Count of subsequences whose product is a difference of square of two integers

Given an array arr[] containing N elements which contain both positive and negative elements, the task is to find the total number of contiguous subsequences whose product can be expressed as the difference of the square of two integers.

Examples:

Input: arr[] = {1, 0, 2, 4, 5}
Output: 14
Explanation:
There are 14 subsequences whose product can be expressed as the difference of the square of two integers.
They are: {1}, {0}, {4}, {5}, {1, 0}, {1, 0, 2}, {1, 0, 2, 4}, {1, 0, 2, 4, 5}, {0, 2}, {0, 2, 4}, {0, 2, 4, 5}, {2, 4}, {2, 4, 5}, {4, 5}
The product of all the subsequences can be expressed as the difference of two squares. For example:
1 -> 1^2 – 0^2
0 -> 1^2 – 1^2
4 -> 2^2 – 0^2
5 -> 3^2 – 2^2
8 -> 3^2 – 1^2 …… and so on.

Input: arr[] = {-2, -7, 8, 9}
Output: 8
Explanation:
There are 8 subsequences whose product can be expressed as the difference of the square of two integers.
They are: {-7}, {8}, {9}, {-2, -7, 8}, {-2, -7, 8, 9}, {-7, 8}, {-7, 8, 9}, {8, 9}
The product of all the subsequences can be expressed as the difference of two squares. For example:
-7 -> 3^2 – 4^2
8 -> 3^2 – 1^2
9 -> 3^2 – 0^2
112 -> 11^2 – 3^2 …… and so on.

Naive approach: The naive approach for this problem is to generate all the contiguous subsequences and compute its product and simply check if that number can be expressed as the difference of two squares or not.



Below is the implementation of the above approach:

C++

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// C++ implementation to count the
// number of contiguous subsequences
// whose product can be expressed as
// the square of difference of two integers
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to count the number
// of contiguous subsequences
// whose product can be expressed
// as square of difference of two integers
int CntcontSubs(int a[], int n)
{
    int c = 0, d = 0, i, sum = 1, j;
  
    // Iterating through the array
    for (i = 0; i < n; i++) {
  
        // Check if that number can be
        // expressed as the square of
        // difference of two numbers
        if (a[i] % 2 != 0 || a[i] % 4 == 0)
            d++;
  
        // Variable to compute the product
        sum = a[i];
  
        // Finding the remaining subsequences
        for (j = i + 1; j < n; j++) {
            sum = sum * a[j];
  
            // Check if that number can be
            // expressed as the square of
            // difference of two numbers
            if (sum % 2 != 0 || sum % 4 == 0)
                c++;
        }
        sum = 1;
    }
  
    // Return the number of subsequences
    return c + d;
}
  
// Driver code
int main()
{
    int arr[] = { 5, 4, 2, 9, 8 };
    int n = sizeof(arr) / sizeof(arr[0]);
  
    cout << CntcontSubs(arr, n);
  
    return 0;
}

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Java

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// Java implementation to count the
// number of contiguous subsequences
// whose product can be expressed as
// the square of difference of two integers
  
class GFG{
   
// Function to count the number
// of contiguous subsequences
// whose product can be expressed
// as square of difference of two integers
static int CntcontSubs(int a[], int n)
{
    int c = 0, d = 0, i, sum = 1, j;
   
    // Iterating through the array
    for (i = 0; i < n; i++) {
   
        // Check if that number can be
        // expressed as the square of
        // difference of two numbers
        if (a[i] % 2 != 0 || a[i] % 4 == 0)
            d++;
   
        // Variable to compute the product
        sum = a[i];
   
        // Finding the remaining subsequences
        for (j = i + 1; j < n; j++) {
            sum = sum * a[j];
   
            // Check if that number can be
            // expressed as the square of
            // difference of two numbers
            if (sum % 2 != 0 || sum % 4 == 0)
                c++;
        }
        sum = 1;
    }
   
    // Return the number of subsequences
    return c + d;
}
   
// Driver code
public static void main(String[] args)
{
    int arr[] = { 5, 4, 2, 9, 8 };
    int n = arr.length;
   
    System.out.print(CntcontSubs(arr, n));
   
}
}
  
// This code contributed by PrinciRaj1992

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Python3

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# Python3 implementation to count the
# number of contiguous subsequences
# whose product can be expressed as
# the square of difference of two integers
  
# Function to count the number
# of contiguous subsequences
# whose product can be expressed
# as square of difference of two integers
def CntcontSubs(a, n):
    c = 0
    d = 0
    sum = 1
  
    # Iterating through the array
    for i in range(n):
  
        # Check if that number can be
        # expressed as the square of
        # difference of two numbers
        if (a[i] % 2 != 0 or a[i] % 4 == 0):
            d += 1
  
        # Variable to compute the product
        sum = a[i]
  
        # Finding the remaining subsequences
        for j in range(i + 1, n):
            sum = sum * a[j]
  
            # Check if that number can be
            # expressed as the square of
            # difference of two numbers
            if (sum % 2 != 0 or sum % 4 == 0):
                c += 1
        sum = 1
  
    # Return the number of subsequences
    return c + d
  
# Driver code
if __name__ == '__main__':
    arr=[5, 4, 2, 9, 8]
    n = len(arr)
  
    print(CntcontSubs(arr, n))
  
