f05-hwex - CS 473G Combinatorial Algorithms Fall 2005...

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CS 473G: Combinatorial Algorithms, Fall 2005 Homework 0 Due Thursday, September 1, 2005, at the beginning of class (12:30pm CDT) 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. Please write the same alias on every homework and exam! 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! 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 course 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 during lecture. Don’t forget to submit this cover sheet with the rest of your homework solutions. This homework tests your familiarity with prerequisite material—big-Oh notation, elemen- tary algorithms and data structures, 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. Most homeworks will also include one extra-credit problem and several practice (no-credit) problems. Each numbered problem is worth 10 points. # 1 2 3 4 5 6 * Total Score Grader
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CS 473G Homework 0 (due September 1, 2005) Fall 2005 1. Solve the following recurrences. State tight asymptotic bounds for each function in the form Θ( f ( n )) for some recognizable function f ( n ). 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! (a) A ( n ) = 2 A ( n/ 4) + n (b) B ( n ) = max n/ 3 <k< 2 n/ 3 ( B ( k ) + B ( n - k ) + n ) (c) C ( n ) = 3 C ( n/ 3) + n/ lg n (d) D ( n ) = 3 D ( n - 1) - 3 D ( n - 2) + D ( n - 3) (e) E ( n ) = E ( n - 1) 3 E ( n - 2) [Hint: This is easy!] (f) F ( n ) = F ( n - 2) + 2 /n (g) G ( n ) = 2 G ( ( n + 3) / 4 - 5 n/ lg n + 6 lg lg n ) + 7 8 n - 9 - lg 10 n/ lg lg n + 11 lg * n - 12 (h) H ( n ) = 4 H ( n/ 2) - 4 H ( n/ 4) + 1 [Hint: Careful!] (i) I ( n ) = I ( n/ 2) + I ( n/ 4) + I ( n/ 8) + I ( n/ 12) + I ( n/ 24) + n (j) J ( n ) = 2 n · J ( n ) + n [Hint: First solve the secondary recurrence j ( n ) = 1 + j ( n ) .] 2. Penn and Teller agree to play the following game. Penn shuffles a standard deck 1 of playing cards so that every permutation is equally likely. Then Teller draws cards from the deck, one at a time without replacement, until he draws the three of clubs (3
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