Problem Description

The problem requires us to find the count of prime numbers less than a given integer n. Remember, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Intuition

The solution is based on an ancient algorithm known as the "Sieve of Eratosthenes". The algorithm works by iteratively marking the multiples of each number starting from 2. Once all multiples of a particular number are marked, we move on to the next unmarked number and repeat the process.

Here's the step-by-step intuition:

1. We start with a boolean array primes where each entry is set to True, meaning we assume all numbers are primes initially.
2. Starting from the first prime number 2, we mark all of its multiples as False, since multiples of 2 cannot be primes.
3. We continue this process for the next unmarked number (which would be the next prime) and mark all of its multiples as False.
4. We repeat the process until we have processed all numbers less than n.
5. During the process, every time we encounter a number that's not marked as False, it means this number is a prime number, and we increment the counter ans.

This solution is efficient because once a number is marked as False, it will not be checked again, which greatly reduces the number of operations needed compared to checking every number individually for primality.

Learn more about Math patterns.

Solution Approach

The implementation of the countPrimes function follows the Sieve of Eratosthenes algorithm:

1. We initialize a list called primes filled with True, representing all numbers from 0 to n-1. Here, True signifies that the number is assumed to be a prime number.

2. We iterate over each number starting from 2 (the smallest prime number) using a for loop for i in range(2, n):. If the number is marked as True in our primes list, it is a prime, as it hasn't been marked as not prime by an earlier iteration (via its multiples).

3. When we find such a prime number i, we increment our answer counter ans by 1, as we've just found a prime.

4. To mark the multiples of i as not prime, we loop through a range starting from i*2 up to n (exclusive), in steps of i, using the inner loop for j in range(i + i, n, i):. The step of i makes sure we only hit the multiples of i.

5. For each multiple j of the prime number i, we set primes[j] to False to denote that j is not a prime.

6. Continue the process until all numbers in our list have been processed.

7. Finally, we return ans, which now holds the count of prime numbers strictly less than n.

Throughout the process, the use of the array primes and the marking of non-prime numbers optimizes the approach and avoids unnecessary checks, making this a classic and time-efficient solution for counting prime numbers.

Here is the final code that implements this approach:

class Solution:
    def countPrimes(self, n: int) -> int:
        if n < 2:
            return 0
        primes = [True] * n
        ans = 0
        for i in range(2, n):
            if primes[i]:
                ans += 1
                for j in range(i * i, n, i):
                    primes[j] = False
        return ans

In this implementation, notice how we optimized the inner loop's starting point from i + i to i * i. Since any multiple k * i (where k < i) would already have been marked False by a prime less than i, it suffices to start marking from i * i.

Example Walkthrough

Let's illustrate the solution approach with a small example where n = 10. We want to find the count of prime numbers less than 10.

Following the steps outlined in the solution approach:

1. We start by initializing a list primes that represents the numbers from 0 to 9. All the values are set to True, indicating we assume they are prime until proven otherwise:

   primes = [True, True, True, True, True, True, True, True, True, True]

2. We start checking numbers from 2 (the smallest prime number). Since primes[2] is True, 2 is a prime number, so we increment our prime count ans.

3. Now, we mark all multiples of 2 as not prime by setting their respective positions in the primes array to False. This will mark 4, 6 and 8 as not prime:

   primes = [True, True, True, True, False, True, False, True, False, True]

   The prime count ans is now 1.

4. The next number is 3, which is also True in the primes list, so we increment ans again. We then mark all multiples of 3 as False, affecting 6 and 9:

   primes = [True, True, True, True, False, True, False, True, False, False]

   The prime count ans is now 2.

5. The next number is 4, which is False in the primes list, so we skip it.

6. Then we check 5, which is True. Therefore, we increment ans and mark its multiples (none within our range, as the first would be 10, which is outside our range).

   The prime count ans is now 3.

7. Continuing this process, we check 6 (marked as not prime), 7 (prime), and 8 (not prime). When we reach 7, we mark it as prime and increment ans.

   The prime count ans is now 4.

8. Finally, we process 9 (marked as not prime) and the primes list won't change anymore as there's no need to mark further multiples.

9. Our final prime count ans is 4. Therefore, there are 4 prime numbers less than 10.

Here's a visualization of the primes list after processing primes:

primes = [True, True, True, True, False, True, False, True, False, False]
             ^      ^      ^                     ^
Indices:     2      3      4     5      6      7     8     9

At the indices where primes list is True (excluding the indices 0 and 1 since we start counting primes from 2), those numbers are the primes less than 10, and we count them up to get our answer, which is 4. This is how the Sieve of Eratosthenes algorithm works and the code from the solution approach implements this efficiently to count the number of prime numbers less than any given integer n.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity for this Sieve of Eratosthenes algorithm is O(n log(log n)). This is because the inner loop for marking the multiples as non-primes runs n/i times for each prime number i, and the sum of these series approximates n multiplied by the harmonic series, which tends to log(log n) as n approaches infinity.

Space Complexity

The space complexity of the algorithm is O(n) due to the primes list which stores a boolean for each number up to n to indicate whether it's a prime number or not.

Learn more about how to find time and space complexity quickly using problem constraints.