ALGORITHMUS-LESENOTIZ

Sieve of Eratosthenes

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

TYPAlgorithmusKATEGORIEMathTIEFEEinsteigerTYPISCHE KOMPLEXITÄTO(n log log n)

KURZE INTERVIEWANTWORT

Beginne mit diesem mentalen Modell

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

EINFACHE ERKLÄRUNG

So funktioniert es – ohne unnötigen Jargon

Start with all candidates marked prime.

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

AUSFÜHRUNG

Lies den Algorithmus als Folge von Entscheidungen

State the mathematical identity/property.

Choose numeric representation and constraints.

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

Ein kleines Beispiel als Orientierung

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.

LESBARE JAVASCRIPT-IMPLEMENTIERUNGJavaScript

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] : []);
}

So liest du diesen Code

Lerne die Syntax nicht auswendig. Folge dem Zustand, der sich in jeder Zeile ändert, und verbinde ihn mit der oben beschriebenen Invariante.

KOMPLEXITÄT

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Do not quote this complexity mechanically.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.

WANN EINSETZEN

Gut geeignet

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

HÄUFIGE FEHLER

Was typischerweise schiefgeht

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

SO ERKLÄRST DU ES IM INTERVIEW

Struktur einer professionellen Antwort

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

Problem→Kernidee→Invariante / Zustand→Big-O→Trade-off

SCHNELLER 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.

Was sollte ich zur Komplexität erwähnen?

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.

Was ist der häufigste Fehler?

Ignoring floating-point precision or integer range.

WeiterlesenNumber 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] : []); }