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Closest product pair in an array

  • Difficulty Level : Easy

Given an array of non-negative integers and a number x, find a pair in the array whose product is closest to x.

Examples:  

Input : arr[] = [2, 3, 5, 9]
        x = 47
Output : {5, 9}

Input : arr[] = [2, 3, 5, 9]
        x = 8
Output : {2, 5} 

Method 1 
A simple solution is to consider every pair and keep track of the closest pair (absolute difference between pair product and x is minimum). Finally, print the closest pair. The time complexity of this solution is O(n^2  )
 

Method 2 O(n Log n) 

  1. Sort the array 
  2. Initialize a variable diff as infinite (Diff is used to store the difference between a pair and x). We need to find the minimum diff. 
  3. Traverse the array and for each i, do the following : 
    • Find the lower bound for x/arr[i] in the sub-array on the right of arr[i], i.e., in subarray arr[i+1..n-1]. Let it be denoted by l. 
    • Find the upper bound for x/arr[i] in the sub array on the right of arr[i], i.e., in sub array arr[i+1..n-1]. Let it be denoted by u. 
    • If min(abs((arr[i] * l) – x), abs((arr[i] * u) – x)) < diff, then update diff and result 
       

Method 3 O(n for sorted) 
An efficient solution can find the pair in O(n) time. The following is the detailed algorithm.  



1) Initialize a variable diff as infinite 
   (Diff is used to store the difference 
   between pair and x).  We need to find 
   the minimum diff.
2) Initialize two index variables l and r
   in the given sorted array.
   (a) Initialize first to the leftmost index:  
        l = 0
   (b) Initialize second  the rightmost index: 
        r = n-1
3) Loop while l < r.
   (a) If  abs((arr[l] * arr[r]) - x) < diff  
       then update diff and result 
   (b) Else if((arr[l] * arr[r]) <  x)  then
       l++
   (c) Else r-- 

The following is the implementation of the above algorithm.  

C++




// Simple C++ program to find the pair with
// product closest to a given no.
#include <bits/stdc++.h>
using namespace std;
 
// Prints the pair with product closest to x
void printClosest(int arr[], int n, int x)
{
    int res_l, res_r; // To store indexes of result pair
 
    // Initialize left and right indexes and
    // difference between pair product and x
    int l = 0, r = n - 1, diff = INT_MAX;
 
    // While there are elements between l and r
    while (r > l) {
 
        // Check if this pair is closer than
        // the closest pair so far
        if (abs(arr[l] * arr[r] - x) < diff) {
            res_l = l;
            res_r = r;
            diff = abs(arr[l] * arr[r] - x);
        }
 
        // If this pair has more product,
        // move to smaller values.
        if (arr[l] * arr[r] > x)
            r--;
 
        else // Move to larger values
            l++;
    }
 
    cout << " The closest pair is "
        << arr[res_l] << " and " << arr[res_r];
}
 
// Driver program to test above functions
int main()
{
    int arr[] = { 2, 3, 5, 9 }, x = 8;
    int n = sizeof(arr) / sizeof(arr[0]);
    printClosest(arr, n, x);
    return 0;
}

Java




// Simple Java program to find
// the pair with product closest
// to a given no.
import java.io.*;
 
class GFG
{
// Prints the pair with
// product closest to x
static void printClosest(int arr[],
                         int n, int x)
{
    // To store indexes of result pair
    int res_l = 0, res_r = 0;
 
    // Initialize left and right
    // indexes and difference
    // between pair product and x
    int l = 0, r = n - 1, diff = Integer.MAX_VALUE;
 
    // While there are
    // elements between l and r
    while (r > l)
    {
 
        // Check if this pair is closer 
        // than the closest pair so far
        if (Math.abs(arr[l] * arr[r] - x) < diff)
        {
            res_l = l;
            res_r = r;
            diff = Math.abs(arr[l] * arr[r] - x);
        }
 
        // If this pair has more product,
        // move to smaller values.
        if (arr[l] * arr[r] > x)
            r--;
 
        // Move to larger values
        else
            l++;
    }
 
    System.out.print("The closest pair is ");
    System.out.print (arr[res_l] +
                         " and " +
                      arr[res_r]);
}
 
// Driver Code
public static void main (String[] args)
{
int arr[] = {2, 3, 5, 9};
int x = 8;
int n = arr.length;
printClosest(arr, n, x);
}
}
 
// This code is contributed by anuj_67.

