#include<bits/stdc++.h>
using namespace std;

// Returns sum of lengths of all subarrays with distinct
// elements.
int sumoflength(int arr[], int n)
{
    // For maintaining distinct elements.
    unordered_set<int> s;

    // Initialize ending point and result
    int j = 0, ans = 0;

    // Fix starting point
    for (int i=0; i<n; i++)
    {
        // Find ending point for current subarray with
        // distinct elements.
        while (j < n && s.find(arr[j]) == s.end())
        {
            s.insert(arr[j]);
            j++;
        }

        // Calculating and adding all possible length
        // subarrays in arr[i..j]
        ans += ((j - i) * (j - i + 1))/2;

        // Remove arr[i] as we pick new starting point
        // from next
        s.erase(arr[i]);
    }

    return ans;
}

// Driven Code
int main()
{
    int arr[] = {1, 2, 3, 4};
    int n = sizeof(arr)/sizeof(arr[0]);
    cout << sumoflength(arr, n) << endl;
    return 0;
}

// Java program to calculate sum of lengths of subarrays 
// of distinct elements. 
import java.util.*;

class geeks
{

    // Returns sum of lengths of all subarrays
    // with distinct elements.
    public static int sumoflength(int[] arr, int n) 
    {

        // For maintaining distinct elements.
        Set<Integer> s = new HashSet<>();

        // Initialize ending point and result
        int j = 0, ans = 0;

        // Fix starting point
        for (int i = 0; i < n; i++)
        {
            while (j < n && !s.contains(arr[j]))
            {
                s.add(arr[j]);
                j++;
            }

            // Calculating and adding all possible length
            // subarrays in arr[i..j]
            ans += ((j - i) * (j - i + 1)) / 2;

            // Remove arr[i] as we pick new starting point
            // from next
            s.remove(arr[i]);
        }

        return ans;
    }

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

        System.out.println(sumoflength(arr, n));
    }
}

// This code is contributed by
// sanjeev2552

# Python 3 program to calculate sum of 
# lengths of subarrays of distinct elements.

# Returns sum of lengths of all subarrays 
# with distinct elements.
def sumoflength(arr, n):

    # For maintaining distinct elements.
    s = []

    # Initialize ending point and result
    j = 0
    ans = 0

    # Fix starting point
    for i in range(n):
        
        # Find ending point for current 
        # subarray with distinct elements.
        while (j < n and (arr[j] not in s)):
            s.append(arr[j])
            j += 1

        # Calculating and adding all possible 
        # length subarrays in arr[i..j]
        ans += ((j - i) * (j - i + 1)) // 2

        # Remove arr[i] as we pick new 
        # starting point from next
        s.remove(arr[i])

    return ans

# Driven Code
if __name__=="__main__":
    
    arr = [1, 2, 3, 4]
    n = len(arr)
    print(sumoflength(arr, n))

# This code is contributed by ita_c

// C# program to calculate sum of lengths of subarrays 
// of distinct elements
using System;
using System.Collections.Generic;
    
public class geeks
{

    // Returns sum of lengths of all subarrays
    // with distinct elements.
    public static int sumoflength(int[] arr, int n) 
    {

        // For maintaining distinct elements.
        HashSet<int> s = new HashSet<int>();

        // Initialize ending point and result
        int j = 0, ans = 0;

        // Fix starting point
        for (int i = 0; i < n; i++)
        {
            while (j < n && !s.Contains(arr[j]))
            {
                s.Add(arr[j]);
                j++;
            }

            // Calculating and adding all possible length
            // subarrays in arr[i..j]
            ans += ((j - i) * (j - i + 1)) / 2;

            // Remove arr[i] as we pick new starting point
            // from next
            s.Remove(arr[i]);
        }

        return ans;
    }

    // Driver Code
    public static void Main(String[] args) 
    {
        int[] arr = { 1, 2, 3, 4 };
        int n = arr.Length;

        Console.WriteLine(sumoflength(arr, n));
    }
}

/* This code is contributed by PrinciRaj1992 */

// Javascript program to calculate sum of lengths of subarrays 
// of distinct elements.     

    // Returns sum of lengths of all subarrays
    // with distinct elements.
    function sumoflength(arr,n)
    {
        // For maintaining distinct elements.
        let s=new Set();
        // Initialize ending point and result
        let j = 0, ans = 0;
  
        // Fix starting point
        for (let i = 0; i < n; i++)
        {
            while (j < n && !s.has(arr[j]))
            {
                s.add(arr[i]);
                j++;
            }
  
            // Calculating and adding all possible length
            // subarrays in arr[i..j]
            ans += Math.floor(((j - i) * (j - i + 1)) / 2);
  
            // Remove arr[i] as we pick new starting point
            // from next
            s.delete(arr[i]);
        }
  
        return ans;
    }
    
    // Driver Code
    let arr=[1, 2, 3, 4];
    let n = arr.length;
    console.log(sumoflength(arr, n));
    
    // This code is contributed by avanitrachhadiya2155
    

20

Time Complexity: O(n^2). Note that the inner loop runs n times in total as j goes from 0 to n across all outer loops.
Space Complexity: O(n), since n extra space has been added