CMPSCI 311: Theory of Algorithms

David Mix Barrington

Fall, 2003

Practice Final Exam

Posted Wednesday 10 December 2003

Solutions to be posted Friday 12 December 2003

Actual final exam (120 min) is Tuesday 16 December 2003

Directions: There are 150 total points. Each of the first four 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) Let A, B, and C be decision problems and let "≤" denote poly-time reducibility. If A ≤ B and B≤ C, then A ≤ C must be true.

• Question 2 (10): (true/false with justification) Let f and g be functions. If f is in the class o(g), then f+g is in the class Θ(f).

• Question 3 (10): (true/false with justification) Let G be a directed graph given by its adjacency matrix. We say that a vertex v in G is special if (u,v) is an edge for every vertex u. Determining whether a given vertex v is special requires Θ(n) time.

• Question 4 (10): (true/false with justification) If A and B are both decision problems in the class NP, then A ≤ B and B ≤ A cannot both be true. (Again, "≤" denotes poly-time reducibility.)

• Question 5 (30): Consider n items stored in a min-heap. The items may only be moved and compared to one another. Moves and comparisons take O(1) time each.

  • (a,10) Explain how to create, from this heap, an array that has all n items in sorted order, using O(n log n) time. Justify your time bound carefully.

  • (b,20) Explain why such a sorted array cannot be created from the heap in o(n log n) time.

• Question 6 (50): Suppose we are given an n by n matrix of booleans called A, denoting a binary relation, and we want to compute the n by n matrix of booleans B that denotes the reflexive transitive closure of this relation. Let m be the number of entries of the matrix that are "true".

  • (a,10) Carefully describe a reinterpretation of this problem in terms of a directed graph -- what information about the graph is given and what information is asked for?

  • (b,10) Explain how to find B by matrix powering. What is the running time of this method, assuming the ordinary and not the Solovay-Strassen method is used to multiply matrices?

  • (c,15) Explain how dynamic programming can be used to improve this time asymptotically (by Warshall's algorithm). What is the new running time?

  • (d,15) Suppose m is in the class o(n²). Explain how to determine B asymptotically faster than by Warshall's algorithm. State and justify the new running time in terms of n and/or m.

• Question 7 (30): Suppose we are given an array of n integer values and we want to decide whether they are an arithmetic progression when viewed as a set. An arithmetic progression is defined to be a set of the form {a,a+b,a+2b,...,a+(n-1)b} for some integers a and b. Thus, our decision problem is to determine whether when sorted, the array is in this form.

  • (a,15) Show that this decision problem is in the class NP by describing an NP-procedure that can possibly return "true" if and only if the actual answer is "true". Your evaluation phase should take only O(n) time.

  • (b,15) Explain how to solve this decision problem in O(n) time (with an ordinary algorithm, not an NP-procedure).

  Last modified 12 December 2003