f06-hwex - CS 473U Undergraduate Algorithms Fall 2006...

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CS 473U: Undergraduate Algorithms, Fall 2006 Homework 0 Due Friday, September 1, 2006 at noon in 3229 Siebel Center 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, and submit this page with your solutions. We will list homework and exam grades on the course web site by alias. For privacy reasons, your alias should not resemble your name, your NetID, your university ID number, or (God forbid) your Social Security number. Please use the same alias for every homework and exam. Federal law forbids us from publishing your grades, even anonymously, without your explicit permission. By providing an alias, you grant us permission to list your grades on the course web site; if you do not provide an alias, your grades will not be listed. Please carefully read the Homework Instructions and FAQ on the course web page, and then check the box above. This page describes what we expect in your homework solutions—start each numbered problem on a new sheet of paper, write your name and NetID on every page, don’t turn in source code, analyze and prove everything, use good English and good logic, and so on—as well as policies on grading standards, regrading, and plagiarism. 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. Each numbered problem is worth 10 points; not all subproblems have equal weight. # 1 2 3 4 5 6 * Total Score Grader
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CS 473U Homework 0 (due September 1, 2006) Fall 2006 Please put your answers to problems 1 and 2 on the same page. 1. Sort the functions listed below from asymptotically smallest to asymptotically largest, indi- cating ties if there are any. Do not turn in proofs , but you should probably do them anyway, just for practice. 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 . lg n ln n n n n lg n n 2 2 n n 1 /n n 1+1 / lg n lg 1000 n 2 lg n ( 2) lg n lg 2 n n 2 (1 + 1 n ) n n 1 / 1000 H n H n 2 H n H 2 n F n F n/ 2 lg F n F lg n In case you’ve forgotten: lg n = log 2 n = ln n = log e n lg 3 n = (lg n ) 3 = lg lg lg n . The harmonic numbers: H n = n i =1 1 /i ln n + 0 . 577215 . . . The Fibonacci numbers: F 0 = 0 , F 1 = 1 , F n = F n - 1 + F n - 2 for all n 2 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. Don’t turn in proofs , but you should do them anyway, just for practice. Assume reasonable but nontrivial base cases.
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