CMPSCI 311: Theory of Algorithms

First Midterm Exam

David Mix Barrington

5 October 2006

Directions:

• Answer the problems on the exam pages.
• There are seven problems for 100 total points. Actual scale was A = 85, C = 55.
• No books, notes, calculators, or collaboration.
• The exam had a time limit of 120 minutes, though it was not intended that you would need all the time.
• Questions 1, 2, and 3 are "true/false with explanation" -- you get five points for a correct boolean answer, and up to five additional points for a convincing justification.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 20 points
  Q5: 15 points
  Q6: 15 points
  Q7: 20 points
 Total: 100 points 

• Questions 1 and 2 concern the weighted interval scheduling problem. Here the input is a set of n jobs, each with a start time , finish time, and reward. Our task is to schedule some of these jobs on a single machine. The output is a non-overlapping subset of the jobs, and the goal is to maximize the total reward from the jobs in the set.

• Question 1 (10): True or false with justification: The following greedy algorithm always gets the maximum possible reward: Sort the jobs by reward and schedule them one by one starting with the highest reward, rejecting any that overlap with jobs already scheduled.

• Question 2 (10): True or false with justification: Now assume that the reward of each job is proportional to its length. The following greedy algorithm always maximizes the total reward: First schedule the job that starts earliest, then keep scheduling the earliest-starting job that does not overlap with jobs already scheduled.

• Question 3 (15): True or false with justification: Suppose that in the Huffman algorithm to create an optimal prefix code, there is one letter of the alphabet that has a frequency probability of 0.5 or greater. Then that letter will definitely be assigned a one-bit code word.

• Question 4 (20): Let G = (V,E) be a connected, undirected graph where each edge has a positive weight and let X be a non-empty proper subset of V (a set of vertices that is not equal to either ∅ or to V). Consider the set Z of edges that have one endpoint in X and one not in X, and let e be an edge in Z that has smaller weight than any other edge in Z. What does the Cut Property say about e and minimum spanning trees for G? Prove that this Cut Property is true.

• Question 5 (15): Let G = (V,E) be a connected undirected graph and let v be a vertex in G. Let T by the depth-first search tree of G starting from v, and let U be the breadth-first search tree of G starting from v. Prove that the depth of T is at least as great as the depth of U.

• Question 6 (15): Recall that an independent set of an undirected graph G is a set X of vertices such that no edge of G has both endpoints in X. Describe an algorithm that takes G and a number k as input, and finds an independent set of size k in G if one exists. Let n be the number of vertices in G. Analyze the big-O running time of your algorithm as a function of both n and k -- for full credit your algorithm should take O(n^kk²) time.

• Question 7 (20): Here we have a scheduling problem slightly different from the others we have seen. Each job has a length and a preferred finish time, and we must schedule all the jobs on a single machine without overlapping. (There is a single start time at which all the jobs become available at once.) If we complete a job before the preferred time, we get a reward equal to the time we have saved. If we complete it after the preferred time, we pay a penalty equal to the amount of time that we are late. Our total reward (which we want to maximize) is the sum of all rewards for early completion minus the sum of all penalties for late completion.

  • (a,5) Argue carefully that there is an optimal schedule that has no idle time (actually, we want to argue that every optimal schedule has no idle time). That is, every optimal schedule always has one job start at the same time the previous job finishes. (Hint: Show how to change any schedule with idle time to another schedule that has no idle time and gets at least more reward.)

  • (b,15) Consider the greedy algorithm where we sort the jobs by length and schedule them in that order, shortest job first, with no idle time. Prove that this achieves a net reward at least as great as that of any schedule. (Note: In lecture we proved that a different greedy algorithm is optimal for a different goal, that of minimizing the maximum lateness. You cannot simply quote that result because it does not apply to this algorithm or to these rewards and penalties. But a similar exchange argument will work in this new case.

Last modified 9 October 2006