CS255 Spring 2015Practice Final

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 at least three times. Second and third time try to see how much 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) It is comprehensive (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) It is 10 problems, 6 problems will be on material since the lecture before the midterm, four problems will come from the topics covered prior to the midterm. (f) Two problems will be exactly (less typos) off of the practice final, and one will be off of practice midterm.

1. Define `C_R^(obl)`. How does our competitiveness lower bound from class for a randomized paging algorithm depend on the cache size? How is the coupon collector problem used in the proof of the lower bound?
2. Give pseudo-code for the Euclidean algorithm then use it to calculate (showing computation steps) the gcd of 124 and 42.
3. Prove the equation `ax equiv b(mod n)` either has `d` distinct solutions modulo `n`, where `d=gcd(a,n)`, or it has no solutions.
4. Using the Chinese remaindering construction from class, find a number `a mod 42` such `a mod 6 equiv 5 mod 6`, `a mod 7 equiv 3 mod 7`. You should first show the prerequisites of the Chinese Remainder Theorem are met.
5. Briefly describe the RSA cryptoscheme.
6. Define the complexity classes `P`, `NP`. Define the terms `p`-time reduction, `NP`-hard, and `NP`-complete.
7. Give a polynomial time reduction from CIRCUIT-SAT to 3SAT.
8. Give an example with proof of a NP-hard optimization problem which cannot be approximated to within a constant factor `r` for any `r >0 `.
9. Give a randomized constant factor approximation algorithm for MAX-5SAT.
10. Give the APPROX-SUBSET-SUM algorithm from class.