Course Schedule
[Note: this webpage last modified Thursday, 21-Apr-2011 11:31:51 EDT]
We will cover somewhere around one third to one half of the required textbook. This web-page will be kept up to date with the required reading to perform before each lecture. I will also list the key concepts that we cover each week, in particular the most important definitions and theorems.
The following gives a tentative outline of the semester, organized by week. This webpage will be updated throughout the semester as things change.
For the key theorems/definitions, it is important for all of them that you understand the statement of the definition/theorem. The book often has examples that can help you understand the definitions and statements of the theorems. For some of the theorems, it will also be important to understand the proof; I will mark these as we cover them in the class.
Week 0 (Jan 11, 13)
- Key Theorems/Definitions: Some of the classic proofs in math and CS: infinitude of primes, sqrt(2) is irrational, uncountability of the real numbers, some proof by induction. Definition of computable functions and running time (Definition 1.3), measurement of this as function of input length, what is input length, Turing machines versus regular computers, universal machine (Theorem 1.9). Uncomputable functions (Theorems 1.10, 1.11).
- Required Reading: Appendix A (in particular A.1). Chapter 0. open courseware readings for weeks 1, 2, and 3, and the asymptotics part of the reading for week 8. Sections 1.1, 1.2, Definition 1.3. Sections 1.4, 1.5.
- Just for fun: video (click on "Part 2") and powerpoint slides for rousing talk by Scott Aaronson about "the future of CS".
Week 1 (Jan 18, 20)
- Key Theorems/Definitions: Definition for P and NP (Definitions 1.13, 2.1), examples of problems that are in P, problems that are in NP. Hierarchy theorems for P and NP (Theorems 3.1, 3.2, 4.8, 6.22). Reinforce input length issue with factoring example. Problems that are in P: all regular languages, all CFG's, sorting, shortest path, minimum spanning tree, ... Reinforce difference between solving a problem on some particular instance and solving the problem on all possible instances. Also, that running time is worst-case running time and not just running time on some particular input - a good example is problem 4 from hw0.
- Required Reading: Sections 1.6, 2.1, 3.1, 3.2.
Week 2 (Jan 25, 27)
- Key Theorems/Definitions: NP-completeness (Definition 2.7). Various and sundry NP-complete problems (Theorems 2.9, 2.10, 2.16, ..., Exercises 2.15, 2.16, ...). Emphasize difference between "not in P" and NP-complete. P versus NP problem. Search versus decision problems (Theorem 2.18).
- Required Reading: Sections 2.2, 2.3, 2.4, 2.5, 2.7.
Week 3 (Feb 1, 3)
- Key Theorems/Definitions: NP-completeness proofs and undecidability proofs. See the hw2 page for which ones are being covered. For Feb 1, all students should be prepared to present their NP-completeness proof.
- Required Reading: Students should provide a link to their source for the proofs. Those will be posted on the hw2 page.
Week 4 (Feb 8, 10)
- Key Theorems/Definitions: Problems that are "even easier" than P - space-bounded (Definition 4.1), NC circuits (Definitions 6.1, 6.27), regular expressions, CFG's, ... Fact that NL=coNL (Theorem 4.20), whereas we think NP!= coNP. PATH is NL-complete (Theorem 4.18).
- Required Reading: Sections 4.1, 4.3, 6.7 (background in sections 6.1, 6.2 as needed)
Week 5 (Feb 15, 17)
- Key Theorems/Definitions: Problems that are "likely even harder" than NP - PH (Definitions 5.1, 5.3, Exercise 2.33), #P (Definition 17.5), PSPACE (Theorem 4.13 TQBF is PSPACE-complete), EXP. Fact that NPSPACE=PSPACE (Theorem 4.14), wherease we think P!=NP. If P=NP then EXP=NEXP (Theorem 2.22).
- Required Reading: Section 2.6, 4.2, 5.1, 5.2, 17.1, 17.2.
