Time Complexity of Sieve of Eratosthenes Algorithm

The Sieve of Eratosthenes algorithm is an efficient method for finding all prime numbers up to a given limit. It works by iteratively marking the multiples of each prime number, starting from 2, as composite (not prime). The remaining unmarked numbers are prime.

The time complexity of the Sieve of Eratosthenes algorithm is O(n), where n is the given limit.

Explanation:

The algorithm starts by assuming that all numbers from 2 to the given limit are prime. Then, it iteratively marks the multiples of each prime number as composite. The algorithm terminates when the square of the current prime number exceeds the given limit.

Let's break down the time complexity of the algorithm:

1. Initialization: The algorithm initializes an array of size n+1 to keep track of the prime numbers. This step takes constant time, i.e., O(1).

2. Iteration: The algorithm iterates through the numbers from 2 to sqrt(n), where sqrt(n) is the square root of the given limit. For each number, it checks if it is prime (i.e., not marked as composite) and marks its multiples as composite. This step takes O(n) time.

3. Counting primes: Finally, the algorithm counts the number of prime numbers found by counting the unmarked numbers in the array. This step takes O(n) time.

Therefore, the overall time complexity of the Sieve of Eratosthenes algorithm is O(n).

Conclusion:

The Sieve of Eratosthenes algorithm has a time complexity of O(n), making it an efficient method for finding prime numbers up to a given limit. It outperforms other methods such as trial division, which has a time complexity of O(sqrt(n)).