Homework #1 -- due Monday Wk 2

#	required problems	topic
1	CLR Exercise 3.1-1 on page 52 Let f(n) and g(n) be asymptotically nonnegative functions. Using the basic definition of θ-notation, prove that max( f(n), g(n) ) = θ( f(n)+g(n) ).	O-notation
2	CLR Problem 3-3(a) on pages 61-62 Rank the following functions by order of growth; that is, find an arrangement g1, g2, ..., g30, of the functions satisfying g1 = Ω(g2), g2 = Ω(g3), ..., g29 = Ω(g30). Partition your list into equivalence classes such that functions f(n) and g(n) are in the same class if and only if f(n) = θ(g(n)).                                                                                                                         lg(lg*n) 2lg*n (√2)lg n n2 n! (lg n)! (3/2)n n3 lg2 n lg(n!) 22n n1/lg n ln ln n lg* n n ⋅ 2n nlg lg n ln n 1 2lg n (lg n)lg n en 4lg n (n+1)! √(lg n) lg*(lg n) 2√(2 lg n) n 2n n lg n 22n+1	O-notation
3	CLR Exercise C.3-3 on page 1200 A carnival game consists of three dice in a cage. A player can bet a dollar on any of the numbers 1 through 6. The cage is shaken, and the payoff is as follows. If the player's number doesn't appear on any of the dice, he loses his dollar. Otherwise, if his number appears on exactly k of the three dice, for k = 1, 2, 3, he keeps his dollar and wins k more dollars. What is his expected gain from playing the carnival game once?	expected value
4	CLR Exercise A.2-1 on page 1156 Show that ∑k=1 to n ( 1/k2 ) is bounded above by a constant.	bounding summations

#	suggested problems	topic
5	CLR Exercise 3.1-3 on page 53	O-notation
6	CLR Problem 3-4(d,e,f,g) on page 62	prove/disprove properties
7	Resolve the following questions, with proof: n log n = O(n2)? n/log n = O(n)? (log n)363 = O(n)? 2n = O(2n/2)? 3n = O(2n)?	O-notation
8	The following is known about functions f and g: f(n) = θ(n) ; f(1) = 1 ; f(2) = 2 g(n) = θ(n3) ; g(1) = 1 ; g(2) = 8 Is f(3) < g(3)?     Explain why or why not.	O-notation
9	Prove by induction that, for all n > 6, the unit square can be partitioned into n squares (not necessarily of identical size).	induction
10*	CLR Exercise 3.2-4 on page 60	polynomially-bounded?
11*	CLR Exercise C.2-9 on pages 1195-1196, explanation required	probability