601.433/633 Introduction to Algorithms Fall 2018

Homework  #1 Due:  September 13, 2018, 1:30pm

Remember: you may work in groups of up to three people, but must write up your solution entirely on your own. Collaboration is limited to discussing the problems – you may not look at, compare, reuse, etc. any text from anyone else in the class. Please include your list of collaborators on the first page of your submission. You may use the internet to look up formulas, definitions, etc., but may not simply look up the answers online.

Please include proofs with all of your answers, unless stated otherwise.

1 Asymptotic Notation (40 points)

For each of the following statements explain if it true or false and prove your answer. The base of log is 2 unless otherwise specified, and ln is loge.

(a) log(n⁷⁰) = O(log(n^1/2))

(b) 2ⁿ = Θ(eⁿ)

(c) 1000(n log² n + ¹ n²) = Θ(n²)

(d) 3ⁿ = Θ(3⁽ⁿ⁻⁴⁾)

(e) n cos n = Θ(n)

(f) Let f, g be positive functions. Then f (n) + g(n) = Ω(min(f (n), g(n)))

(g) Let f, g be positive functions, and let g(n) = o(f (n)). Then f (n) + g(n) = Θ(f (n))

(h) 2^5 log n = O(n²)

2 Recurrences (35 pts)

Solve the following recurrences, giving your answer in Θ notation. For each of them you may assume T (x) = 1 for x ≤ 5 (or if it makes the base case easier you may assume T (x) is any other constant for x ≤ 5). Justify your answer (formal proof not necessary, but recommended).

(a) T (n) = 3T (n − 5)

(b) T (n) = n^2/3T (n^1/3) + n (c) T (n) = 4T (n/3) + n

(d) T (n) = 4T (n/4) + n log4 n (e) T (n) = T (n − 3) + 5

3 Basic Proofs (25 pts)

(a) Let A, B, C, D be sets. Prove that

(A ∪ B) ∩ (C ∪ D) = (A ∩ C) ∪ (B ∩ C) ∪ (A ∩ D) ∪ (B ∩ D)

(b) There are 130 students registered for this class. Prove that there are at least 11 students who were all born in the same month.

(c) Prove by induction that Σn

(2i − 1) = n² for all positive integers n.