# This code is contributed by mohit kumar 29

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C#

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// C# implementation to count the 
// number of contiguous subsequences 
// whose product can be expressed as 
// the square of difference of two integers 
using System;
  
class GFG{ 
  
// Function to count the number 
// of contiguous subsequences 
// whose product can be expressed 
// as square of difference of two integers 
static int CntcontSubs(int []a, int n) 
    int c = 0, d = 0, i, sum = 1, j; 
  
    // Iterating through the array 
    for(i = 0; i < n; i++) 
    
          
       // Check if that number can be 
       // expressed as the square of 
       // difference of two numbers 
       if (a[i] % 2 != 0 || a[i] % 4 == 0) 
           d++; 
             
       // Variable to compute the product 
       sum = a[i]; 
         
       // Finding the remaining subsequences 
       for(j = i + 1; j < n; j++)
       
          sum = sum * a[j]; 
            
          // Check if that number can be 
          // expressed as the square of 
          // difference of two numbers 
          if (sum % 2 != 0 || sum % 4 == 0) 
              c++; 
       
       sum = 1; 
    
  
    // Return the number of subsequences 
    return c + d; 
  
// Driver code 
static void Main() 
    int []arr = { 5, 4, 2, 9, 8 }; 
    int n = arr.Length;
      
    Console.Write(CntcontSubs(arr, n)); 
  
// This code is contributed by grand_master 

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Output:

13

Time Complexity: O(N2) where N is the length of the array.

Efficient Approach: The idea lies behind the identity that a number cannot be expressed as the difference of two squares if it gives a remainder 2 when divided by 4. Therefore, the idea is to find all the subsequences which yield a product of 2 and subtract these from the total possible subsequences for the array. The total number of contiguous subsequences can be obtained by the formula: (N * (N + 1)) / 2

  • If the array contains the element 0, then the product of all the subsequences containing this element becomes 0. Therefore, all those subsequences can be expressed as the difference of two squares.
  • If any element gives the remainder 2 when the number is divided by 4, then all the subsequences up to the nearest 2 or 0 have to be avoided because the remainder becomes 4 when a 2 is encountered and it becomes 0 when a 0 is encountered.

Example:

  • Let arr[] = {6, 5, 13, 10, 4, 8, 14, 17}.
  • We compute the remainders when the elements are divided by 4. So, {2, 1, 1, 2, 0, 0, 2, 1} are the remainders of the given array.
  • Here we get the remainder 2 at the index 1, 4, 7. Let’s observe the 2 at the first index.
  • The following are the subsequences of the remainder array {2}, {2, 1}, {2, 1, 1}, {2, 1, 1, 2}, {2, 1, 1, 2, 0} … {2, 1, 1, 2, 0, 0, 2, 1}.
  • The product of the subsequences is {2, 2, 2, 4, 0 …. 0}.
  • So we get the product as 2 from index 1 to index 3. Hence, the total contiguous subsequences from index 1 to index 3 whose product is 2 are 2 (i.e.) [index-3 – index-1].
  • So, clearly, we find the nearest index of the element 2 or 0 and ignore all the subsequences until this index.

Below is the implementation of the above approach:

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// C++ implementation to count all the
// contiguous subsequences whose
// product is expressed as the square
// of the difference of two integers
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to count all the
// contiguous subsequences whose
// product is expressed as the square
// of the difference of two integers
int CntcontSubs(int a[], int n)
{
    int prod = 1;
  
    // Creating vectors to store
    // the remainders and the
    // subsequences
    vector<pair<int, int> > vect;
  
    vect.push_back(make_pair(0, 2));
  
    vector<int> two, zero;
  
    // Iterating through the array
    for (int i = 0; i < n; i++) {
  
        // Finding the remainder when the
        // element is divided by 4
        a[i] = a[i] % 4;
  
        // Bringing all the elements in
        // the range [0, 3]
        if (a[i] < 0)
            a[i] = a[i] + 4;
  
        // If the remainder is 2, store
        // the index of the
        if (a[i] == 2)
            two.push_back(i + 1);
  
        // If the remainder is 2, store
        // the index of the
        if (a[i] == 0)
            zero.push_back(i + 1);
  
        if (a[i] == 0 || a[i] == 2)
            vect.push_back(make_pair(i + 1, a[i]));
    }
    vect.push_back(make_pair(n + 1, 2));
  
    // Finding the total number of subsequences
    int total = (n * (n + 1)) / 2;
  
    // If there are no numbers which
    // yield the remainder 2
    if (two.empty())
        return total;
    else {
        int sum = 0;
  
        int pos1 = -1, pos2 = -1, pos3 = -1;
  
        int sz = vect.size();
  
        // Iterating through the vector
        for (int i = 1; i + 1 < sz; i++) {
  
            // If the element is 2, find the nearest
            // 2 or 0 and find the number of
            // elements between them
            if (vect[i].second == 2) {
                sum += (vect[i].first
                        - vect[i - 1].first)
                           * (vect[i + 1].first
                              - vect[i].first)
                       - 1;
            }
        }
  
        // Returning the count
        return total - sum - two.size();
    }
}
  
// Driver code
int main()
{
    int a[] = { 5, 4, 2, 9, 8 };
    int n = sizeof(a) / sizeof(a[0]);
  
    cout << CntcontSubs(a, n);
  
    return 0;
}

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Output:

13

Time Complexity: O(N) where N is the length of the array.

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