Math 115B Homework 8solution - MAT 115B HOMEWORK 8 SOLUTION...

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MAT 115B HOMEWORK 8 SOLUTION 2.3.2: 2 n 4 + 3 n 3 + 17 2 n 4 + 3 n 4 + 17 n 4 22 n 4 . It follows 2 n 4 + 3 n 3 + 17 is O ( n 4 ). 2.3.10 Let r R , r 1. If f is O (log 2 n ), then K R + such that f ( n ) K log 2 n for all n . We have f ( n ) K log 2 n = ( K log 2 r )(log r n ), where K log 2 r > 0. So f is O (log r n ). Conversely, if f is O (log r n ), then K R + such that f ( n ) K log r n for all n . It follows f ( n ) K log r n = ( K log r 2)(log 2 n ), where K log r 2 > 0. So f is O (log 2 n ). 2.3.14: (a) Each addition takes O ( n ) time and there are N - 1 = O ( N ) additions to perform. The total complexity is O ( nN ). (b) Multiplication takes O ( n 2 ) time (or ˜ O ( n ) using the Sch¨onhage-Strassen algorithm). And the single shift takes O (1). So the total complexity is O ( n 2 ) (or ˜ O ( n )). GK 8.1: (a) Using the standard approach, it takes: four 100-digit multiplications (for AC, AD, BC and BD ), one 200-digit addition (for AD + BC ) and two 100-digit additions (for the the overlaps in AC 10 200 + ( AD + BC )10 100 + BD ). The total charge is $4 . 04.
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