Virtual Labs

What is the main advantage of Karatsuba's algorithm over traditional multiplication?

a: It uses less memory Explanation

Explanation

b: It is easier to implement Explanation

Explanation

c: It has better asymptotic time complexity Explanation

Explanation

d: It works better with small numbers Explanation

Explanation

In Karatsuba's algorithm, when we split x = 1234 at position 2, what are the high and low parts?

a: High = 12, Low = 34 Explanation

Explanation

b: High = 34, Low = 12 Explanation

Explanation

c: High = 1, Low = 234 Explanation

Explanation

d: High = 123, Low = 4 Explanation

Explanation

How many multiplication operations does Karatsuba's algorithm require at each level of recursion?

a: 2 multiplications Explanation

Explanation

b: 3 multiplications Explanation

Explanation

c: 4 multiplications Explanation

Explanation

d: 5 multiplications Explanation

Explanation

What is the recurrence relation for Karatsuba's time complexity?

a: T(n) = 2T(n/2) + O(n) Explanation

Explanation

b: T(n) = 3T(n/2) + O(n) Explanation

Explanation

c: T(n) = 4T(n/2) + O(n) Explanation

Explanation

d: T(n) = 3T(n/2) + O(n²) Explanation

Explanation

For the multiplication 1234 × 5678 using Karatsuba with split position 2, what is the value of (a+b)(c+d)?

a: 46 × 134 = 6164 Explanation

Explanation

b: 134 × 46 = 6164 Explanation

Explanation

c: 78 × 34 = 2652 Explanation

Explanation

d: 56 × 12 = 672 Explanation

Explanation

In the final combination step of Karatsuba's algorithm, what is the formula for computing the result?

a: z₂ + z₁ + z₀ Explanation

Explanation

b: z₂ × 10^m + z₁ + z₀ Explanation

Explanation

c: z₂ × 10^(2m) + z₁ × 10^m + z₀ Explanation

Explanation

d: z₂ × 10^(2m) + z₁ × 10^(2m) + z₀ Explanation

Explanation

What is the primary reason Karatsuba's algorithm becomes more efficient as input size increases?

a: The constant factors become negligible Explanation

Explanation

b: The algorithm switches to faster hardware operations Explanation

Explanation

c: The recursive overhead decreases proportionally Explanation

Explanation

d: The asymptotic advantage of O(n^1.585) over O(n²) becomes more significant Explanation

Explanation

Consider implementing Karatsuba for very large numbers. What is the most critical optimization consideration?

a: Choosing the optimal threshold for switching to traditional multiplication Explanation

Explanation

b: Using the fastest possible addition algorithm Explanation

Explanation

c: Minimizing memory allocation in recursive calls Explanation

Explanation

d: Implementing parallel processing for the three recursive calls Explanation

Explanation

In a practical implementation of Karatsuba's algorithm, what happens if the input numbers have different lengths?

a: The algorithm fails and cannot proceed Explanation

Explanation

b: The shorter number is padded with leading zeros Explanation

Explanation

c: The algorithm automatically switches to traditional multiplication Explanation

Explanation

d: The longer number is truncated to match the shorter one Explanation

Explanation

Karatsuba's Integer Multiplication