Homework 6

[Note: this webpage last modified Wednesday, 16-Mar-2011 08:56:31 EDT]

This homework assignment is due before class starts on Tuesday, March 22. Your solutions should be explained in complete sentences such that your classmates will understand the solution (and can verify your proofs are correct) even if they have not solved the problems themselves.

  1. HW Polices.

    1. Type your HW (in word, latex, text file, etc.) and send to me by email.

    2. Do NOT share electronically. You must type your own solutions. You can discuss the problems with each other, but you may only discuss them. You may not write out solutions together.

    3. You MAY NOT search the Internet, textbooks, etc. for solutions to the problems. The following are the ONLY sources of information that you may use in solving the problems: the textbook for this course, wikipedia articles on basic math/probability/etc., and mathematical review material at the following MIT opencourseware page: 6.042J / 18.062J Mathematics for Computer Science, Fall 2005. You may discuss the problems with each other and with myself, but must obey the previous item in doing so.

      If you do find the solution in one of these three sources, you still MUST cite the source in your document.

      You may use NOTHING ELSE that is online or other textbooks.

    4. You MAY NOT copy word-for-word from any source, even the three you are allowed to consult. If you feel it is necessary, you should put the quotation in quotes and provide a reference/citation.

  2. (-2 points if left blank) List who you collaborated with on this assignment, "none" if none.

  3. (5 Points) Initial project idea. In one or two pages, describe what you plan to work on for the project. If you plan to study a hard problem (e.g., NP-complete problems), say what the problem is, what is the complexity of the problem (NP-complete, PSPACE, etc.), what is the problem useful for, and what are the best-known algorithms. Also indicate which aspects you plan to study in more depth and what your sources are for information on the problem. If you plan to focus more on writing programs for the hard problem, then you could begin by giving a short description of the problem and include a program that solves the problem by brute force.

    See the project webpage for more ideas, and further details.

    For the project, you may look at any source. You must list your sources at the beginning of your document. You may NOT copy word for word from any source (except for short quotations if needed, and they should be clearly marked as quotations).

  4. (5 Points) Determine the smallest value of n such that the expected number of pairs of people with the same birthday in a group of n "random" people is at least 1 (and show your work). Also determine the smallest value of n such that the expected number of triples and groups of four all having the same birthday is at least 1.

    So you will have three different values for n: the one where the expected number of groups of two is at least 1, the one where the expected number of groups of three is at least 1, and the one where the expected number of groups of four is at least 1. As a "reality/dummy check", the value for groups of two should be less than the value for groups of three, which should be less than the value for groups of four.

  5. (Extra credit, 2 Points) Let ZPPZPP be the set of problems that can be decided with a "ZPP type" randomized algorithm (i.e., zero error) when the randomized algorithm is given free ( and always correct) answers to some ZPP problem. Show that ZPPZPP = ZPP.

    By contrast, we do not know if RPRP=RP. Explain why this situation is different than ZPPZPP.

    Note. Here are some other containments, for your information only - you do not need to prove these. RPRP is contained in BPP. BPPBPP=BPP. RPZPP is contained in RP.