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Sum of f(a[i], a[j]) over all pairs in an array of n integers

• Difficulty Level : Medium
• Last Updated : 29 May, 2021

Given an array of n integers, find the sum of f(a[i], a[j]) of all pairs (i, j) such that (1 <= i < j <= n).

f(a[i], a[j]):

```If |a[j]-a[i]| > 1
f(a[i], a[j]) = a[j] - a[i]
Else // if |a[j]-a[i]| <= 1
f(a[i], a[j]) = 0```

Examples:

```Input : 6 6 4 4
Output : -8
Explanation:
All pairs are: (6 - 6) + (6 - 6) +
(6 - 6) + (4 - 6) + (4 - 6) + (4 - 6) +
(4 - 6) + (4 - 4) + (4 - 4) = -8

Input: 1 2 3 1 3
Output: 4
Explanation: the pairs that add up are:
(3, 1), (3, 1) to give 4, rest all pairs
according to condition gives 0.  ```

A naive approach is to iterate through all pairs and calculate f(a[i], a[j]), and summing it up while traversing two nested loops will give us our answer.
Time Complexity: O(n^2)

An efficient approach would be to use a map/hash function to keep a count of every occurring number and then traverse through the list. While traversing through the list, we multiply the count of numbers that are before it and the number itself. Then subtract this result with the pre-sum of the number before that number to get the sum of the difference of all pairs possible with that number. To remove all pairs whose absolute difference is <=1, simply subtract the count of occurrence of (number-1) and (number+1) from the previously computed sum. Here we subtract the count of (number-1) from the computed sum as it had been previously added to the sum, and we add the (number+1) count since the negative has been added to the pre-computed sum of all pairs.
Time Complexity: O(n)

Below is the implementation of the above approach:

C++

 `// CPP program to calculate the``// sum of f(a[i], aj])``#include ``using` `namespace` `std;` `// Function to calculate the sum``int` `sum(``int` `a[], ``int` `n)``{``    ``// map to keep a count of occurrences``    ``unordered_map<``int``, ``int``> cnt;` `    ``// Traverse in the list from start to end``    ``// number of times a[i] can be in a pair and``    ``// to get the difference we subtract pre_sum.``    ``int` `ans = 0, pre_sum = 0;``    ``for` `(``int` `i = 0; i < n; i++) {``        ``ans += (i * a[i]) - pre_sum;``        ``pre_sum += a[i];` `        ``// if the (a[i]-1) is present then``        ``// subtract that value as f(a[i], a[i]-1)=0``        ``if` `(cnt[a[i] - 1])``            ``ans -= cnt[a[i] - 1];` `        ``// if the (a[i]+1) is present then``        ``// add that value as f(a[i], a[i]-1)=0``        ``// here we add as a[i]-(a[i]-1)<0 which would``        ``// have been added as negative sum, so we add ``        ``// to remove this pair from the sum value``        ``if` `(cnt[a[i] + 1])``            ``ans += cnt[a[i] + 1];` `        ``// keeping a counter for every element``        ``cnt[a[i]]++;``    ``}``    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``int` `a[] = { 1, 2, 3, 1, 3 };``    ``int` `n = ``sizeof``(a) / ``sizeof``(a[0]);``    ``cout << sum(a, n);``    ``return` `0;``}`

Java

 `// Java program to calculate ``// the sum of f(a[i], aj])``import` `java.util.*;``public` `class` `GfG {` `    ``// Function to calculate the sum``    ``public` `static` `int` `sum(``int` `a[], ``int` `n)``    ``{``        ``// Map to keep a count of occurrences``        ``Map cnt = ``new` `HashMap();``        ` `        ``// Traverse in the list from start to end``        ``// number of times a[i] can be in a pair and``        ``// to get the difference we subtract pre_sum``        ``int` `ans = ``0``, pre_sum = ``0``;``        ``for` `(``int` `i = ``0``; i < n; i++) {``            ``ans += (i * a[i]) - pre_sum;``            ``pre_sum += a[i];``    ` `            ``// If the (a[i]-1) is present then subtract``            ``// that value as f(a[i], a[i]-1) = 0``            ``if` `(cnt.containsKey(a[i] - ``1``))``                ``ans -= cnt.get(a[i] - ``1``);` `            ``// If the (a[i]+1) is present then``            ``// add that value as f(a[i], a[i]-1)=0``            ``// here we add as a[i]-(a[i]-1)<0 which would``            ``// have been added as negative sum, so we add``            ``// to remove this pair from the sum value``            ``if` `(cnt.containsKey(a[i] + ``1``))``                ``ans += cnt.get(a[i] + ``1``);` `            ``// keeping a counter for every element``            ``if``(cnt.containsKey(a[i])) {``                ``cnt.put(a[i], cnt.get(a[i]) + ``1``);``            ``}``            ``else` `{``                ``cnt.put(a[i], ``1``);``            ``}``        ``}``        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String args[])``    ``{``        ``int` `a[] = { ``1``, ``2``, ``3``, ``1``, ``3` `};``        ``int` `n = a.length;``        ``System.out.println(sum(a, n));``    ``}``}` `// This code is contributed by Swetank Modi`

