# Mid-Term Exam Study Guide

## Plan

The mid-term exam is scheduled in the Testing Center (see the course schedule). You will have a 3 hour time limit. The exam is closed book, and you may have one page of notes handwritten or typed by you – both sides of the page can be used. Your notes may not include any material pasted in from other resources. No calculators or digital assistants (you won't need one). I think a well-prepared student will be able to complete the exam in two hours. You’re going to do well!

The format is short answers, worked mathematical solutions, and possibly some T/F or multiple choice.

## Study

I recommend the following activities to study the topics covered by the exam:

• Review the lecture notes and identify the topics we emphasized in class. Focus on those listed below. Prepare your one page of notes, handwritten or typed by you. This process will probably be your best preparation.
• Compare the homework solution keys to your homework assignments, and make sure that you understand the major principles covered in the homework problems. The difficulty of homework problems is generally a good gauge for exam problems.

## Topics

The exam will cover a subset of the following topics:

1. Main ideas: Problems, problem instances, solutions, algorithms
2. Orders of growth
3. Distinctions among best case, average case, and worst case
4. The roles of empirical analysis and theoretical analysis
5. Asymptotic notation: formal, mathematical definitions of big-O, big-Theta and big-Omega
• A “formal, mathematical” definition is a precise definition employing mathematical notation (not English or some other natural language).
6. Applications of the definitions of asymptotic notation
7. The Limit and Max rules. L'Hopital's rule will also be useful sometimes when using the Limit rule.
8. Modular exponentiation, esp. the key ideas that make modular exponentiation efficient
9. Euclid’s Algorithm for Greatest Common Divisor and its efficiency
10. Fermat primality tester and its efficiency
11. RSA Public-key Cryptography
• how algorithm analysis informs its security
• NO manual computation using the Extended Euclid Algorithm, including NO computation of multiplicative inverses modulo N
12. Analyzing divide and conquer algorithms using the “Master Theorem”
13. Analysis of Mergesort
14. Quicksort, the pivot selection problem and the worst-case behavior of quicksort
15. Deriving the big-O (upper) bounds for certain kinds of divide-and-conquer algorithms using the methods for solving recurrence relations covered in class. This will involve knowing some logarithm and exponent identities.
• Homogeneous Recurrence Relations
• Non-homogeneous Recurrence Relations (only with geometric forcing functions)
• The characteristic equation theorem
• Roots with multiplicity
• Divide and Conquer-style Recurrence Relations requiring Change of Variable.
16. Formulating problems as graphs
17. Depth-first search and connected components; the broader usefulness of DFS; its efficiency
18. Linearization of a DAG (i.e., topological sorting)
19. Finding strongly connected components
20. Shortest path(s) problem 