Questions are in black, answers in blue.
Yes, if f = Ω(g) then there must exist a positive real number c such that f(n) is at least cg(n) for sufficiently large n.
Actually it is the maximum possible number of bits in the answer -- the number of coefficients in the polynomial C(x). Is this the same "n" as the "n"'th roots of unity? Is that the same "n" as in "p = kn + 1"?
Yes. You're going to multiply two polynomials, whose product has degree at most n-1, using FFT and thus using n'th roots of unity. You're told that these roots of unity exist if p is equal to kn+1 for some k.
No, you are asked to multiply two integers in ordinary binary notation and get a binary result. You're going to use FFT to help you do this, and FFT needs roots of unity. Rather than use the roots of unity in the complex numbers, you're going to use the roots of unity in the field of integers modulo p. That's where p comes in -- when you add and multiply "numbers" in the FFT algorithm, these are going to be additions and multiplications modulo p.
The snarky answer is that you should get software that supports standard HTML. But failing that, using "View Source" will tell you which symbol goes in which blank space.
Actually the new conditions on f make that O(nk) bound irrelevant. The reason you now know bounds on g is that g is increasing, g agrees with f on the input values bi, and f is either Θ(nα) or Θ(nα log n). These conditions prevent g from being too small.
You're right, it wasn't -- it is now. Corrections are in green. Sorry.
The point of the problem is that FFT gives us a fast way to multiply
polynomials, and we would like to use it to help us multiply integers
quickly. Here's an example. We have two integers A = 13 and B = 7.
We form polynomials A(x) = 1 + x2
+ x3 and B(x) = 1 + x +
x2, so that A(2) = 13 and B(2) = 7.
The product of these two polynomials is C(x) = 1 +
x + 2x2 +
2x3 + 2x4 + x5. C(2) is equal to 91, the
product of 13 and 7, but the coefficients of C(x) do not immediately
give us the bits of the binary representation of 91.
In part (a) you have to figure out how to get from the coefficients of C
to a binary representation of C(2), and how much time this will take. In
part (b) you have to look at how many bit operations (such as AND and OR)
you will need to carry out the FFT algorithm.
Sorry, of course I meant for you to solve it in O(n) time -- this has been corrected on the assignment page.
You're right, and I have corrected the assignment sheet. You can get up to five extra points for decscribing your counterexample to the statement I originally asked you to prove.
font color=blue>But (as you realized in another email a few minutes later)
you don't know how to find the median if the elements are not
Comparable -- it isn't even defined!
You're right, I corrected this as well.
And in Problem 1.4 (b), do you really mean to solve the "element distinctness problem" in O(n) time? I read that this requires Ω(n log n).
Another mistake, sorry, I meant the same "majority element problem" from the rest of problem 1.4.
Yes. Corrected on the assignment page.
Also, is the variable i an integer like i0 or may it be any real?
It must be an integer. Also corrected on the assignment page.
Last modified 22 September 2005