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hbs0(1) - CS 473 Head-Banging Session 0(January 2021 2009...

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CS 473 Head-Banging Session 0 (January 20–21, 2009) Spring 2009 CS 473: Undergraduate Algorithms, Spring 2009 Head Banging Session 0 January 20 and 21, 2009 1. Solve the following recurrences. If base cases are provided, find an exact closed-form solution. Otherwise, find a solution of the form Θ( f ( n )) for some function f . Warmup: You should be able to solve these almost as fast as you can write down the answers. (a) A ( n ) = A ( n - 1 ) + 1, where A ( 0 ) = 0. (b) B ( n ) = B ( n - 5 ) + 2, where B ( 0 ) = 17. (c) C ( n ) = C ( n - 1 ) + n 2 (d) D ( n ) = 3 D ( n / 2 ) + n 2 (e) E ( n ) = 4 E ( n / 2 ) + n 2 (f) F ( n ) = 5 F ( n / 2 ) + n 2 Real practice: (a) A ( n ) = A ( n / 3 ) + 3 A ( n / 5 ) + A ( n / 15 ) + n (b) B ( n ) = min 0 < k < n ( B ( k ) + B ( n - k ) + n ) (c) C ( n ) = max n / 4 < k < 3 n / 4 ( C ( k ) + C ( n - k ) + n ) (d) D ( n ) = max 0 < k < n D ( k ) + D ( n - k ) + k ( n - k ) , where D ( 1 ) = 0 (e) E ( n ) = 2 E ( n - 1 ) + E ( n - 2 ) , where E ( 0 ) = 1 and E ( 1 ) = 2 (f) F ( n ) = 1 F ( n - 1 ) F ( n - 2 ) , where F ( 0 ) = 1 and F ( 2 ) = 2 ? (g) G ( n ) = n G ( p n ) + n 2 2. The Fibonacci numbers F n are defined recursively as follows: F 0 = 0, F 1 = 1, and F n = F n - 1 + F n - 2 for every integer n 2. The first few Fibonacci numbers are 0,1,1,2,3,5,8,13,21,34,55, .... Prove that any non-negative integer can be written as the sum of distinct non-consecutive Fibonacci numbers. That is, if any Fibonacci number F n appears in the sum, then its neighbors
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