Check if an element is present in an array using at most floor(N / 2) + 2 comparisons

Given an array A[] of size N and an integer X, the task is to check if X exists in A[] with no more than floor(N/2) + 2 comparisons. 
Note: For any index i, (i < N) or (A[i] == X) are considered as separate comparisons.

Examples:

Input: A[] = {-3, 5, 11, 3, 100, 2, 88, 22, 7, 900, 23, 4, 1}, X = 88
Output: Yes 8
Explanation: X = 88 exists in the given array, A[] and is detected with 8 comparisons.

Input: A[]= {-3, 5, 11, 3, 100, 2, 88, 22, 7, 900, 23, 4, 1}, X = 6
Output: No
Explanation: X = 6 doesn't exist in the given array, A[].

Approach: Follow the steps to solve the problem:

• Initialize a variable, say T as 1, to store product of all array elements - X i.e (A[i] - X)
• Initialize a variable, say comparisons as 0, to store the number of comparisons required.
• Initialize pointer, i as 0 to traverse the array.
• If the value of N is odd, increment comparisons by 1 because parity of N is checked and update T to T * (A[0] - X) and i to 1.
• Therefore, the number of elements in the range [i, N - 1] i.e N - i is always even.
• Traverse the array, A[] in range [i, N-1] and perform the following steps:
  • Update the value of T to T * (A[i] - X) * (A[i + 1] - X).
  • Update i to i + 2 and increment comparisons by 1 because condition i < N is checked.
• If the value of T is 0, increment comparisons by 1 because the equality of T is compared. Therefore, X exists in A[] and print "Yes" and number of comparisons.
• Otherwise, Print "No" as the result.
• The algorithm guarantees that the number of comparisons ? floor(N / 2) + 2.

Below is the implementation of the above approach:

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to check whether X
// is present in the array A[]
void findElement(int A[], int N, int X)
{
    // Initialise a pointer
    int i = 0;

    // Store the number
    // of comparisons
    int Comparisons = 0;

    // Variable to store product
    int T = 1;

    string Found = "No";

    // Check is N is odd
    Comparisons++;
    if (N % 2 == 1) {

        // Update i and T
        i = 1;
        T *= (A[0] - X);
    }

    // Traverse the array
    for (; i < N; i += 2) {

        // Check if i < N
        Comparisons += 1;

        // Update T
        T *= (A[i] - X);
        T *= (A[i + 1] - X);
    }

    // Check if T is equal to 0
    Comparisons += 1;
    if (T == 0) {
        cout << "Yes " << Comparisons;
    }
    else {
        cout << "No";
    }
}

// Driver Code
int main()
{
    // Given Input
    int A[] = { -3, 5, 11, 3, 100, 2, 88,
                22, 7, 900, 23, 4, 1 };
    int N = sizeof(A) / sizeof(A[0]);
    int X = 1;

    // Function Call
    findElement(A, N, X);

    return 0;
}

// Java program for the above approach
public class GFG {

    // Function to check whether X
    // is present in the array A[]
    static void findElement(int[] A, int N, int X)
    {

        // Initialise a pointer
        int i = 0;

        // Store the number
        // of comparisons
        int Comparisons = 0;

        // Variable to store product
        int T = 1;

        // Check is N is odd
        Comparisons++;
        if (N % 2 == 1) {

            // Update i and T
            i = 1;
            T *= (A[0] - X);
        }

        // Traverse the array
        for (; i < N; i += 2) {

            // Check if i < N
            Comparisons += 1;

            // Update T
            T *= (A[i] - X);
            T *= (A[i + 1] - X);
        }

        // Check if T is equal to 0
        Comparisons += 1;

        if (T == 0) {
            System.out.println("Yes " + Comparisons);
        }
        else {
            System.out.println("No");
        }
    }

    // Driver code
    public static void main(String[] args)
    { 
      // Given Input
        // Given Input
        int[] A = { -3, 5, 11,  3,  100, 2, 88,
                    22, 7, 900, 23, 4,   1 };
        int N = A.length;
        int X = 1;

        // Function Call
        findElement(A, N, X);
    }
}

// This code is contributed by abhinavjain194

# Python 3 program for the above approach

# Function to check whether X
# is present in the array A[]
def findElement(A, N, X):
  
    # Initialise a pointer
    i = 0

    # Store the number
    # of comparisons
    Comparisons = 0

    # Variable to store product
    T = 1

    Found = "No"

    # Check is N is odd
    Comparisons += 1
    if (N % 2 == 1):

        # Update i and T
        i = 1
        T *= (A[0] - X)

    # Traverse the array
    while(i< N):
      
        # Check if i < N
        Comparisons += 1

        # Update T
        T *= (A[i] - X)
        T *= (A[i + 1] - X)
        i += 2

    # Check if T is equal to 0
    Comparisons += 1
    if (T == 0):
        print("Yes",Comparisons)
    else:
        print("No")

# Driver Code
if __name__ == '__main__':
  
    # Given Input
    A = [-3, 5, 11, 3, 100, 2, 88, 22, 7, 900, 23, 4, 1]
    N = len(A)
    X = 1

    # Function Call
    findElement(A, N, X)
    
    # This code is contributed by bgangwar59.

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

class GFG{
    
// Function to check whether X
// is present in the array A[]
static void findElement(int[] A, int N, int X)
{
    
    // Initialise a pointer
    int i = 0;

    // Store the number
    // of comparisons
    int Comparisons = 0;

    // Variable to store product
    int T = 1;

    // Check is N is odd
    Comparisons++;
    if (N % 2 == 1)
    {
        
        // Update i and T
        i = 1;
        T *= (A[0] - X);
    }

    // Traverse the array
    for(; i < N; i += 2)
    {
        
        // Check if i < N
        Comparisons += 1;

        // Update T
        T *= (A[i] - X);
        T *= (A[i + 1] - X);
    }

    // Check if T is equal to 0
    Comparions += 1;
    
    if (T == 0) 
    {
        Console.Write("Yes " + Comparisons);
    }
    else
    {
        Console.Write("No");
    }
}

// Driver Code
public static void Main()
{
    
    // Given Input
    int[] A = { -3, 5, 11, 3, 100, 2, 88,
                22, 7, 900, 23, 4, 1 };
    int N = A.Length;
    int X = 1;

    // Function Call
    findElement(A, N, X);
}
}

// This code is contributed by ukasp

<script>

// JavaScript program for the above approach

// Function to check whether X
// is present in the array A[]
function findElement(A, N, X)
{
    // Initialise a pointer
    var i = 0;

    // Store the number
    // of comparisons
    var Comparisons = 0;

    // Variable to store product
    var T = 1;

    var Found = "No";

    // Check is N is odd
    Comparisons += 1;
    if (N % 2 == 1) {

        // Update i and T
        i = 1;
        T *= (A[0] - X);
    }

    // Traverse the array
    for (; i < N; i += 2) {

        // Check if i < N
        Comparisons += 1;

        // Update T
        T *= (A[i] - X);
        T *= (A[i + 1] - X);
    }

    // Check if T is equal to 0
    Comparisons += 1;
    if (T == 0) {
        document.write("Yes " + Comparisons);
    }
    else {
        document.write("No");
    }
}

// Driver Code
    // Given Input
    var A = [-3, 5, 11, 3, 100, 2, 88,
                22, 7, 900, 23, 4, 1];
    var N = A.length;
    var X = 1;

    // Function Call
    findElement(A, N, X);

</script>

Yes 8

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