# h1-sol - Homework 1 Solutions Fundamental Algorithms Spring...

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Homework 1 Solutions Fundamental Algorithms, Spring 2008, Professor Yap Due: Wed Feb 6, in class. HOMEWORK with SOLUTION, prepared by Instructor and T.A.s INSTRUCTIONS: Please read questions carefully. When in doubt, please ask. You may also post general questions to the homework discussion forum in class website. Also, bring your questions to recitation on Monday. There are links from the homework page to the old homeworks from previous classes, including solu- tions. Feel free to study these. 1. (10 Points) Exercise 7.1 in Lecture 1. Assume f ( n ) 1 (ev.). (a) Show that f ( n ) = n O (1) i± there exists k > 0 such that f ( n ) = O ( n k ). This is mainly an exercise in unraveling our notations! (b) Show a counter example to (a) in case f ( n ) 1 (ev.) is false. SOLUTION: a) Let f ( n ) 1 (ev) and assume that f ( n ) = n O (1) . Therefore there exists a g ∈ O (1) such that g ( n ) k (ev) and f ( n ) n g ( n ) n k (ev). This shows that f ( n ) ∈ O ( n k ). In the other direction, suppose f = O ( n k ). Then f Cn k (ev) for some C > 1. Thus f n k + ǫ (ev) for any ǫ > 0 (we This shows f = n O (1) . b) To ²nd a counterexample, let f ( n ) = 1 / 2. Clearly f = O ( n ). But f ( n ) n = n O (1) because if f ( n ) = n g ( n ) for some function g ( n ), then clearly g ( n ) < 0 (ev). But this means g ( n ) n = O (1). Recall that g ∈ O (1) implies g 0 (ev). 2. (25 Points) Do Exercise 7.5, Lecture 1 (in 8 parts). Provide either a counter-example when false or a proof when true. The base b of logarithms is arbitrary but ²xed, and b > 1. Assume the functions f, g are arbitrary (do not assume that f and g are 0 eventually). (a) f = O ( g ) implies g = O ( f ). (b) max { f, g } = Θ( f + g ). (c) If g > 1 and f = O ( g ) then ln f = O (ln g ). HINT: careful! (d)

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h1-sol - Homework 1 Solutions Fundamental Algorithms Spring...

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