Algorithm: Sieve of Eratosthenes to Calculate First 1000 Prime Numbers

1. Initialize Variables:
   - Let n be an upper bound estimate for the 1000th prime number. A safe estimate is n = 7920, since the 1000th prime is 7919.
   - Create a boolean array `isPrime` of size n+1, initialized to true. This array will help track prime numbers.
   - Set `isPrime[0]` and `isPrime[1]` to false, as 0 and 1 are not prime numbers.

2. Implement the Sieve of Eratosthenes:
   - For each integer p from 2 to √n:
     a. If `isPrime[p]` is true, it means p is a prime number.
     b. Mark all multiples of p (starting from p^2) as false in the `isPrime` array, as they are not prime.

3. Collect Prime Numbers:
   - Initialize an empty list `primes` to store prime numbers.
   - Iterate over the `isPrime` array:
     a. For each index i where `isPrime[i]` is true, add i to the `primes` list.
     b. Stop collecting once the list contains 1000 prime numbers.

4. Return the List of Primes:
   - The `primes` list now contains the first 1000 prime numbers.

Validation:
- Ensure that the length of the `primes` list is exactly 1000.
- Verify that the last element in the `primes` list is 7919, the 1000th prime number.

This algorithm efficiently calculates the first 1000 prime numbers using the Sieve of Eratosthenes, which has a time complexity of O(n log log n).