cs 4440 exam study guide

Theory Of Algorithnms (CS 4440) Exam Study Guide

Chapter Seven -- Divide-and-Conquer

Understand the definition of divide-and-conquer algorithm.
Understand how to calculate the big-Θ of portions of a divide-and-conquer algorithm and how to solve a recurrence to obtain information about the work function of a divide-and-conquer algorithm.
Understand what the algorithms in chapter seven do (all the steps).
Be familiar enough with the steps of the algorithms in the chapter so you are able to say what happens to data structures as an algorithm progresses.
- For example, if our task is to use divide-and-conquer multiplication to calculate the product of 981 X 1234, then to what three 'smaller' multiplications will we reduce the problem? {Ans: 9X12, 81X34, and (9+81)X(12+34)=90X46}.
- Another example: suppose I tell you that left=1, right=17, piv=75, s=7, and the elements of array A in slots 'left' through 'right' are
```
        [51 70 87 68 94 59 81 95 37 80 26 75 90 03 52 51 03]
	
```
  just before the call to pivotbis in the repeat loop of the selection2 algorithm. Which of those elements of A will be in slots 'left' through 'right' after the call to pivotbis executes, and after the part of the code that updates 'left' or 'right' executes? { Ans: 51, 70, 68, 59, 37, 26, 03, 52, 51, and 03 (in some order)}

Chapter Eight -- Dynamic Programming

Understand the definition of dynamic programming algorithm.
Understand how to compute the big-Θ of the work function of part or all of a dynamic programming algorithm.
Understand what the algorithms in the chapter do (all the steps).
Be familiar enough with the steps of the algorithms in the chapter so you are able to say what happens to data structures as an algorithm progresses.
Understand and be able to explain the principle of optimality.
Understand memory functions, the advantages of combining dynamic programming with top-down methods using a table as a memory function, and how to program such algorithms.

Chapter Nine -- Exploring Graphs

Know how to set up and mark a game graph for a simple game
- Understand how to make the nodes and edges.
- Know how to mark winning and losing positions
- Know how to find winning moves and losing moves.
Understand the backtracking technique -- be able to trace out steps of the backpack(i,r) function of page 308.
Understand how to do the assignment problem of pages 312-315.
Understand the role of the bounding function in the assignment problem.
Understand the "General Considerations" in section 9.7.3

Be able to disuss the version of the knapsack problem described in section 9.6.1, and the basic characteristics of the algorithm given there for finding a solution.

Be familiar with the meaning of the branch-and-bound technique and with how it is applied to the assignment problem.

Chapter Ten -- Probabalistic Algorithms

Introduction (to Probabilistic Algorithms) pp. 328-329
Probabilistic Does Not Imply Uncertain pp. 329-330
Numerical Probabilistic Algorithms p. 333
Numerical Integration pp. 336-338
Monte Carlo Algorithms p. 341

Know the differences among the concepts of Numerical Algorithm, Monte Carlo Algorithm, and Monte Carlo Algorithm.

Chapter Twelve -- Computational Complexity

Read the chapter. Know the material. Use the notes in the class web space as a guide for figuring out what I consider important. Also, use the notes to help figure out and understand the concepts.
What is the study of computational complexity? What kinds of questions are addressed by this study?
What is a decision tree? What is a verdict in a decision tree? What is a trip through a decision tree? How does a decision tree correspond to an algorithm? What is the relationship between the height of a decision tree and the number of questions asked by the corresponding algorithm? What is the connection between the height of a decision tree and the number of possible verdicts in the tree?
What is an adversary argument? Explain the role of the "malevolent daemon" (the adversary) in the example of 20 questions, finding the maximum of an array, or testing graph connectivity.
What are the formal definitions of linear reducibility and linear equivalence? What is meant by the transitivity of linear reducibility and linear equivalence?
Be prepared to verify whether or not an alleged "proof" of linear reducibility is correct.
What is a polynomial time algorithm?
What is a decision problem?
What is the class P?
What is the class NP?
Be able to prove that P is contained in NP.
What are the formal definitions of polynomially Turing reducible and polynomially Turing equivalent? What is meant by the transitivity of polynomial Turing reducibility and polynomial Turing equivalence?
Be prepared to verify whether or not an alleged "proof" of polynomial Turing reducibility is correct.
Be familiar with the statement and proof of theorem 12.5.6: "Consider two problems A and B. If A is polynomially Turing reducible to B and if B can be solved in polynomial time, then A can also be solved in polynomial time."
What does it mean to say that a problem is decision-reducible?
What is the importance of decision-reducibility?
Be familiar with the specific problems discussed in chapter 12. For example, be familiar with the travelling salesperson problem, the problem of determining if an undirected graph has a Hamiltonian circuit, the problem of multiplying symmetric matrices, the problem of coloring a graph, and so forth.
What is an NP-complete decision problem?
Be familiar with Theorem 12.5.11, and how it can be used to find "new" examples of NP-complete problems.
What is the satisfiability problem SAT? What are the problems SAT-CNF and SAT-k-CNF?
What does Cook's theorem say? What is the importance of Cook's theorem?
Be prepared to give some examples of NP-complete problems.
What is an NP-hard problem?
Is every NP-hard problem NP-complete? Explain.
Is every NP-complete problem NP-hard?
What does NP actually stand for?
What is a non-deterministic decision algorithm? Specifically how is a non-deterministic algorithm allowed to differ from a deterministic one?