CS 422,  Fall 2002  Instructor:  Jeffrey Horn

Final Exam Part 1:  Topics

ANALYSIS

• Big Oh Notation (including Big Omega, Big Theta, little Omega, little theta, little oh)
• Example questions:  Section 1.3, page 11, numbers 1.2-2 and 1.2-5 are good examples.
• Be familiar with the two common data structures for representing graphs:  ADJACENCY MATRIX and ADJACENCY LIST.  I will likey ask a question along the lines of "which would be more efficient to use here?"  (e.g., with a sparse graph, or perhaps with a "dense" graph)
• Also I might ask a question about binary search, and if it would speed up a given sorting algorithm.  So be familiar with binary search.

UNDECIDABILITY

• Example Questions:
  1. You are asked to a write a program that reads in the Java source code of any other program, P, and answers the following questions.  Which of the following are undecidable?
     1. Are all of the variable names in P syntactically correct?
     2. Does every method defined in P get called at some point?
     3. Will P ever cause an "array-out-of-bounds" exception to be thrown?
     4. etc.

NP-COMPLETENESS

• Know what an "independent set" (of vertices) is.  This NP-Complete problem is closely related to Clique, and is described in our text on page 961.
• Have some idea what the hierarchy means, what a polynomial transformation is, and why it must be polynomial, etc. 
• Might ask questions like:  "Which of the following are instances of NP-Complete problems?  Identify the NP-Complete problem."   Only worry about CLIQUE, VERTEX-COVER, SATISFIABILITY, HAMILITONIAN-CIRCUIT, and maybe SET-COVER.

APPROXIMATION ALGORITHMS

• Example questions:
  1. Explain how the maximum degree of a graph G (i.e., the degree of the vertex with the highest degree, which is the number of incident edges) can be used to compute an upper bound on the max. clique size.  Give pseudocode if that helps you to explain.  Is this computable in polynomial time (in n, the number of vertices in G)?
  2. So, then, given the answer to the (1) above, consider the restricted MAX-CLIQUE problem, "What is the maximum size clique in G, where G has vertices <= k?" .  Is this restricted problem also NP-Complete or not?  Why?