Comments on the final exam: [PDF]

Thanks everyone for a great semester, and enjoy your summer!

]]>Comments on the final exam: [PDF]

Thanks everyone for a great semester, and enjoy your summer!

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It covers the whole course:

- (in theory) anything in Problem Sets 1-7, Exams 1-2, Classes 1-28, Sipser Ch 1-5+7, Liskov’s talk, assigned readings
- (in practice) emphasizes most important things that have been covered many times; slight emphasis on material since Exam 2

It will be similar to Exams 1 and 2 except that no resources may be used (no notes at all!).

I will be out of town Thursday [More...]]]>
**Thursday, May 13, 9am-noon** (as scheduled by the registrar).

It covers the whole course:

- (in theory) anything in Problem Sets 1-7, Exams 1-2, Classes 1-28, Sipser Ch 1-5+7, Liskov’s talk, assigned readings
- (in practice) emphasizes most important things that have been covered many times; slight emphasis on material since Exam 2

It will be similar to Exams 1 and 2 except that no resources may be used (no notes at all!).

I will be out of town Thursday and Friday so will not hold office hours this Thursday. Next week there will be two scheduled office hours:

- Office Hours, Monday (10 May) 1:30-3pm
- Office Hours, Wednesday (12 May) 4-5:30pm

I am willing to entertain any questions you want during these times, but may decide not to some answer questions directly related to the exam preview handout [PDF].

]]>Thanks everyone for your contributions to the class!

]]>Thanks everyone for your contributions to the class!

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*n*! is **not** in *O*(2*n*).

It is easy to see that *n*! < *n**n*, since all the numbers in the product to compute *n*! are less than (or equal to) *n*, but this doesn’t prove it is not in *O*(2*n*). For that, we can use Sterling’s approximation which gives a tight approximation of the value of *n*! as
which is definitely not in *O*(2*n*) since the base of [More...]]]>

*n*! is **not** in *O*(2^{n}).

It is easy to see that *n*! < *n*^{n}, since all the numbers in the product to compute *n*! are less than (or equal to) *n*, but this doesn’t prove it is not in *O*(2^{n}). For that, we can use Sterling’s approximation which gives a tight approximation of the value of *n*! as

which is definitely not in

For details, see this wikipedia page.

Sorry I was confused on this today! I was thinking of Fibonacci, which is approximated by φ^{n} where φ is the golden ratio (1.618…). This is in *O*(2^{n}) since 1.618… < 2.

I hope it didn’t change the outcome of the game. As my penance, maybe I should add a question related to this on the final to redeem myself.

]]>Some reminders from the PS7 handout are below, see that handout for full details.

If you work in a team, your team should jointly post a single submission with all of your names on it.

Your post should include:

Some reminders from the PS7 handout are below, see that handout for full details.

If you work in a team, your team should jointly post a single submission with all of your names on it.

Your post should include:

- the names of everyone on the team
- a description of your target audience
- either the artifact itself or a link to it if it is hosted elsewhere
- (optionally) a poetic license statement

Note: if you log into your account on this blog, you can post a comment without needing moderation. Otherwise, you can post a comment but it won’t appear until I approve it.

Remember to send me email before 5pm today (Monday, May 3) if you would like to present or perform your artifact in class tomorrow.

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In class, someone asked about languages known to be in BQP that are not known to be in NP. There is at least one such language, known as the Recursive Fourier Sampling problem. It is described in this paper: *Quantum complexity theory* by Ethan Bernstein and Umesh Vazirani, ACM Symposium on Theory of Computing, 1993. [Full version: PDF] Section 8.4 of this paper describes the problem, but it [More...]]]>

In class, someone asked about languages known to be in BQP that are not known to be in NP. There is at least one such language, known as the Recursive Fourier Sampling problem. It is described in this paper: *Quantum complexity theory* by Ethan Bernstein and Umesh Vazirani, ACM Symposium on Theory of Computing, 1993. [Full version: PDF] Section 8.4 of this paper describes the problem, but it is definitely not light reading! Before tackling this paper, I would recommend reading Scott Aaronson’s *Scientific American* article: *The Limits of Quantum Computers*.

Since I badly messed up the reduction from SUBSET-SUM to KNAPSACK, and don’t see an obvious way to fix this, you can solve this question in place of Problem 5 on PS6. If you get an especially elegant and convincing answer that you would like to present in class Tuesday, send it to me by Monday afternoon.

The reading handed out today is available here: *The Status of the P versus NP Problem*, [More...]]]>

Since I badly messed up the reduction from SUBSET-SUM to KNAPSACK, and don’t see an obvious way to fix this, you can solve this question in place of Problem 5 on PS6. If you get an especially elegant and convincing answer that you would like to present in class Tuesday, send it to me by Monday afternoon.

The reading handed out today is available here: *The Status of the P versus NP Problem*, Lance Fortnow, *Communications of the ACM*, September 2009.