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Unformatted text preview: use we threw away
information (taking the ﬂoor) before performing another operation. Instead, we ﬁrst
calculate B = A3 , then compute C = B 1/4 using the formula from part (a) to
continually improve our estimate. Because we have a quadratic rate of convergence,
we will improve our estimate Θ(log d) times, and each time we will perform a constant
number of arithmetic operations.
Note: This is equivalent to applying Newton’s Method to the function f (x) = x4 − A3 ,
but does not require you to rederive the equation for xi+1 in terms of xi .
Θ(d) Solution: Use the comparison method from part (c) to do binary search. This
requires us to do Θ(1) operations for each of the bits in the result, for a total of Θ(d)
operations. 10 6.006 Quiz 2 Solutions Name 11 Problem 5. The Tourist [15 points]
Your new startup, Bird Tours, brings people around Boston in a new aviation car that can both drive
and ﬂy. You’ve constructed a weighted directed graph G = (V, E, w) representing the best time to
drive or ﬂy between various...
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 Fall '11
 ErikDemaine

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