Two elements whose sum is closest to zero

2.3

Question: An Array of integers is given, both +ve and -ve. You need to find the two elements such that their sum is closest to zero.

For the below array, program should print -80 and 85.

METHOD 1 (Simple)
For each element, find the sum of it with every other element in the array and compare sums. Finally, return the minimum sum.

Implementation:

C

# include <stdio.h>
# include <stdlib.h> /* for abs() */
# include <math.h>
void minAbsSumPair(int arr[], int arr_size)
{
  int inv_count = 0;
  int l, r, min_sum, sum, min_l, min_r;

  /* Array should have at least two elements*/
  if(arr_size < 2)
  {
    printf("Invalid Input");
    return;
  }

  /* Initialization of values */
  min_l = 0;
  min_r = 1;
  min_sum = arr[0] + arr[1];

  for(l = 0; l < arr_size - 1; l++)
  {
    for(r = l+1; r < arr_size; r++)
    {
      sum = arr[l] + arr[r];
      if(abs(min_sum) > abs(sum))
      {
        min_sum = sum;
        min_l = l;
        min_r = r;
      }
    }
  }

  printf(" The two elements whose sum is minimum are %d and %d",
          arr[min_l], arr[min_r]);
}

/* Driver program to test above function */
int main()
{
  int arr[] = {1, 60, -10, 70, -80, 85};
  minAbsSumPair(arr, 6);
  getchar();
  return 0;
}

Java

import java.util.*;
import java.lang.*;
class Main
{
    static void minAbsSumPair(int arr[], int arr_size)
    {
      int inv_count = 0;
      int l, r, min_sum, sum, min_l, min_r;
     
      /* Array should have at least two elements*/
      if(arr_size < 2)
      {
        System.out.println("Invalid Input");
        return;
      }
     
      /* Initialization of values */
      min_l = 0;
      min_r = 1;
      min_sum = arr[0] + arr[1];
     
      for(l = 0; l < arr_size - 1; l++)
      {
        for(r = l+1; r < arr_size; r++)
        {
          sum = arr[l] + arr[r];
          if(Math.abs(min_sum) > Math.abs(sum))
          {
            min_sum = sum;
            min_l = l;
            min_r = r;
          }
        }
      }
     
      System.out.println(" The two elements whose "+
                              "sum is minimum are "+
                        arr[min_l]+ " and "+arr[min_r]);
    }
    
    // main function
    public static void main (String[] args) 
    {
        int arr[] = {1, 60, -10, 70, -80, 85};
        minAbsSumPair(arr, 6);
    }
    
}


Output:

The two elements whose sum is minimum are -80 and 85

Time complexity: O(n^2)



METHOD 2 (Use Sorting)
Thanks to baskin for suggesting this approach. We recommend to read this post for background of this approach.

Algorithm
1) Sort all the elements of the input array.
2) Use two index variables l and r to traverse from left and right ends respectively. Initialize l as 0 and r as n-1.
3) sum = a[l] + a[r]
4) If sum is -ve, then l++
5) If sum is +ve, then r–
6) Keep track of abs min sum.
7) Repeat steps 3, 4, 5 and 6 while l < r Implementation

C

# include <stdio.h>
# include <math.h>
# include <limits.h>

void quickSort(int *, int, int);

/* Function to print pair of elements having minimum sum */
void minAbsSumPair(int arr[], int n)
{
  // Variables to keep track of current sum and minimum sum
  int sum, min_sum = INT_MAX;

  // left and right index variables
  int l = 0, r = n-1;

  // variable to keep track of the left and right pair for min_sum
  int min_l = l, min_r = n-1;

  /* Array should have at least two elements*/
  if(n < 2)
  {
    printf("Invalid Input");
    return;
  }

  /* Sort the elements */
  quickSort(arr, l, r);

  while(l < r)
  {
    sum = arr[l] + arr[r];

    /*If abs(sum) is less then update the result items*/
    if(abs(sum) < abs(min_sum))
    {
      min_sum = sum;
      min_l = l;
      min_r = r;
    }
    if(sum < 0)
      l++;
    else
      r--;
  }

  printf(" The two elements whose sum is minimum are %d and %d",
          arr[min_l], arr[min_r]);
}

