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 i^th 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: 

C++

#include<bits/stdc++.h>
using namespace std;

int* createPrefixSum(int ar[], int n) {
    // Initialize prefix sum array
    int* prefix_sum = new int[n];

    // Initialize prefix_sum[0] by ar[0]
    prefix_sum[0] = ar[0];

    // Compute prefix sum for all indices
    for (int i = 1; i < n; i++) {
        prefix_sum[i] = prefix_sum[i-1] + ar[i];
    }

    return prefix_sum;
}

int sumLessThanX(int ar[], int x, int prefix_sum[], int n) {
    // Index of the last element which is less than x
    int i = 0;
    while (i < n && ar[i] < x) {
        i++;
    }

    // If no element is less than x
    if (i == n) {
        return 0;
    }

    return prefix_sum[i-1];
}

int sumGreaterThanY(int ar[], int y, int prefix_sum[], int n) {
    // Index of the first element which is greater than y
    int i = n-1;
    while (i >= 0 && y < ar[i]) {
        i--;
    }

    // If no element is greater than y
    if (i < 0) {
        return 0;
    }

    return prefix_sum[n-1] - prefix_sum[i];
}

void solve(int ar[], int x, int y, int prefix_sum[], int n) {
    int ltx = sumLessThanX(ar, x, prefix_sum, n);
    int gty = sumGreaterThanY(ar, y, prefix_sum, n);

    // printing the final sum
    cout << ltx + gty << endl;
}

void print_l(int lb, int ub) {
    cout << "sum of integers less than " << lb << " and greater than " << ub << " is ";
}

int main() {
    // example 1
    int ar[] = {3, 6, 6, 12, 15};
    int n = sizeof(ar) / sizeof(ar[0]);
    int* prefix_sum = createPrefixSum(ar, n);

    // for query 1
    int q1x = 5;
    int q1y = 12;
    print_l(q1x, q1y);
    solve(ar, q1x, q1y, prefix_sum, n);

    // for query 2
    int q2x = 7;
    int q2y = 8;
    print_l(q2x, q2y);
    solve(ar, q2x, q2y, prefix_sum, n);

    return 0;
}

Java

/*package whatever //do not write package name here */
import java.util.*;
class GFG {

static int[] createPrefixSum(int []ar,int n){

   // Initialize prefix sum
    // array
    int prefix_sum[]=  new int[n];

    // Initialize prefix_sum[0]
    // by ar[0]
    prefix_sum[0] = ar[0];

    // Compute prefix sum for
    // all indices

    for(int i=1;i<n;i++){
        prefix_sum[i] = prefix_sum[i-1]+ar[i];
    }

    return prefix_sum;
}

// Function to return sum of all
// elements less than X
static int sumLessThanX(int []ar,int x,int []prefix_sum){

    // Index of the last element
    // which is less than x

  int i = 0;

    while(i<ar.length && ar[i]<x){

      i++;

    }
   // If no element is less than x
  if(i==ar.length) return 0;


    return prefix_sum[i-1];
}

// Function to return sum of all
// elements greater than Y
static int sumGreaterThanY(int []ar, int y,int[] prefix_sum){

    // Index of the first element
    // which is greater than y

  int i=ar.length-1;

  while(i>=0 && y<ar[i]){
      i--;
  }

  // If no element is greater than y
  if(i<0){
      return 0;
  }


   return prefix_sum[prefix_sum.length-1]-prefix_sum[i];

}
static void solve(int []ar, int x, int y, int []prefix_sum){
    int ltx = sumLessThanX(ar, x, prefix_sum);
    int gty = sumGreaterThanY(ar, y, prefix_sum);

    // printing the final sum
    System.out.println(ltx + gty);
 }

static void print_l(int lb, int ub){
    System.out.print("sum of integers less than " + lb + " and greater than " + ub + " is ");
 }


    public static void main (String[] args) {
        // example 1
    int ar[] = {3, 6, 6, 12, 15};
    int n = ar.length;
    int []prefix_sum = createPrefixSum(ar, n);

    // for query 1
    int q1x = 5;
    int q1y = 12;
    print_l(q1x, q1y);
    solve(ar, q1x, q1y, prefix_sum);

    // for query 2
    int q2x = 7;
    int q2y = 8;
    print_l(q2x, q2y);
    solve(ar, q2x, q2y, prefix_sum);
    }
}

// This code is contributed by aadityaburujwale.

