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KaratsubaRecurrence

# KaratsubaRecurrence - In class we looked at a recursive...

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In class, we looked at a recursive algorithm for multiplying two natural numbers. The algorithm was discovered by A. A. Karatsuba in 1960. Here is the time bound for that algorithm. Let T ( n ) be the worst-case time for multiplying two n -bit numbers. We derived the recurrence: T ( n ) is O (1) for n 3, and T ( n ) T ( n 2 ) + T ( n 2 ) + T ( n 2 + 1) + an for n > 3. (In the above, a is a constant.) I claimed in class that T ( n ) is O ( n log 2 3 ). Here is a proof of that claim. You might first try to prove that T ( n ) cn log 2 3 (for some constant c ). Unfortunately, if you try this, you will see that the induction hypothesis is not strong enough for the induction step to work. So, to strengthen the induction hypothesis, we prove a stronger claim. (This is the same trick as described on page 85 of the textbook.) Let c = max { T ( n )+2 an ( n - 3) log 2 3 : n ∈ { 4 , 5 , 6 , 7 }} . Claim : for all n 4, T ( n ) c ( n - 3) log 2 3 - 2 an . Base case ( n = 4 , 5 , 6 , 7): We chose c precisely so that the claim holds for these values of n .
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