Skip to content

Latest commit

 

History

History
241 lines (203 loc) · 8.25 KB

explanation.md

File metadata and controls

241 lines (203 loc) · 8.25 KB
Raw

Prime Subtraction Operation - Step-by-Step Explanation

In this problem, we are given an array of integers, and our goal is to check if we can perform a series of operations to make the array strictly increasing by subtracting the largest prime number less than the difference between consecutive elements.

Problem Breakdown

  1. Prime Numbers: The main operation revolves around subtracting prime numbers from elements in the array.
  2. Strictly Increasing Array: We need to ensure that after each operation, the sequence remains strictly increasing.
  3. Prime Subtraction: For each element in the array, we subtract the largest prime number that is less than the difference between the current element and the next element.

Below, we explain the code solutions in different programming languages (C++, Java, JavaScript, Python, Go).

C++ Code Explanation

  1. Prime Precomputation using Sieve of Eratosthenes:

    • We use a bitset to mark numbers as prime or not, up to 1001 (to cover the potential range of differences).
    • For each number p, if it is prime, we mark its multiples as non-prime.
    • After applying the sieve, we store the primes in a vector prime.
    void sieve(int M) {
    bitset<1001> isPrime;
    isPrime.set();
    isPrime[0] = isPrime[1] = 0;
    for (int p = 2; p * p <= M; ++p) {
        if (isPrime[p]) {
            for (int j = p * p; j <= M; j += p) {
                isPrime[j] = 0;
            }
        }
    }
    for (int p = 2; p <= M; ++p) {
        if (isPrime[p]) prime.push_back(p);
    }
}

  2. Prime Subtraction Logic:

    • The primeSubOperation function iterates backward through the array.
    • For each element, it checks if the difference with the next element is positive.
    • It then finds the largest prime less than the difference using upper_bound to ensure the sequence remains strictly increasing.
    • If no such prime exists or the updated value becomes zero or negative, the function returns false.
    bool primeSubOperation(vector<int>& nums) {
    int n = nums.size(), M = *max_element(nums.begin(), nums.end());
    sieve(M);
    for (int i = n - 2; i >= 0; i--) {
        if (nums[i] >= nums[i + 1]) {
            auto it = upper_bound(prime.begin(), prime.end(), nums[i] - nums[i + 1]);
            if (it == prime.end()) return false;
            nums[i] -= *it;
        }
        if (nums[i] <= 0) return false;
    }
    return true;
}

Java Code Explanation

  1. Prime Checking:

    • The isPrime method checks if a number is prime by checking divisibility up to its square root.
    private boolean isPrime(int number) {
    if (number < 2) return false;
    for (int i = 2; i <= Math.sqrt(number); i++) {
        if (number % i == 0) {
            return false;
        }
    }
    return true;
}

  2. Prime Subtraction Logic:

    • The primeSubOperation function iterates through the array from left to right.
    • It calculates the bound for each element and checks if the current element is greater than the next element.
    • Then, it subtracts the largest prime less than the difference to make the sequence strictly increasing.
    • If the value becomes zero or negative, it returns false.
    public boolean primeSubOperation(int[] nums) {
    for (int i = 0; i < nums.length; i++) {
        int bound = (i == 0) ? nums[0] : nums[i] - nums[i - 1];
        if (bound <= 0) {
            return false;
        }
        int largestPrime = 0;
        for (int j = bound - 1; j >= 2; j--) {
            if (isPrime(j)) {
                largestPrime = j;
                break;
            }
        }
        nums[i] -= largestPrime;
    }
    return true;
}

JavaScript Code Explanation

  1. Prime Precomputation:

    • An array sieve of size 1001 is created, and all values are set to true to assume that they are prime.
    • We then mark the multiples of each prime number as non-prime.
    const sieve = new Array(1001).fill(true);
const primes = [];
for (let i = 2; i <= 1000; i++) {
    if (sieve[i]) {
        primes.push(i);
        for (let j = i * 2; j <= 1000; j += i) sieve[j] = false;
    }
}

  2. Prime Subtraction Logic:

    • The function iterates through the array in reverse order and checks the difference between consecutive elements.
    • It finds the smallest prime greater than the difference and subtracts it from the current element.
    • If the result is non-positive, it returns false.
    const primeSubOperation = function(nums) {
    for (let i = nums.length - 2; i >= 0; i--) {
        if (nums[i] < nums[i + 1]) continue;
        let target = nums[i] - nums[i + 1];
        let prime = primes.find(p => p > target);
        if (!prime) return false;
        nums[i] -= prime;
        if (nums[i] <= 0) return false;
    }
    return true;
};

Python Code Explanation

  1. Prime Precomputation using Sieve of Eratosthenes:

    • Similar to the C++ and JavaScript versions, we use the Sieve of Eratosthenes to compute all primes up to 1000.
    def sieve_of_eratosthenes(self, max_num):
    is_prime = [True] * (max_num + 1)
    is_prime[0] = is_prime[1] = False
    for i in range(2, int(math.sqrt(max_num)) + 1):
        if is_prime[i]:
            for j in range(i * i, max_num + 1, i):
                is_prime[j] = False
    return [i for i in range(2, max_num + 1) if is_prime[i]]

  2. Prime Subtraction Logic:

    • The function iterates backward through the array and checks whether the difference between consecutive elements is non-positive.
    • It finds the smallest prime greater than the difference and subtracts it from the current element.
    def primeSubOperation(self, nums):
    primes = self.sieve_of_eratosthenes(1000)
    for i in range(len(nums) - 2, -1, -1):
        if nums[i] < nums[i + 1]: continue
        target = nums[i] - nums[i + 1]
        prime = next((p for p in primes if p > target), None)
        if not prime: return False
        nums[i] -= prime
        if nums[i] <= 0: return False
    return True

Go Code Explanation

  1. Prime Precomputation:

    • The generatePrimes function uses the Sieve of Eratosthenes to find all primes up to a given maximum number.
    func generatePrimes(maxNum int) []int {
    isPrime := make([]bool, maxNum+1)
    for i := 2; i <= maxNum; i++ {
        isPrime[i] = true
    }
    for i := 2; i*i <= maxNum; i++ {
        if isPrime[i] {
            for j := i * i; j <= maxNum; j += i {
                isPrime[j] = false
            }
        }
    }
    var primes []int
    for i := 2; i <= maxNum; i++ {
        if isPrime[i] {
            primes = append(primes, i)
        }
    }
    return primes
}

  2. Prime Subtraction Logic:

    • The function iterates backward through the array, ensuring the sequence remains strictly increasing by subtracting the largest prime less than the difference.
    func primeSubOperation(nums []int) bool {
    primes := generatePrimes(1000)
    for i := len(nums) - 2; i >= 0; i-- {
        if nums[i] < nums[i+1] {
            continue
        }
        target := nums[i] - nums[i+1]
        for _, p := range primes {
            if p > target {
                nums[i] -= p
                break
            }
        }
        if nums[i] <= 0 || nums[i] >= nums[i+1] {
            return false
        }
    }
    return true
}

Conclusion

These implementations follow the same core logic, using the Sieve of Eratosthenes to precompute

the primes and applying the prime subtraction operation in reverse to ensure the array becomes strictly increasing. The solution works efficiently within the constraints and ensures correctness across different programming languages.