Sum of all array elements less than X and greater than Y for Q queries

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.

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

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# 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)

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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))
Auxillary Space complexity: O(N)

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