s04-hwex - CS 373U Combinatorial Algorithms Spring 2004...

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CS 373U: Combinatorial Algorithms, Spring 2004 Homework 0 Due January 28, 2004 at noon Name: Net ID: Alias: I understand the Homework Instructions and FAQ. Neatly print your full name, your NetID, and an alias of your choice in the boxes above. Grades will be listed on the course web site by alias; for privacy reasons, your alias should not resemble your name or NetID. By providing an alias, you agree to let us list your grades; if you do not provide an alias, your grades will not be listed. Never give us your Social Security number! Before you do anything else, read the Homework Instructions and FAQ on the course web page, and then check the box above. This web page gives instructions on how to write and submit homeworks—staple your solutions together in order, start each numbered problem on a new sheet of paper, write your name and NetID one every page, don’t turn in source code, analyze and prove everything, use good English and good logic, and so on. See especially the policies regarding the magic phrases “I don’t know” and “and so on”. If you have any questions, post them to the course newsgroup or ask in lecture. This homework tests your familiarity with prerequisite material—basic data structures, big- Oh notation, recurrences, discrete probability, and most importantly, induction—to help you identify gaps in your knowledge. You are responsible for filling those gaps on your own. Chapters 1–10 of CLRS should be sufficient review, but you may also want consult your discrete mathematics and data structures textbooks. Every homework will have five required problems and one extra-credit problem. Each num- bered problem is worth 10 points. # 1 2 3 4 5 6 * Total Score Grader
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CS 373U Homework 0 (due January 28, 2004) Spring 2004 1. Sort the functions in each box from asymptotically smallest to asymptotically largest, indi- cating ties if there are any. You do not need to turn in proofs (in fact, please don’t turn in proofs), but you should do them anyway, just for practice. Don’t merge the lists together. To simplify your answers, write f ( n ) g ( n ) to mean f ( n ) = o ( g ( n )), and write f ( n ) g ( n ) to mean f ( n ) = Θ( g ( n )). For example, the functions n 2 , n, ( n 2 ) , n 3 could be sorted either as n n 2 ( n 2 ) n 3 or as n ( n 2 ) n 2 n 3 . (a) 2 lg n 2 lg n 2 lg n 2 lg n lg 2 n lg 2 n lg 2 n lg 2 n lg n 2 lg n 2 lg n 2 lg n 2 lg 2 n lg 2 n lg 2 n lg n 2 (b) lg( n !) lg( n !) lg( n !) (lg n )! ( lg n )! (lg n )! [Hint: Use Stirling’s approximation for factorials: n ! n n +1 / 2 /e n ] 2. Solve the following recurrences. State tight asymptotic bounds for each function in the form Θ( f ( n )) for some recognizable function f ( n ). Proofs are not required; just give us the list of answers. You do not need to turn in proofs (in fact, please don’t turn in proofs), but you should do them anyway, just for practice. Assume reasonable but nontrivial base cases. If your solution requires specific base cases, state them! Extra credit will be awarded for more exact solutions.
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