ملاحظة قراءة الخوارزمية

Sieve of Eratosthenes

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

النوعخوارزميةالفئةMathالعمقمبتدئالتعقيد المعتادO(n log log n)

إجابة مقابلة قصيرة

ابدأ بهذا النموذج الذهني

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

شرح مبسط

كيف يعمل، بلا مصطلحات زائدة

Start with all candidates marked prime.

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

التنفيذ

اقرأ الخوارزمية كسلسلة من القرارات

State the mathematical identity/property.

Choose numeric representation and constraints.

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

مثال صغير لتحتفظ به في ذهنك

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.

تطبيق JavaScript قابل للقراءةJavaScript

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] : []);
}

كيف تقرأ هذا الكود

لا تحفظ الصياغة. تتبع الحالة التي تتغير في كل سطر واربطها بالثابت الموضح أعلاه.

التعقيد

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Do not quote this complexity mechanically.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.

متى تستخدمه

مناسب لـ

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

أخطاء شائعة

ما الذي يسوء عادةً

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

كيف تشرحه في المقابلة

صيغة إجابة احترافية

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

المشكلة→الفكرة الأساسية→الثابت / الحالة→Big-O→المفاضلة

فحص سريع للمقابلة

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.

ماذا يجب أن أذكر عن التعقيد؟

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.

ما الخطأ الأكثر شيوعاً؟

Ignoring floating-point precision or integer range.

متابعة القراءةNumber 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] : []); }