Python3

 `# Python3 program to calculate the``# sum of f(a[i], aj])` `# Function to calculate the sum``def` `sum``(a, n):` `    ``# map to keep a count of occurrences``    ``cnt ``=` `dict``()` `    ``# Traverse in the list from start to end``    ``# number of times a[i] can be in a pair and``    ``# to get the difference we subtract pre_sum.``    ``ans ``=` `0``    ``pre_sum ``=` `0``    ``for` `i ``in` `range``(n):``        ``ans ``+``=` `(i ``*` `a[i]) ``-` `pre_sum``        ``pre_sum ``+``=` `a[i]` `        ``# if the (a[i]-1) is present then``        ``# subtract that value as f(a[i], a[i]-1)=0``        ``if` `(a[i] ``-` `1``) ``in` `cnt:``            ``ans ``-``=` `cnt[a[i] ``-` `1``]` `        ``# if the (a[i]+1) is present then add that ``        ``# value as f(a[i], a[i]-1)=0 here we add``        ``# as a[i]-(a[i]-1)<0 which would have been``        ``# added as negative sum, so we add to remove ``        ``# this pair from the sum value``        ``if` `(a[i] ``+` `1``) ``in` `cnt:``            ``ans ``+``=` `cnt[a[i] ``+` `1``]` `        ``# keeping a counter for every element``        ``if` `a[i] ``not` `in` `cnt:``            ``cnt[a[i]] ``=` `0``        ``cnt[a[i]] ``+``=` `1``    ` `    ``return` `ans` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``a ``=` `[``1``, ``2``, ``3``, ``1``, ``3``]``    ``n ``=` `len``(a)``    ``print``(``sum``(a, n))``        ` `# This code is contributed by``# SHUBHAMSINGH10`

C#

 `using` `System;``using` `System.Collections.Generic;` `// C#  program to calculate  ``// the sum of f(a[i], aj])``public` `class` `GfG``{` `    ``// Function to calculate the sum``    ``public` `static` `int` `sum(``int``[] a, ``int` `n)``    ``{``        ``// Map to keep a count of occurrences``        ``IDictionary<``int``, ``int``> cnt = ``new` `Dictionary<``int``, ``int``>();` `        ``// Traverse in the list from start to end``        ``// number of times a[i] can be in a pair and``        ``// to get the difference we subtract pre_sum``        ``int` `ans = 0, pre_sum = 0;``        ``for` `(``int` `i = 0; i < n; i++)``        ``{``            ``ans += (i * a[i]) - pre_sum;``            ``pre_sum += a[i];` `            ``// If the (a[i]-1) is present then subtract``            ``// that value as f(a[i], a[i]-1) = 0``            ``if` `(cnt.ContainsKey(a[i] - 1))``            ``{``                ``ans -= cnt[a[i] - 1];``            ``}` `            ``// If the (a[i]+1) is present then``            ``// add that value as f(a[i], a[i]-1)=0``            ``// here we add as a[i]-(a[i]-1)<0 which would ``            ``// have been added as negative sum, so we add ``            ``// to remove this pair from the sum value``            ``if` `(cnt.ContainsKey(a[i] + 1))``            ``{``                ``ans += cnt[a[i] + 1];``            ``}` `            ``// keeping a counter for every element``            ``if` `(cnt.ContainsKey(a[i]))``            ``{``                ``cnt[a[i]] = cnt[a[i]] + 1;``            ``}``            ``else``            ``{``                ``cnt[a[i]] = 1;``            ``}``        ``}``        ``return` `ans;``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main(``string``[] args)``    ``{``        ``int``[] a = ``new` `int``[] {1, 2, 3, 1, 3};``        ``int` `n = a.Length;``        ``Console.WriteLine(sum(a, n));``    ``}``}` `// This code is contributed by Shrikant13`

Javascript

 ``

Output:

`4`

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