Python3




# Simple Python3 program to find
# the pair with product closest
# to a given no.
import sys
 
# Prints the pair with
# product closest to x
def printClosest(arr, n, x):
     
    # To store indexes
    # of result pair
    res_l = 0;
    res_r = 0;
 
    # Initialize left and right
    # indexes and difference
    # between pair product and x
    l = 0;
    r = n - 1;
    diff = sys.maxsize;
 
    # While there are elements
    # between l and r
    while (r > l):
 
        # Check if this pair is
        # closer than the closest
        # pair so far
        if (abs(arr[l] *
                arr[r] - x) < diff):
            res_l = l;
            res_r = r;
            diff = abs(arr[l] *
                       arr[r] - x);
 
        # If this pair has more
        # product, move to smaller
        # values.
        if (arr[l] * arr[r] > x):
            r = r - 1;
 
        # Move to larger values
        else:
            l = l + 1;
 
    print("The closest pair is", arr[res_l] ,
                          "and", arr[res_r]);
 
# Driver Code
arr = [2, 3, 5, 9];
x = 8;
n = len(arr);
printClosest(arr, n, x);
 
# This code is contributed
# by rahul

C#




// Simple C# program to find
// the pair with product closest
// to a given no.
using System;
 
class GFG
{
// Prints the pair with
// product closest to x
static void printClosest(int []arr,
                         int n, int x)
{
    // To store indexes of result pair
    int res_l = 0, res_r = 0;
 
    // Initialize left and right
    // indexes and difference
    // between pair product and x
    int l = 0, r = n - 1,
        diff = int.MaxValue;
 
    // While there are
    // elements between l and r
    while (r > l)
    {
 
        // Check if this pair is closer
        // than the closest pair so far
        if (Math.Abs(arr[l] *
                     arr[r] - x) < diff)
        {
            res_l = l;
            res_r = r;
            diff = Math.Abs(arr[l] *
                            arr[r] - x);
        }
 
        // If this pair has more product,
        // move to smaller values.
        if (arr[l] * arr[r] > x)
            r--;
 
        // Move to larger values
        else
            l++;
    }
 
    Console.Write("The closest pair is ");
    Console.Write (arr[res_l] +
                      " and " +
                   arr[res_r]);
}
 
// Driver Code
public static void Main ()
{
int []arr = {2, 3, 5, 9};
int x = 8;
int n = arr.Length;
printClosest(arr, n, x);
}
}
 
// This code is contributed by anuj_67.

PHP




<?php
// Simple PHP program to find
// the pair with product closest
// to a given no.
 
// Prints the pair with
// product closest to x
function printClosest($arr, $n, $x)
{
    // To store indexes
    // of result pair
    $res_l; $res_r;
 
    // Initialize left and right
    // indexes and difference
    // between pair product and x
    $l = 0; $r = $n - 1; $diff = PHP_INT_MAX;
 
    // While there are elements
    // between l and r
    while ($r > $l)
    {
 
        // Check if this pair is
        // closer than the closest
        // pair so far
        if (abs($arr[$l] *
                $arr[$r] - $x) < $diff)
        {
            $res_l = $l;
            $res_r = $r;
            $diff = abs($arr[$l] *
                        $arr[$r] - $x);
        }
 
        // If this pair has more 
        // product, move to smaller
        // values.
        if ($arr[$l] * $arr[$r] > $x)
            $r--;
 
        // Move to larger values
        else
            $l++;
    }
 
    echo " The closest pair is " ,
          $arr[$res_l] , " and " ,
                     $arr[$res_r];
}
 
// Driver Code
$arr = array(2, 3, 5, 9);
$x = 8;
$n = count($arr);
printClosest($arr, $n, $x);
 
// This code is contributed by anuj_67.
?>

Javascript




<script>
 
// Simple Javascript program to find the pair with
// product closest to a given no.
 
// Prints the pair with product closest to x
function printClosest(arr, n, x)
{
    var res_l, res_r; // To store indexes of result pair
 
    // Initialize left and right indexes and
    // difference between pair product and x
    var l = 0, r = n - 1, diff = 10000000000;
 
    // While there are elements between l and r
    while (r > l) {
 
        // Check if this pair is closer than
        // the closest pair so far
        if (Math.abs(arr[l] * arr[r] - x) < diff) {
            res_l = l;
            res_r = r;
            diff = Math.abs(arr[l] * arr[r] - x);
        }
 
        // If this pair has more product,
        // move to smaller values.
        if (arr[l] * arr[r] > x)
            r--;
 
        else // Move to larger values
            l++;
    }
 
    document.write( " The closest pair is "
        + arr[res_l]  + " and " + arr[res_r]);
}
 
// Driver program to test above functions
var arr = [2, 3, 5, 9 ], x = 8;
var n = arr.length;
printClosest(arr, n, x);
 
 
</script>

Output :  

The closest pair is 2 and 5

 

 

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