ALGORITHM READING NOTE

Sieve of Eratosthenes

The Sieve of Eratosthenes finds all primes up to n by marking multiples of each discovered prime as composite.

TYPEAlgorithmCATEGORYMathDEPTHBeginnerTYPICAL COMPLEXITYO(n log log n)

SHORT INTERVIEW ANSWER

Start with this mental model

The Sieve of Eratosthenes finds all primes up to n by marking multiples of each discovered prime as composite.

SIMPLE EXPLANATION

How it works, without unnecessary jargon

Start with all candidates marked prime.

For each prime p up to √n, mark multiples starting at p²; smaller multiples were already handled.

EXECUTION

Read the algorithm as a sequence of decisions

State the mathematical identity/property.

Choose numeric representation and constraints.

Translate the property into an iterative/recursive computation and analyze precision.

A small example to keep in mind

Read a tiny input by hand and write down the state after every important operation. For Sieve of Eratosthenes, the goal is to make the invariant visible rather than to memorize code.

READABLE JAVASCRIPT IMPLEMENTATIONJavaScript

function primesUpTo(n) {
  const prime = Array(n + 1).fill(true);
  prime[0] = prime[1] = false;
  for (let p = 2; p * p <= n; p++) {
    if (!prime[p]) continue;
    for (let x = p * p; x <= n; x += p) prime[x] = false;
  }
  return prime.flatMap((isPrime, i) => isPrime ? [i] : []);
}

How to read this code

Do not memorize the syntax. Follow the state that changes on each line and connect it to the invariant described above.

COMPLEXITY

O(n log log n)

Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.

WHEN TO USE IT

Good fit

Use the direct mathematical property when it removes unnecessary brute force.

COMMON MISTAKES

What usually goes wrong

Ignoring floating-point precision or integer range.
Using a formula outside its valid assumptions.

HOW TO EXPLAIN IT IN AN INTERVIEW

A professional answer shape

Mention O(n log log n) time and O(n) space, and explain why marking can start from p².

Problem→Core idea→Invariant / state→Big-O→Trade-off

QUICK INTERVIEW CHECK

What problem does Sieve of Eratosthenes solve?

The Sieve of Eratosthenes finds all primes up to n by marking multiples of each discovered prime as composite.

What should I mention about complexity?

Its typical cost here is O(n log log n), but a complete answer should connect that cost to the representation, input constraints and the dominant operations.

What is the most common mistake?

Ignoring floating-point precision or integer range.

Continue readingNumber theoryNumerical precisionRecurrences

function primesUpTo(n) { const prime = Array(n + 1).fill(true); prime[0] = prime[1] = false; for (let p = 2; p * p <= n; p++) { if (!prime[p]) continue; for (let x = p * p; x <= n; x += p) prime[x] = false; } return prime.flatMap((isPrime, i) => isPrime ? [i] : []); }