Unformatted text preview: Solutions Name (b) [5 points] Mal decides to use the technique from part (a) to compute the value B =
A1/4 . She then plans to compute A3/4 by calculating the value C = B 3 = B ·B ·B .
Provide an explanation of why this technique does not work.
Hint: Deﬁne α to be the fractional part of A1/4 , so that B = A1/4 − α. What happens
when you compute C = B 3 ?
Solution: When you expand out the formula for C , you get
C = B 3 = (A1/4 − α)3 = A3/4 − 3A2/4 α + 3A1/4 α2 − α3 . If A is large, then γ = 3A2/4 α − 3A1/4 α2 + α3 will be signiﬁcantly greater than 1,
and so we’ll have C = A3/4 − γ with γ > 1. Hence, C will not be A3/4 . (c) [5 points] Mal clearly needs a way to check her answer for A3/4 , using only integers.
Given a pair of positive integers A and C , explain how to check whether C ≤ A3/4
using O(1) additions, subtractions, multiplications, and comparisons.
Solution: The equation C ≤ A3/4 is equivalent to the equation C 4 ≤ A3 , which is
equivalent to C · C · C · C ≤ A · A ...
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This document was uploaded on 03/17/2014 for the course ELECTRICAL 6.006 at MIT.
 Fall '11
 ErikDemaine

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