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CS255 Spring 2006Practice Final

The practice exam will appear one week before the exam.

To study for the final I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes three times. Second and third time try to see how huch you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice final is below. Here are some facts about the actual final: (a) The final will be in class Monday, May 22 from 5:15pm-7:30pm. (b) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (c) You should bring photo ID. (d) There will be more than one version of the test. Each version will be of comparable difficulty. (e) If your cell-phone or beeper goes off you will be excused from the test at that point and graded on what you have done till your excusal. (f) One problem (less typos) on the actual test will be from the practice test.

[Practice Final Student Solutions-PDF]

1. Find the unique number x mod 30 using the Chinese Remaindering technique from class such that x mod 2 = 1, x mod 5 =4, and x mod 3 = 2.

2. Where in the proof of correctness of Rabin-Miller do we use the following theorem:

Theorem. The values of n > 1 for which Z_n^* is cycle are 2, 4, p^e, and 2p^e for all primes p > 2 and for all positive integers e.

3. Give the RSA transformation P and S and prove that they are inverse of each other.

4. Prove VERTEX-COVER is in NP.

5. Consider the language {F | F is a boolean formula whose smallest satisfying assignment is unique}. Here size of an assignment is the number of variables set to true. Show this language is coNP-HARD.

6. Prove TAUTOLOGY is coNP-complete.

7. Prove the TSP where the cost function satisfies the triangle inequality has a 2-approximation scheme.

8. Give an 8/7-approximation scheme for MAX-3-CNF.

9. What is a Carmichael number? What is a pseudo-prime?

10. Prove that {x | x is a PRIME} is in coNP.