I went straight for the Sieve of Eratosthenes and it felt fine.
Use the Sieve of Eratosthenes to efficiently find all primes up to n, leveraging a boolean array to mark composites. Optimize memory by using a bit array or byte array and only iterating up to sqrt(n).
Pro tip: Mention that for n=10^7, a boolean array of size n+1 uses about 10 MB, which is acceptable, but you can further reduce memory by using a bitset or by only storing odd numbers. Also, discuss the time complexity O(n log log n) and how it scales.
Confirm that n can be up to 10^7, output should be in ascending order, and discuss time/memory constraints. Ask if the function should return a list or print.
Select the Sieve of Eratosthenes for its efficiency. Explain why it's better than trial division for large n.
Initialize a boolean array of size n+1 with True, then mark multiples of each prime starting from 2 up to sqrt(n) as False. Collect primes in order.
Use a bytearray or bitset to reduce memory. Skip even numbers after 2 to halve the work. Consider segmented sieve if memory is extremely tight.
State time complexity O(n log log n) and space O(n). Test with edge cases like n=1, n=2, and n=10^7.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the original problem and constraints, then propose algorithmic adaptations that reduce time and space complexity, such as streaming, sampling, or approximation. Walk through the complexity analysis for each option, emphasizing trade-offs between accuracy, memory, and speed.
Pro tip: Always tie your solution back to business impact—at Amazon, showing how your approach scales cost-effectively and meets customer needs is as important as the algorithm itself.
Restate the problem to ensure you understand the input size (n up to 10^12), memory limits, and required output precision. Ask clarifying questions about data distribution, access patterns, and whether approximate results are acceptable.
Analyze the time and space complexity of the initial solution. Point out why it fails for n=10^12, such as O(n) memory or O(n log n) time.
Suggest techniques like streaming algorithms, probabilistic data structures (e.g., Bloom filters, HyperLogLog), sampling, or divide-and-conquer. Explain how each reduces memory or time complexity.
For each proposed approach, walk through the new time and space complexity. Discuss trade-offs: accuracy vs. memory, latency vs. throughput, and implementation complexity.
Choose the best approach based on constraints and business needs. Explain how it scales and why it's suitable for Amazon's context (e.g., cost, customer impact).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify the constraints (n, array size, memory) and discuss trade-offs between precomputing primes up to n with a sieve and testing each element individually. Then propose an efficient solution that leverages the fact that the array is a subset of 1..n, such as sieving up to n and filtering the array, or using a segmented sieve if n is huge. Finally, analyze time and space complexity and consider edge cases like small n or large memory requirements.
Pro tip: Mention that if the array is much smaller than n, testing each element with a fast primality test (e.g., Miller-Rabin for large n) may be more memory-efficient than a full sieve, showing you balance time vs. space. Also, note that Amazon values customer obsession, so tie your choice to practical constraints like memory limits in production.
Ask about the range of n, the size of the array, memory limits, and whether the array can be modified. This shows you consider practical constraints before diving into algorithms.
Discuss at least two methods: (a) Sieve of Eratosthenes up to n, then check each array element; (b) Test each element individually with a primality test (e.g., trial division up to sqrt, or Miller-Rabin for large n).
Compare time and space complexity: Sieve is O(n log log n) time and O(n) space; individual testing is O(k sqrt(n)) or O(k log^3 n) for Miller-Rabin, where k is array size. Choose based on n vs. k and memory availability.
If n is moderate and memory allows, use a sieve and then filter the array. If n is very large or memory constrained, use a segmented sieve or a probabilistic primality test for each element.
Address edge cases: n < 2 (no primes), array containing 1 (not prime), and duplicate handling (though distinct). Summarize the chosen approach and its complexity.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.