/* Driver program to test above function */
int main()
{
  int arr[] = {1, 60, -10, 70, -80, 85};
  int n = sizeof(arr)/sizeof(arr[0]);
  minAbsSumPair(arr, n);
  getchar();
  return 0;
}

/* FOLLOWING FUNCTIONS ARE ONLY FOR SORTING
    PURPOSE */
void exchange(int *a, int *b)
{
  int temp;
  temp = *a;
  *a   = *b;
  *b   = temp;
}

int partition(int arr[], int si, int ei)
{
  int x = arr[ei];
  int i = (si - 1);
  int j;

  for (j = si; j <= ei - 1; j++)
  {
    if(arr[j] <= x)
    {
      i++;
      exchange(&arr[i], &arr[j]);
    }
  }

  exchange (&arr[i + 1], &arr[ei]);
  return (i + 1);
}

/* Implementation of Quick Sort
arr[] --> Array to be sorted
si  --> Starting index
ei  --> Ending index
*/
void quickSort(int arr[], int si, int ei)
{
  int pi;    /* Partitioning index */
  if(si < ei)
  {
    pi = partition(arr, si, ei);
    quickSort(arr, si, pi - 1);
    quickSort(arr, pi + 1, ei);
  }
}

Java

import java.util.*;
import java.lang.*;
class Main
{
    static void minAbsSumPair(int arr[], int n)
    {
      // Variables to keep track of current sum and minimum sum
      int sum, min_sum = 999999;
     
      // left and right index variables
      int l = 0, r = n-1;
     
      // variable to keep track of the left and right pair for min_sum
      int min_l = l, min_r = n-1;
     
      /* Array should have at least two elements*/
      if(n < 2)
      {
        System.out.println("Invalid Input");
        return;
      }
     
      /* Sort the elements */
      sort(arr, l, r);
     
      while(l < r)
      {
        sum = arr[l] + arr[r];
     
        /*If abs(sum) is less then update the result items*/
        if(Math.abs(sum) < Math.abs(min_sum))
        {
          min_sum = sum;
          min_l = l;
          min_r = r;
        }
        if(sum < 0)
          l++;
        else
          r--;
      }
     
      
      System.out.println(" The two elements whose "+
                              "sum is minimum are "+
                        arr[min_l]+ " and "+arr[min_r]);
    }
     
    // main function
    public static void main (String[] args) 
    {
        int arr[] = {1, 60, -10, 70, -80, 85};
        int n = arr.length;
        minAbsSumPair(arr, n);
    }
    
    /* Functions for QuickSort */
    
    /* This function takes last element as pivot,
       places the pivot element at its correct
       position in sorted array, and places all
       smaller (smaller than pivot) to left of
       pivot and all greater elements to right
       of pivot */
    static int partition(int arr[], int low, int high)
    {
        int pivot = arr[high]; 
        int i = (low-1); // index of smaller element
        for (int j=low; j<high; j++)
        {
            // If current element is smaller than or
            // equal to pivot
            if (arr[j] <= pivot)
            {
                i++;

                // swap arr[i] and arr[j]
                int temp = arr[i];
                arr[i] = arr[j];
                arr[j] = temp;
            }
        }

        // swap arr[i+1] and arr[high] (or pivot)
        int temp = arr[i+1];
        arr[i+1] = arr[high];
        arr[high] = temp;

        return i+1;
    }


    /* The main function that implements QuickSort()
      arr[] --> Array to be sorted,
      low  --> Starting index,
      high  --> Ending index */
    static void sort(int arr[], int low, int high)
    {
        if (low < high)
        {
            /* pi is partitioning index, arr[pi] is 
              now at right place */
            int pi = partition(arr, low, high);

            // Recursively sort elements before
            // partition and after partition
            sort(arr, low, pi-1);
            sort(arr, pi+1, high);
        }
    }
}

Output:

The two elements whose sum is minimum are -80 and 85


Time Complexity:
complexity to sort + complexity of finding the optimum pair = O(nlogn) + O(n) = O(nlogn)

Asked by Vineet.

Please write comments if you find any bug in the above program/algorithm or other ways to solve the same problem.

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