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)

C#

// C# code for the above program
using System;

class GFG {

static int[] createPrefixSum(int[] ar,int n){

// Initialize prefix sum
    // array
    int[] prefix_sum= new int[n];

    // Initialize prefix_sum[0]
    // by ar[0]
    prefix_sum[0] = ar[0];

    // Compute prefix sum for
    // all indices

    for(int i=1;i<n;i++){
        prefix_sum[i] = prefix_sum[i-1]+ar[i];
    }

    return prefix_sum;
}

// Function to return sum of all
// elements less than X
static int sumLessThanX(int[] ar,int x,int[] prefix_sum){

    // Index of the last element
    // which is less than x

int i = 0;

    while(i<ar.Length && ar[i]<x){

    i++;

    }
// If no element is less than x
if(i==ar.Length) return 0;


    return prefix_sum[i-1];
}

// Function to return sum of all
// elements greater than Y
static int sumGreaterThanY(int[] ar, int y,int[] prefix_sum){

    // Index of the first element
    // which is greater than y

int i=ar.Length-1;

while(i>=0 && y<ar[i]){
    i--;
}

// If no element is greater than y
if(i<0){
    return 0;
}


return prefix_sum[prefix_sum.Length-1]-prefix_sum[i];

}
static void solve(int []ar, int x, int y, int []prefix_sum){
    int ltx = sumLessThanX(ar, x, prefix_sum);
    int gty = sumGreaterThanY(ar, y, prefix_sum);

    // printing the final sum
    Console.WriteLine(ltx + gty);
}

static void print_l(int lb, int ub){
    Console.Write("sum of integers less than " + lb + " and greater than " + ub + " is ");
}


    static public void Main () {
        // example 1
    int[] ar = {3, 6, 6, 12, 15};
    int n = ar.Length;
    int[] prefix_sum = createPrefixSum(ar, n);

    // for query 1
    int q1x = 5;
    int q1y = 12;
    print_l(q1x, q1y);
    solve(ar, q1x, q1y, prefix_sum);

    // for query 2
    int q2x = 7;
    int q2y = 8;
    print_l(q2x, q2y);
    solve(ar, q2x, q2y, prefix_sum);
    }
}

// This code is contributed by Aman Kumar.

Javascript

function createPrefixSum(ar) {
  // Initialize prefix sum array
  const prefix_sum = new Array(ar.length);

  // Initialize prefix_sum[0] by ar[0]
  prefix_sum[0] = ar[0];

  // Compute prefix sum for all indices
  for (let i = 1; i < ar.length; i++) {
    prefix_sum[i] = prefix_sum[i-1] + ar[i];
  }

  return prefix_sum;
}

// Function to return sum of all elements less than X
function sumLessThanX(ar, x, prefix_sum) {
  // Index of the last element which is less than x
  let i = 0;

  while (i < ar.length && ar[i] < x) {
    i++;
  }

  // If no element is less than x
  if (i == ar.length) {
    return 0;
  }

  return prefix_sum[i-1];
}

// Function to return sum of all elements greater than Y
function sumGreaterThanY(ar, y, prefix_sum) {
  // Index of the first element which is greater than y
  let i = ar.length-1;

  while (i >= 0 && y < ar[i]) {
    i--;
  }

  // If no element is greater than y
  if (i < 0) {
    return 0;
  }

  return prefix_sum[prefix_sum.length-1] - prefix_sum[i];
}

function solve(ar, x, y, prefix_sum) {
  const ltx = sumLessThanX(ar, x, prefix_sum);
  const gty = sumGreaterThanY(ar, y, prefix_sum);

  // Printing the final sum
  console.log(ltx + gty);
}

function print_l(lb, ub) {
  console.log("sum of integers less than " + lb + " and greater than " + ub + " is ");
}

// example 1
const ar = [3, 6, 6, 12, 15];
const prefix_sum = createPrefixSum(ar);

// for query 1
const q1x = 5;
const q1y = 12;
print_l(q1x, q1y);
solve(ar, q1x, q1y, prefix_sum);

// for query 2
const q2x = 7;
const 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: O(N)