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Sum of all array elements less than X and greater than Y for Q queries
• Difficulty Level : Hard
• Last Updated : 14 Aug, 2020

Given a sorted array arr[], and a set Q having M queries, where each query has values X and Y, the task is to find the sum of all integers less than X and greater than Y present in the array.
Note: X and Y may or may not be present in the array.

Examples:

Input: arr[] = [3 5 8 12 15], Q = {{5, 12}, {4, 8}}
Output:
18
30
Explanation:
For query 1, X = 5 and Y = 12. Sum = 3( < 5) + 15( > 12) = 18.
For query 2, X = 4 and Y = 8. Sum = 3( < 4) + 15 ( > 8) + 12 ( > 8) = 30.

Input: arr[] = [4 7 7 12 15], Q = {{3, 8}, {4, 8}}
Output:
27
27
Explanation:
For query 1, X = 3 and Y = 8. Sum = 12( > 8) + 15 ( > 8) = 27.
For query 2, Sum = 12 + 15 = 27.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

1. Build a prefix sum array where prefix_sum[i] denotes the sum of arr[0] + arr[1] + … arr[i].
2. Find the last index i which has a value less than X and extract the prefix sum up to the ith index. The index can be obtained in O(logN) complexity by using bisect_left() or lower_bound() in Python and C++ respectively..
3. Similarly, find the first index i in the array with a value greater than Y and calculate the sum prefix_sum[n-1] – prefix_sum[i-1]. Inbuilt functions bisect() and upper_bound() in Python and C++ respectively, perform the required operation in O(logN).
4. Sum of the two results calculated in the above two steps is the required answer. Keep repeating these steps for every query.

Below is the implementation of the above approach:

## Python3

 `# Python code for the above program``from` `bisect ``import` `bisect, bisect_left`` ` `def` `createPrefixSum(ar, n):`` ` `    ``# Initialize prefix sum ``    ``# array``    ``prefix_sum ``=` `[``0``]``*``n`` ` `    ``# Initialize prefix_sum[0]``    ``# by ar[0]``    ``prefix_sum[``0``] ``=` `ar[``0``]``     ` `    ``# Compute prefix sum for``    ``# all indices    ``    ``for` `i ``in` `range``(``1``, n):``        ``prefix_sum[i] ``=` `prefix_sum[i``-``1``]``+``ar[i]``    ``return` `prefix_sum`` ` `# Function to return sum of all``# elements less than X``def` `sumLessThanX(ar, x, prefix_xum):`` ` `    ``# Index of the last element ``    ``# which is less than x``    ``pos_x ``=` `bisect_left(ar, x) ``-` `1`` ` `     ` `    ``if` `pos_x >``=` `0` `:``        ``return` `prefix_sum[pos_x]``    ``# If no element is less than x``    ``else``:``        ``return` `0`` ` `# Function to return sum of all``# elements greater than Y``def` `sumGreaterThanY(ar, y, prefix_sum):`` ` `    ``# Index of the first element ``    ``# which is greater than y``    ``pos_y ``=` `bisect(ar, y)``    ``pos_y ``-``=` `1`` ` `    ``if` `pos_y < ``len``(ar)``-``1` `:``        ``return` `prefix_sum[``-``1``]``-``prefix_sum[pos_y]``    ``# If no element is greater than y``    ``else``:``        ``return` `0`` ` ` ` `def` `solve(ar, x, y, prefix_sum):``    ``ltx ``=` `sumLessThanX(ar, x, prefix_sum)``    ``gty ``=` `sumGreaterThanY(ar, y, prefix_sum)`` ` `    ``# printing the final sum``    ``print``(ltx ``+` `gty) `` ` ` ` `def` `print_l(lb, ub):``    ``print``(``"sum of integers less than {}"``.``format``(lb)``    ``+` `" and greater than {} is "``.``format``(ub),``    ``end ``=` `'')`` ` ` ` `if` `__name__ ``=``=` `'__main__'``:`` ` `    ``# example 1``    ``ar ``=` `[``3``, ``6``, ``6``, ``12``, ``15``]``    ``n ``=` `len``(ar)``    ``prefix_sum ``=` `createPrefixSum(ar, n)`` ` `    ``# for query 1``    ``q1x ``=` `5``    ``q1y ``=` `12``    ``print_l(q1x, q1y)``    ``solve(ar, q1x, q1y, prefix_sum)`` ` `    ``# for query 2``    ``q2x ``=` `7``    ``q2y ``=` `8``    ``print_l(q2x, q2y)``    ``solve(ar, q2x, q2y, prefix_sum)`
Output:
```sum of integers less than 5 and greater than 12 is 18
sum of integers less than 7 and greater than 8 is 42
```

Time Complexity: O(N + (M * logN))
Auxiliary Space complexity: O(N)

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