CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Midterm Exam #1

Given in class Friday 3 October 2003

Directions: Each of the first two questions is a statement that may be true or false. Please state whether it is true or false -- you will get five points for a correct boolean answer and there is no penalty for a wrong guess. Then justify your answer -- the remaining points will depend on the quality and validity of your justification.

• Question 1 (10): (true/false with justification) If f, g, and h are positive increasing functions with f in O(h) and g in Ω(h)$, then the function f+g must be in Θ(h).

• Question 2 (10): (true/false with justification) Let W(n), A(n), and B(n) be the worst-case, average-case, and best-case costs respectively of a particular algorithm on inputs of size n. Then A(n) is in Θ(W(n)+B(n)).

• Question 3 (20): Let T(n) be the function defined by the following recurrence:

  T(0) = 3

  T(1) = 4

  T(n) = (T(n-1)*T(n-2) - 2)/n, for n≥2

  Give an exact solution for T(n) and prove your solution correct by induction.

• Question 4 (40): Let A be an array of n positive int values, where n≥3. The best triple of A is the set of three distinct indices i, j, and k, with i≤j≤k, that makes the product A[i]*A[j]*A[k] as large as possible. You may assume that this product never causes an overflow.

  • (a,20) Write a brute-force algorithm that takes A as a parameter and returns an array B such that B[0]=i, B[1]=j, and B[2]=k where i, j, and k form the best triple of A. Analyze the worst-case running time of your algorithm in terms of n (the length of A), finding the correct Θ-class.

  • (b,10) State and justify a lower bound (in Ω form) on the time needed to solve this problem.

  • (c,10) Informally describe and analyze an algorithm for this problem that is asymptotically faster than your brute-force algorithm in (a). For full credit, its time should match the lower bound in (b), though you may get up to eight points for any asymptotic improvement.

• Question 5 (20): Two questions about a variant of Mergesort:

  • (a,10) Suppose you have three sorted arrays A, B, and C, each of length n/3, and you want to merge them into a single sorted array D of length n containing the same elements. Indicate how you will do this (pseudocode or even English is fine) and determine the Θ-class of the running time of your algorithm.

  • (b,15) Consider the variant of Mergesort where you divide the given array into three equal parts, sort each part, and merge them together as in (a). State and justify a recurrence for the running time T(n) of this algorithm on an input array of length n. Solve this recurrence for the case when n is a power of three. (You may quote the Master Theorem if it is applicable.) In what Θ-class is the running time of this algorithm for all n?

  Last modified 3 October 2003