Pair with given product | Set 1 (Find if any pair exists)

Given an array of distinct elements and a number x, find if there is a pair with a product equal to x. 

Examples : 

Input : arr[] = {10, 20, 9, 40};
        int x = 400;
Output : Yes
Input : arr[] = {10, 20, 9, 40};
        int x = 190;
Output : No
Input : arr[] = {-10, 20, 9, -40};
        int x = 400;
Output : Yes
Input : arr[] = {-10, 20, 9, 40};
        int x = -400;
Output : Yes
Input : arr[] = {0, 20, 9, 40};
        int x = 0;
Output : Yes

Naive approach:  Run two loops to consider all possible pairs. For every pair, check if the product is equal to x or not.  

Yes
No

Time Complexity: O(n²)
Auxiliary Space: O(1)

Better Solution: We sort the given array. After sorting, we traverse the array and for every element arr[i], we do binary search for x/arr[i] in the subarray on the right of arr[i], i.e., in subarray arr[i+1..n-1]

Yes
No

Time Complexity: O(n log n)
Auxiliary Space: O(1)

Another Approach ( Using two pointer technique):

This approach to solve the problem is to sort the array in ascending order and then use Two pointer approach ( l = 0, r = arr.size()-1) to traverse that sorted array. If product of arr[l] and arr[r] is equal to x, then return true. If product is less than k then increase l else decrease r.

Algorithm:

1.  Define a function isProduct that takes an integer array arr, an integer n, and an integer x as inputs and returns a boolean value. The        function first sorts the array arr in ascending order. It then initializes two indices l and r to the beginning and end of the array,                      respectively. While l < r, the function calculates the product of the elements at indices l and r, compares it with x, and adjusts l and r    accordingly. If the product equals x, it returns true. Otherwise, if the product is less than x, it increments l to increase the value of the        product, and if the product is greater than x, it decrements r to decrease the value of the product. If no such pair exists in the array, the  function returns false.
2.  In the main function:
      a. Create an integer array according to the input specification.
      b. Initialize an integer n as the size of the array.
      c. Initialize an integer x according to the input specification.
      d. Call the isProduct function with the array, n, and x as inputs.
      e. Print “Yes” if the function returns true, and “No” otherwise.

Below is the implementation of the above idea:

Yes
No

Time Complexity: O(N * logN) where N is size of input array. This is because sort has been called which takes N*logN time.

Space Complexity: O(1) as no extra space has been taken.

Efficient Solution: We can improve time complexity to O(n) using hashing. Below are the steps.  

1. Create an empty hash table
2. Traverse array elements and do the following for every element arr[i]. 
   • If arr[i] is 0 and x is also 0, return true, else ignore arr[i].
   • If x % arr[i] is 0 and x/arr[i] exists in the table, it returns true.
   • Insert arr[i] into the hash table.
3. Return false

Below is the implementation of the above idea.  

Yes
No

Time Complexity : O(n)
Auxiliary Space: O(n)

In the next set, we will be discussing approaches to print all pairs with products equal to 0.