373
Homework CS 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 (rst name rst, 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 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 benet, 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 lling 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 = 25 + 23 + 21 , 25 = 24 + 23 + 20 , 17 = 24 + 20 .) (b) Prove that any positive integer can be written as the sum of distinct nonconsecutive Fibonacci numbersif Fn appears in the sum, then neither Fn+1 nor Fn1 will. (For example: 42 = F9 + F6 , 25 = F8 + F4 + F2 , 17 = F7 + F4 + F2 .) (c) Prove that any integer can be written in the form i 3i , where the exponents i are distinct non-negative integers. (For example: 42 = 34 33 32 31 , 25 = 33 31 + 30 , 17 = 33 32 30 .)
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), nn/ lg n , nlg n , (lg n)n , 1 1 (lg n)lg n , 2 lg n lg lg n , 2n , nlg lg n , 1000 n, (1 + 1000 )n , (1 1000 )n , lg1000 n, lg(1000) n, log1000 n, n lg 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 n2 , n, n , n3 could be sorted as follows: 2 n n2 n n3 .] 2 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. Assume reasonable (nontrivial) base cases. Extra credit will be given for more exact solutions.
(a) A(n) = A(n/2) + n
(b) B (n) = 2B (n/2) + n
(c) C (n) = 3C (n/2) + n
(d) D(n) =
n/3<k<2n/3 0<k<n
max
D(k) + D(n k) + n
(e) E (n) = min E (k) + E (n k) + 1 (g) G(n) = G(n 1) + 1/n
(h)
(f) F (n) = 4F (n/2 + 5) + n H (n) = H (n/2) + H (n/4) + H (n/6) + H (n/12) + n [Hint:
1 2
+
1 4
+
1 6
+
1 12
= 1.]
(i)
I (n) = 2I (n/2) + n/ lg n (j) J (n) = J (n 1) J (n 2)
4. [273] Alice and Bob each have a fair n-sided die. Alice rolls her die once. Bob then repeatedly
throws his die until the number he rolls is at least as big as the number Alice Each rolled. time Bob rolls, he pays Alice $1. (For example, if Alice rolls a 5, and Bob rolls a 4, then a 3, then a 1, then a 5, the game ends and Alice gets $4. If Alice rolls a 1, then no matter what Bob rolls, the game will end immediately, and Alice will get $1.) Exactly how much money does Alice expect to win at this game? Prove that your answer is correct. (If you have to appeal to intuition or common sense, your answer is probably wrong.)
5. [225] George has a 26-node binary tree, with each node labeled by a unique letter of the
alphabet. The preorder and postorder sequences of nodes are as follows: preorder: M N H C R S K W T G D X I Y A J P O E Z V B U L Q F postorder: C W T K S G R H D N A O E P J Y Z I B Q L F U V X M Draw Georges binary tree.
Only 1U Grad Problems
1. [225/273] A tournament is a directed graph with exactly one edge between every pair of
vertices. (Think of the nodes as players in a round-robin tournament, where each edge points from the winner to the loser.) A Hamiltonian path is a sequence of directed edges, joined end to end, that visits every vertex exactly once. Prove that every tournament contains at least one Hamiltonian path. 2
CS 373
Homework 0 (due 1/26/99)
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Spring 1999
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A six-vertex tournament containing the Hamiltonian path 6 4 5 2 3 1.
Practice Problems
1. [173/273] Recall the standard recursive denition of the Fibonacci numbers: F0 = 0, F1 = 1, and Fn = Fn1 + Fn2 for all n 2. Prove the following identities for all positive integers n and m. (a) Fn is even if and only if n is divisible by 3.
n
(b) (c)
(d)
If n is an integer multiple of m, then Fn is an integer multiple of Fm .
Fi = Fn+2 1 i=0 2 Fn Fn+1 Fn1 =
(1)n+1
2. [225/273] (a) Prove that 2lg n+lg n /n = (n). (b) Is 2lg n = 2lg n ? Justify your answer. (c) Is 22 3. [273] (a) A domino is a 2 1 or 1 2 rectangle. How many different ways are there to completely ll a 2 n rectangle with n dominos?
lg lg n
= 22
lg lg n
? Justify your answer.
(b) A slab is a three-dimensional box with dimensions 1 2 2, 2 1 2, or 2 2 1. How many different ways are there to ll a 2 2 n box with n slabs? Set up a recurrence relation and give an exact closed-form solution.
A 2 10 rectangle lled with ten dominos, and a 2 2 10 box lled with ten slabs.
4. [273] Penn and Teller have a special deck of fty-two cards, with no face cards and nothing but clubsthe ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . . , 52 of clubs. (Theyre big cards.) Penn shufes the deck until each each of the 52! possible orderings of the cards is equally likely. He then takes cards one at a time from the top of the deck and gives them to Teller, stopping as soon as he gives Teller the three of clubs. 3
CS 373
Homework 0 (due 1/26/99)
Spring 1999
(a) On average, how many cards does Penn give Teller? (b) On average, what is the smallest-numbered card that Penn gives Teller?
(c)
On average, what is the largest-numbered card that Penn gives Teller?
Prove that your answers are correct. (If you have to appeal to intuition or common sense, your answers are probably wrong.) [Hint: Solve for an n-card deck, and then set n to 52.]
5. [273/225] Prove that for any nonnegative parameters a and b, the following algorithms terminate and produce identical output. S LOW E UCLID(a, b) : if b > a return S LOW E UCLID(b, a) else if b == 0 return a else return S LOW E UCLID(a, b a)
FAST E UCLID(a, b) : if b == 0 return a else return FAST E UCLID(b, a mod b)
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