Virtual Labs

What is the time complexity of traditional long multiplication for multiplying two n-digit numbers?

a: O(n) Explanation

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b: O(n²) Explanation

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c: O(n log n) Explanation

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d: O(n^1.585) Explanation

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What fundamental technique does Karatsuba's algorithm use to achieve faster multiplication?

a: Dynamic Programming Explanation

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b: Greedy Method Explanation

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c: Divide and Conquer Explanation

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d: Branch and Bound Explanation

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In Karatsuba's algorithm, how many parts is each input number split into?

a: 3 parts Explanation

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b: 4 parts Explanation

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c: 2 parts Explanation

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d: It varies based on the number size Explanation

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What is the key insight that makes Karatsuba's algorithm more efficient than traditional multiplication?

a: It uses fewer addition operations Explanation

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b: It reduces 4 multiplications to 3 multiplications Explanation

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c: It processes digits in parallel Explanation

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d: It uses lookup tables for small multiplications Explanation

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For multiplying two 8-digit numbers, approximately how many levels of recursion will Karatsuba's algorithm have?

a: 2 levels Explanation

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b: 3 levels Explanation

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c: 4 levels Explanation

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d: 8 levels Explanation

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In the Karatsuba algorithm, if we split x = a·10^m + b and y = c·10^m + d, what does the formula z₁ = (a+b)(c+d) - z₂ - z₀ actually compute?

a: The sum ac + bd Explanation

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b: The difference ac - bd Explanation

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c: The sum ad + bc Explanation

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d: The product (a+b)(c+d) Explanation

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What is the exact time complexity of Karatsuba's multiplication algorithm?

a: O(n^log₂3) ≈ O(n^1.585) Explanation

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b: O(n^1.5) Explanation

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c: O(n² log n) Explanation

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d: O(n log² n) Explanation

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When implementing Karatsuba's algorithm, what is the most important consideration for the base case?

a: Stop recursion when numbers become negative Explanation

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b: Stop recursion when numbers have different lengths Explanation

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c: Stop recursion when numbers are small enough that traditional multiplication is faster Explanation

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d: Always recurse until numbers are single digits Explanation

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Consider multiplying x = 5678 and y = 1234 using Karatsuba's algorithm. If we split at position 2, what are the values of the three key multiplications z₀, z₁, and z₂?

a: z₀ = 78×34, z₁ = 56×12, z₂ = (56+78)×(12+34) Explanation

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b: z₀ = 78×34, z₁ = (56+78)×(12+34) - 56×12 - 78×34, z₂ = 56×12 Explanation

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c: z₀ = 56×12, z₁ = 78×34, z₂ = (56+78)×(12+34) Explanation

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d: z₀ = 78×34, z₁ = 56×78, z₂ = 12×34 Explanation

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Karatsuba's Integer Multiplication