Week 6 (Feb 22, 24)
- Key Theorems/Definitions: What can we do with NP-complete problems -- approximation algorithm (Definition 11.1, Examples 11.2, 11.3), parameterized complexity (fixed-parameter tractable algorithm for vertex cover, see for example these lecture notes from MIT), approximation algorithm for metric TSP (see topic 9 on Approximation Algorithms from these lecture ntoes), solving sub-cases of problems (e.g. independent set on a tree in section 10.2 in Algorithm Design by Kleinberg and Tardos), approximation algorithm for vertex cover using linear programming (see topic 10 on LP relaxation, rounding from these lecture ntoes), average-case complexity.
- Required Reading: Section 11.1.
Week 7 (Mar 1, 3)
- Key Theorems/Definitions: Alternative stand-in for P as "efficiently computable" - randomized algorithms (BPP, RP, ZPP) (Definitions 7.2, 7.3, 7.6, 7.7). Examples of randomized algorithms and error-reduction (Lemma 7.9, Theorem 7.10, Exercise 7.4). Basics of probability theory.
- Required Reading: Sections 7.1, 7.2, 7.3, 7.4. open courseware readings for weeks 9, 10, 12, 13.
Week 8 (Mar 15, 17)
- Key Theorems/Definitions: More on randomized algorithms, some of them approximation algorithms for hard problems. More basics of probability theory. Randomized algorithm for approximating 3-SAT (Exercise 11.3). Examples of using probability: birthday problem, Monty hall 3-door problem, BPP in P/poly, BPP in Sigma2, randomized algorithm for max-3sat (with 7/8 approximation ratio), randomized algorithm for min-cut (see Topic 6 from these lecture notes), randomized algorithm for median-finding (see these lecture notes).
- Required Reading: Sections 7.5, 7.7.
Week 9 (Mar 22, 24)
- Key Theorems/Definitions: What if there are hard languages? What can we do with them? One-way functions (Definition 9.4) - weak and strong one-way functions, hard-core predicate. If P=NP then no one-way functions. If P=NP then no cryptography (Lemma 9.2). Important point - many of the arguments are much easier when dealing with randomized algorithms, and we don't know how to make them work for deterministic algorithms - that is one motivation for looking at randomized algorithms. Theorems 9.11, 9.12, 9.13.
- Required Reading: Sections 9.1, 9.2, 9.3.
Week 10 (Mar 29, 31)
- Key Theorems/Definitions: Using one-way functions for: PRG's (Definition 9.8, Theorem 9.9), pseudorandom functions (Definition 9.16, Theorem 9.17), bit-committment, private-key encryption (Theorem 9.6), zero-knowledge (Definition 9.14, Example 9.15), secure multi-party computation. Fact that each of those almost trivially implies one-way functions. Examples of these.
- Required Reading: Sections 9.4, 9.5.
Week 11 (Apr 5, 7)
- Key Theorems/Definitions: Special-purpose PRG's - t-wise independent, small-bias, use of that for derandomized algorithms (Exercise 11.4).
- Required Reading:
Week 12 (Apr 12, 14)
- Key Theorems/Definitions: Quantum algorithms (Definitions 10.6, 10.9) - Grover's search algorithm (Theorem 10.13), Shor's factoring algorithm (Theorem 10.15). BPP is in BQP (Corollary 10.11). BQP is in PSPACE (Theorem 10.23).
- Required Reading: Sections 10.1, 10.2, 10.3, 10.4, 10.6, 10.7.
Week 13 (Apr 19, 21)
- Key Theorems/Definitions: Other complexity measures - decision tree/query complexity (Definition 12.1), communication complexity (Definition 13.1, deterministic lower bound of n for equality, randomized equality testing with O(log n) bits using fingerprinting). Existence proofs and probabilitist method - most functions have high communication complexity, most functions require large circuits, most strings are "uncompressible", ...
- Required Reading: Sections 12.1, 13.1.
Week 14 (Apr 26, 28)
- Key Theorems/Definitions: Mention hardness of approximation results (Theorem 11.9, Corollary 11.10, Theorems 11.15)
- Required Reading: Sections 11.2, 11.4.
Final Exam Slot
Tuesday, May 3 3:00-5:00pm. We will have some of the final project presentations.