CS255 Spring 2018Practice 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 (3pts each), 6 problems will be on materials since the midterm, 4 problems will be from the topics of the midterm. (f) Two problems will be exactly (less typos) off of the practice final, and one will be off of the practice midterm.

1. Use Euclid's algorithm step-by-step, showing work, to compute the gcd of 56 and 76.
2. Prove the equation `ax=b mod n` either has no solutions or it has `d = gcd(a, n)` distinct solutions mod `n`.
3. Find a number `a mod 42` such that `a mod 2 = 0`, `a mod 3 = 2`, and `a mod 7 = 5`.
4. Give the Miller Rabin primality checking algorithm.
5. Define NP, NP-complete, and NP-hard. Argue the problem of determine where every assignment to a 3-SAT instance is satisfying is NP-hard.
6. Give a p-time reduction from CLIQUE to 3-SAT. Give a p-time reduction from 3-SAT to CLIQUE.
7. Prove APPROX-VERTEX-COVER is a p-time 2 approximation algorithm.
8. Give a p-time 16/15-approximation algorithm for MAX 4-SAT (prove your algorithm has the desired properties).
9. Support we had the list `L= langle 50,51,55,70,71,82,83,84,99 rangle`. Give the TRIM(L, `delta`) from class used in our APPROX-SUBSET-SUM algorithm. Then apply it to this list with `delta = 0.1`.
10. What is the probabilistic method? Use it to show for any set of `m` clauses, there is a truth assignment for the variables that satisfies at least `m/2` clauses.