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Unformatted text preview: CS 373 Homework 0 (due 1/26/99) Spring 1999 CS 373: Combinatorial Algorithms, Spring 1999 http://www-courses.cs.uiuc.edu/ cs373 Homework 0 (due January 26, 1999 by the beginning of class) Name: Net ID: Alias: Neatly print your name (first name first, with no comma), your network ID, and a short alias into the boxes above. Do not sign your name. Do not write your Social Security number. Staple this sheet of paper to the top of your homework. Grades will be listed on the course web site by alias, so your alias should not resemble your name (or your Net ID). If you do not give yourself an alias, you will be stuck with one we give you, no matter how much you hate it. Everyone must do the problems marked . Problems marked a3 are for 1-unit grad students and others who want extra credit. (Theres no such thing as partial extra credit!) Unmarked problems are extra practice problems for your benefit, which will not be graded. Think of them as potential exam questions. Hard problems are marked with a star; the bigger the star, the harder the problem. This homework tests your familiarity with the prerequisite material from CS 225 and CS 273 (and their prerequisites)many of these problems appeared on homeworks and/or exams in those classesprimarily to help you identify gaps in your knowledge. You are responsible for filling those gaps on your own. Undergrad/.75U Grad/1U Grad Problems 1. [173/273] (a) Prove that any positive integer can be written as the sum of distinct powers of 2 . (For example: 42 = 2 5 + 2 3 + 2 1 , 25 = 2 4 + 2 3 + 2 , 17 = 2 4 + 2 .) (b) Prove that any positive integer can be written as the sum of distinct nonconsecutive Fi- bonacci numbersif F n appears in the sum, then neither F n +1 nor F n 1 will. (For example: 42 = F 9 + F 6 , 25 = F 8 + F 4 + F 2 , 17 = F 7 + F 4 + F 2 .) (c) Prove that any integer can be written in the form i 3 i , where the exponents i are distinct non-negative integers. (For example: 42 = 3 4 3 3 3 2 3 1 , 25 = 3 3 3 1 + 3 , 17 = 3 3 3 2 3 .) 2. [225/273] Sort the following functions from asymptotically smallest to largest, indicating ties if there are any: n , lg n , lg lg n , lg lg n , lg n , n lg n , lg( n lg n ) , n n/ lg n , n lg n , (lg n ) n , (lg n ) lg n , 2 lg n lg lg n , 2 n , n lg lg n , 1000 n , (1 + 1 1000 ) n , (1 1 1000 ) n , lg 1000 n , lg (1000) n , log 1000 n , lg n 1000 , 1 . [To simplify notation, write f ( n ) g ( n ) to mean f ( n ) = o ( g ( n )) and f ( n ) g ( n ) to mean f ( n ) = ( g ( n )) . For example, the functions n 2 , n , ( n 2 ) , n 3 could be sorted as follows: n n 2 ( n 2 ) n 3 .] 1 CS 373 Homework 0 (due 1/26/99) Spring 1999 3. [273/225] Solve the following recurrences. State tight asymptotic bounds for each function in the form ( f ( n )) for some recognizable function f ( n ) . You do not need to turn in proofs (in fact, please dont turn in proofs), but you should do them anyway just for practice. Assumeturn in proofs), but you should do them anyway just for practice....
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This note was uploaded on 12/15/2009 for the course 942 cs taught by Professor A during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09