Euclidean Algorithm
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
ابدأ بهذا النموذج الذهني
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
كيف يعمل، بلا مصطلحات زائدة
The key property is gcd(a, b) = gcd(b, a mod b).
It is much faster than checking every possible divisor.
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.
Read a tiny input by hand and write down the state after every important operation. For Euclidean Algorithm, the goal is to make the invariant visible rather than to memorize code.
function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b !== 0) {
[a, b] = [b, a % b];
}
return a;
}لا تحفظ الصياغة. تتبع الحالة التي تتغير في كل سطر واربطها بالثابت الموضح أعلاه.
O(log min(a,b))
Do not quote this complexity mechanically.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.
مناسب لـ
- Use the direct mathematical property when it removes unnecessary brute force.
ما الذي يسوء عادةً
- Ignoring floating-point precision or integer range.
- Using a formula outside its valid assumptions.
صيغة إجابة احترافية
Explain the invariant/property behind the modulo step; do not present it as a magic loop.
What problem does Euclidean Algorithm solve?
Euclid's algorithm finds gcd(a, b) by repeatedly replacing (a, b) with (b, a mod b) until the remainder becomes zero.
ماذا يجب أن أذكر عن التعقيد؟
Its typical cost here is O(log min(a,b)), 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.