Algorithmic Approach for Sifting Prime Numbers Using the Eratosthenes Method in C++
Cheeky Code Corner:
Wanna find all the prime numbers less than or equal to a given number, n? No biggie! Here's a little funky C++ program with just the right touch of sass to do the trick. This beauty goes by the name of the Sieve of Eratosthenes, but we like to call it the Prime Puncher 'cause it packs a punch, boy!
```cpp
void primePuncher(int n) { // Create a Boolean array, prime, of size n+1 and init all as true std::vector
}
int main() { int n; std::cout << "Enter a number to find all prime numbers up to: "; std::cin >> n; primePuncher(n); return 0;}```
Breakdown:
- Initiation:
- Create a boolean vector of size and initialize all elements as .
- Set and to , as they are not prime numbers.
- Siege Time:
- Start punching primes from 2 and keep going until you reach the square root of .
- For each prime , if it's still standing tall, make all its multiples fall with grace by marking them as false.
- Parade of Champions:
- Round up the remaining survivors! Print all the prime numbers below or equal to n.
Caveat:
- Time Complexity: (O(n \log(\log n))), thanks to our iterative loop and marking multiples one by one.
- Space Complexity: (O(n)), as we need a vector of size n to keep count of all the prime numbers.
Quick Side Note:We busted out O(1) auxiliary space as it's a hot mess with the Sieve of Eratosthenes. It's just one of those things that inherently requires more space to keep track of prime numbers. So, we spared you the confusion!
- In the given C++ program, the vector is a trie, where each position represents a number, and the boolean value or indicates whether that number is prime.
- Although the vector acts as a trie, it is used for a different purpose in the context of the program - to store the status of the prime numbers from 0 to n instead of creating a data structure for words. It should be noted that this is not a standard use of a trie in computer science, as this particular implementation doesn't follow the tree-like structure of a nucleotide.
