CISC 320 Algorithms and Advanced Programming

Final Exam review info, Fall, 2002

Wed, Dec 18, 1:00pm to 3:00pm Final exam Final exam will cover the last 2/3rds of the course material.

GT chapter 13 sections 1, 2, 3, and 4.2:
Definitions of problem class P, class NP, and class NP-complete.

P: solve in polynomial time, i.e. in O(n^k) time, for a problem dependent, but input independent k.
NP: solve, if given a correct hint, in polynomial time.
NP-complete: as hard as any problem in NP. Known NP-complete problem can be polynomially reduced to this problem.
Polynomial reduction -- problem A to problem B: Shows B is hard. If A is NP-complete and B is in NP, a polynomial reduction of A to B proves B is NP-complete too.
Method: Take any input x for A. Use it to construct an input y for B, such that (1) the size of y is a polynomial in the size of x, and (2) the correct A answer for x is true if the correct B answer for y is true and (3) the correct B answer for y is true if the correct A answer for x is true.

How to prove a problem is in NP. How to prove a problem is NP-complete
Familiaritity with some NP-complete problems
Circuit SAT, CNF SAT, 3 SAT, Vertex-Cover, Set-Cover, Subset-Sum, Knapsack, Hamiltonian-Cycle, TSP (Traveling Salesman Problem).
Approximation for vertex cover.

GT section 12.5:
Convex hull via gift wrapping and via Graham scan. O(nh), O(n log(n)).

GT section 9.1:
Substring search. Boyer-Moore and Knuth-Morris-Pratt

GT chapter 10 sections 1, 2, 3, and 4.
Fast Fourier Transform. Do n point transform in O(n log(n)) ops over a field containing n-th roots of unity.
RSA Cryptography and integer algorithms. RSA script for maple, and it's output.
Karatsuba divide and conquer multiplication. O(n^sqrt(3).
Shoenhage/Strassen FFT based multiplication. O(n log(n) log log (n))
Exponentiation and inverse mod n. Compute a^e mod n in O(log(e)) steps, for exponent e, Inverse, use Extended Euclidean Algorithm.

GT chapter 5, sections 2 and 3. General strategies.
Divide and Conquer, Dynamic Programming, (you do not have to prepare vis a vis 5.1 Greedy methods.)
Divide and Conquer: Karatsuba's O(n^1.59) integer and polynomial multiplication algorithm Strassen's O(n^2.81) matrix multiplication algorithm.
Dynamic Programming: matrix chains, football scores
Handout: football score program and memoized version.

GT section 2.5.
Hash tables.
separate chaining and open addressing
For open addressing: linear probing, double hashing.

Kinds of questions: As in the first midterm, a mix of multiple choice, short answer, short essay, algorithm to read and analyze, algorithm to write.

Homework based questions or similar can be expected, vis a vis hw3, hw4.