Prime Subtraction Operation

class Solution {
public:
    bool isSubOperationPrime(vector<int>& elements) {
        int maximum = *max_element(elements.begin(), elements.end());
    
        // Prepare the prime check array.
        vector<int> primeCheck(maximum + 1, 1);
        primeCheck[1] = 0;
        for (int i = 2; i <= sqrt(maximum); i++) {
            if (primeCheck[i] == 1) {
                for (int j = i * i; j <= maximum; j += i) {
                    primeCheck[j] = 0;
                }
            }
        }
    
        // Initialize necessary values and index.
        int desiredValue = 1;
        int index = 0;
        while (index < elements.size()) {
            // Calculate the difference required to match the desiredValue.
            int requiredDifference = elements[index] - desiredValue;
    
            // Check if the requiredDifference is negative, implying we cannot match.
            if (requiredDifference < 0) {
                return false;
            }
    
            // If the requiredDifference is either prime or zero, proceed to update.
            if (primeCheck[requiredDifference] == 1 or requiredDifference == 0) {
                index++;
                desiredValue++;
            } else {
                // Advance to the next desiredValue if the current is unmatched.
                desiredValue++;
            }
        }
        return true;
    }
};

Summarize the "Prime Subtraction Operation" code written in C++:

• The code defines a function isSubOperationPrime that takes a vector of integers as input.
• The function computes the maximum value in the given vector to determine the range for creating a sieve of prime numbers.
• A vector named primeCheck is utilized to maintain the status (prime or not) for numbers up to the computed maximum. It initializes all indices to 1 (indicating prime) except for index 1, which is set to 0 (not prime).
• A Sieve of Eratosthenes algorithm follows, efficiently marking non-prime numbers (using inner loops starting from squares of each prime).
• The function iteratively checks if the differences (subtraction of a desired value from each element of the vector) are prime. This is done in a loop where:
  • If the differences are negative, the function returns false since the target cannot be achieved.
  • For non-negative differences, if the difference is zero or a prime number, the function successfully moves to the next element.
  • If a non-prime difference is found, the desired value is incremented without moving to the next index, aiming to find a prime difference in subsequent checks.
• If the end of the vector is reached while successfully finding prime differences or zeros for all elements, the function returns true, indicating that it is possible to perform the operation as per the given conditions.

This implementation effectively checks whether a series of prime substractions can meet specified conditions using the vector elements to sequentially reach incremented target values.

class Solution {
    
    public boolean isSequenceValid(int[] sequence) {
        int maximum = findMaximum(sequence);
    
        // Initialize sieve for prime checking.
        boolean[] isPrime = new boolean[maximum + 1];
        initializeArray(isPrime, true);
        isPrime[1] = false;
    
        for (int p = 2; p * p <= maximum; p++) {
            if (isPrime[p]) {
                for (int multiple = p * p; multiple <= maximum; multiple += p) {
                    isPrime[multiple] = false;
                }
            }
        }
    
        // Iterate through the input array.
        int expectedValue = 1;
        int index = 0;
        while (index < sequence.length) {
            int diff = sequence[index] - expectedValue;
    
            // Verify if the current number is smaller than expected.
            if (diff < 0) {
                return false;
            }
    
            // Check if the difference is zero or a prime number.
            if (isPrime[diff] || diff == 0) {
                index++;
                expectedValue++;
            } else {
                expectedValue++;
            }
        }
        return true;
    }
    
    private int findMaximum(int[] data) {
        int maxVal = Integer.MIN_VALUE;
        for (int value : data) {
            if (value > maxVal) {
                maxVal = value;
            }
        }
        return maxVal;
    }
    
    private void initializeArray(boolean[] arr, boolean value) {
        for (int i = 0; i < arr.length; i++) {
            arr[i] = value;
        }
    }
}

The Java program titled "Prime Subtraction Operation" focuses on determining the validity of a sequence based on specific criteria involving prime numbers and subtraction operations. The main functionality is encapsulated within the isSequenceValid method of the Solution class.

• The method begins by determining the highest value in the given integer array sequence using the helper method findMaximum. This maximum value establishes the range for a prime number sieve.
• A boolean array, isPrime, is initialized to keep track of prime numbers up to the maximum value found. This involves setting all entries to true initially and applying the Sieve of Eratosthenes algorithm to filter out non-prime numbers.
• The primary logic sequentially checks each number in sequence. For each number, it computes the difference between the number and an expectedValue, which starts at 1 and increments either when the difference is zero or a prime number.
• The loop continues to verify if each subtraction occurs with either a prime number or is zero. If the difference is negative or if it's not zero and a non-prime, the sequence is deemed invalid.
• The auxiliary methods, findMaximum and initializeArray, support maximum value determination and array initialization, respectively.

This carefully structured implementation ensures that the sequence adheres to the required conditions of formation based on prime numbers, making it efficient and easy to understand.

class Solution:
    def checkPrimeSub(self, sequence):
        highest_value = max(sequence)
    
        # Initialize the prime check array.
        is_prime = [1] * (highest_value + 1)
        is_prime[1] = 0
        for x in range(2, int(math.sqrt(highest_value + 1)) + 1):
            if is_prime[x] == 1:
                for k in range(x * x, highest_value + 1, x):
                    is_prime[k] = 0
    
        # Set initial conditions for the process.
        target_value = 1
        idx = 0
        while idx < len(sequence):
            # Calculate the difference for adjustments.
            diff = sequence[idx] - target_value
    
            # If the sequence value is smaller, not possible, return false.
            if diff < 0:
                return False
    
            # Check if the difference is prime or if there's no need to adjust.
            if is_prime[diff] or diff == 0:
                idx += 1
                target_value += 1
            else:
                # Increment target for the next iteration attempt.
                target_value += 1
        return True

The problem, Prime Subtraction Operation, is solved using a Python class Solution which includes the method checkPrimeSub. This method determines if a sequence of integers can be adjusted to a target sequence where each integer is incremented by 1 from the previous one, solely using differences that are prime numbers.

To achieve this, follow the steps below:

1. Identify and store the maximum value within the sequence. This is used to set boundaries for the prime number determination.
2. Implement the Sieve of Eratosthenes algorithm to create an array called is_prime. This array flags whether numbers up to the highest value in the sequence are prime.
3. Initialize the target starting point from which to compare sequence values—start at 1.
4. Traverse the sequence, comparing each value to the corresponding target value:
   • Compute the difference between the current sequence value and the target.
   • If the difference is negative, return False because adjusting to a lower value isn't feasible.
   • If the difference is prime or zero (no adjustment needed), move to the next integer and increment the target by 1.
   • If the difference isn't prime, increment the target without checking further numbers in the sequence.
5. Return True if all adjustments in the sequence are via prime number differences or require no change (difference of zero).

This approach ensures that the solution checks both the feasibility of achieving target differences using prime numbers and efficiently validates prime numbers through the sieve algorithm. This method efficiently handles sequences of varying lengths and values, making it robust for any input adhered to the provided constraints.