GCD and LCM Euclidean Algorithm in DSA C++ - Time & Space Complexity

Time Complexity: GCD and LCM Euclidean Algorithm

O(log(min(a, b)))

Understanding Time Complexity

We want to understand how fast the Euclidean algorithm finds the greatest common divisor (GCD) of two numbers.

How does the number of steps change as the input numbers get bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int gcd(int a, int b) {
  while (b != 0) {
    int temp = b;
    b = a % b;
    a = temp;
  }
  return a;
}

int lcm(int a, int b) {
  return (a / gcd(a, b)) * b;
}
    

This code finds the GCD of two numbers using the Euclidean algorithm and then calculates their LCM using the GCD.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The while loop repeatedly calculates the remainder of division (a % b).
• How many times: The loop runs until the second number becomes zero, which depends on the size of the inputs.

How Execution Grows With Input

Each step reduces the size of the numbers significantly, so the number of steps grows slowly even if inputs get large.

Input Size (n)	Approx. Operations
10	About 4-5 steps
100	About 7-8 steps
1000	About 10-12 steps

Pattern observation: The number of steps grows roughly proportional to the number of digits, not the number itself.

Final Time Complexity

Time Complexity: O(log(min(a, b)))

This means the steps grow slowly, roughly with the number of digits in the smaller input number.

Common Mistake

[X] Wrong: "The Euclidean algorithm takes as many steps as the size of the input numbers."

[OK] Correct: Actually, each step reduces the problem size quickly, so the number of steps grows with the number of digits, not the number itself.

Interview Connect

Understanding this algorithm's time complexity shows you can analyze efficient number operations, a useful skill in many coding problems.

Self-Check

"What if we used subtraction instead of modulo in the GCD calculation? How would the time complexity change?"

Practice

1. What does the Euclidean algorithm primarily calculate between two numbers a and b?

easy

A. The difference between the two numbers

B. The Least Common Multiple (LCM)

C. The sum of the two numbers

D. The Greatest Common Divisor (GCD)

Solution

1. Step 1: Understand the Euclidean algorithm purpose

   The Euclidean algorithm is designed to find the greatest common divisor (GCD) of two integers by repeatedly applying the remainder operation.

2. Step 2: Differentiate GCD from LCM

   While GCD finds the largest number dividing both, LCM finds the smallest number divisible by both. The Euclidean algorithm is for GCD.

3. Final Answer:

   The Greatest Common Divisor (GCD) -> Option D

4. Quick Check:

   Euclidean algorithm = GCD [OK]

Hint: Euclidean algorithm always finds GCD, not LCM [OK]

Common Mistakes:

• Confusing GCD with LCM
• Thinking it calculates sum or difference
• Assuming it finds prime factors

2. Which of the following is the correct syntax for the base case in a recursive Euclidean algorithm function in C++?

easy

A. if (a == b) return 0;

B. if (a == 0) return b;

C. if (b == 0) return a;

D. if (b == a) return b;

Solution

1. Step 1: Identify the base case in Euclidean recursion

   The base case occurs when the second number b becomes zero, then the GCD is the first number a.

2. Step 2: Match the correct syntax

   The correct syntax is if (b == 0) return a; which stops recursion and returns the GCD.

3. Final Answer:

   if (b == 0) return a; -> Option C

4. Quick Check:

   Base case: b == 0 returns a [OK]

Hint: Base case returns a when b is zero [OK]

Common Mistakes:

• Returning 0 instead of a
• Checking if a == b instead of b == 0
• Swapping a and b incorrectly

3. What is the output of the following C++ code snippet?

int gcd(int a, int b) {
  if (b == 0) return a;
  return gcd(b, a % b);
}

int main() {
  std::cout << gcd(48, 18);
  return 0;
}

medium

A. 6

B. 12

C. 18

D. 3

Solution

1. Step 1: Apply Euclidean algorithm steps for gcd(48, 18)

   Calculate gcd(48, 18): 48 % 18 = 12, so next call gcd(18, 12). Then gcd(18, 12): 18 % 12 = 6, next gcd(12, 6). Then gcd(12, 6): 12 % 6 = 0, next gcd(6, 0).

2. Step 2: Reach base case and return result

   When b=0, return a=6. So gcd(48, 18) = 6.

3. Final Answer:

   6 -> Option A

4. Quick Check:

   gcd(48,18) = 6 [OK]

Hint: Use remainder until zero to find GCD [OK]

Common Mistakes:

• Returning 12 instead of 6
• Confusing modulo operation
• Stopping recursion too early

4. Identify the error in this Euclidean algorithm code snippet:

int gcd(int a, int b) {
  if (a == 0) return b;
  return gcd(a % b, b);
}

medium

A. The base case condition is incorrect

B. The recursive call parameters are swapped incorrectly

C. Modulo operator is used wrongly

D. No error, code is correct

Solution

1. Step 1: Check base case correctness

   The base case if (a == 0) return b; is valid because when a is zero, gcd is b.

2. Step 2: Analyze recursive call parameters

   The recursive call uses gcd(a % b, b), but Euclidean algorithm requires gcd(b, a % b). The parameters are swapped incorrectly.

3. Final Answer:

   The recursive call parameters are swapped incorrectly -> Option B

4. Quick Check:

   Correct recursion order: gcd(b, a % b) [OK]

Hint: Always call gcd(b, a % b) in recursion [OK]

Common Mistakes:

• Swapping parameters in recursive call
• Confusing base case condition
• Using modulo with wrong variable

5. Given two positive integers 15 and 20, which of the following expressions correctly calculates their Least Common Multiple (LCM) using the Euclidean algorithm for GCD?

hard

A. LCM = (15 * 20) / gcd(15, 20);

B. LCM = gcd(15, 20) / (15 * 20);

C. LCM = gcd(15, 20) * (15 + 20);

D. LCM = (15 + 20) / gcd(15, 20);

Solution

1. Step 1: Recall the relationship between LCM and GCD

   LCM(a, b) * GCD(a, b) = a * b. So LCM = (a * b) / GCD(a, b).

2. Step 2: Apply formula for 15 and 20

   Calculate gcd(15, 20) = 5, then LCM = (15 * 20) / 5 = 60.

3. Final Answer:

   LCM = (15 * 20) / gcd(15, 20); -> Option A

4. Quick Check:

   LCM * GCD = 15 * 20 [OK]

Hint: LCM = (a*b)/GCD(a,b) always holds [OK]

Common Mistakes:

• Dividing GCD by product instead of product by GCD
• Adding numbers instead of multiplying
• Multiplying GCD by sum instead of dividing