I've been trying for the better part of an hour to find reference to the following:
f = Ω(g)
But I have had no luck at all. I need to answer a question for an assignment and I can't find references.
The assignment is basically asking me to indicate what it (f = Ω(g)) means, in the context of the following choices:
f = Ω(g(n))
g = o(ln n)
g = o(g(n))
g = O(f)
f = O(g)
Initially, I thought that perhaps there is an error in the question.
I know option 1 is wrong and assume option 5 is also wrong, but after an hour online I couldn't figure out which one is the answer.
Can someone please explain to me how to figure this out? I realize that might mean giving me the answer so it can be explained, but I'm more interested in why one of these answers are correct.
"f = Ω(g) means "f is bounded below by g asymptotically". f = O(g) means "f is bounded above by g asymptotically" as per the comments.
If a river's upper bound is a bridge, what's a bridge's lower bound? The river.
I would suggest d
(For completeness, the "little" versions of these imply a very strong difference in growth.)
Related
I have always been confused with this and I wanted to clarify this.
How exactly do you prove that an asymptotic notation is true?
Example 1:
sin(n) = Ω(cos(n))
What I have been doing is to rewrite the question into the big-omega
form which says:
f(n) >= c * g(n) for all n >= n0
Which is: sin(n) >= c * cos(n)
I then proceed to randomly find an n and c that makes the equation
true.
In this case, I found c = 5, n = 3 which makes:
sin(3) >= 5 x cos(3) true
Another example 2:
2^squareroot(lgn) = ω(lgn)
I tried n = 5 and c = 6 as well as n = 6 and c = 5, and both yielded
false, however, the answer is actually true.
However, I am not sure if this is the correct way to do it, because I managed to make it true while the answer says that it is false.
I believe I've been doing it all wrong by 'guessing the n and c' values to make the equation true. When I watch some youtube videos, they seem to be shifting equations around using algebra, before guessing a value for n, and I am not sure how to do it for this instance.
What is the correct way to do it?
How would i find the smallest positive integer for which algorithm B out-performs algorithm A?
A = n/4, B = 8×log2n(Base of 2 Not 2n)
A = n^3/10, B = 5×n2
A = n^2/2, B = 20×n×log2n(Base of 2 Not 2n)
A = n^4, B = 16×n2×n
It would be greatly appreciated if someone could help me find answers to these :)
You are really asking whether A(n) > B(n).
It is simple to answer to those questions:
solve the inequalities for n
You can also plot the two functions on the same plane and see how they behave and what is the relation among them. The following is for the first of your questions. As you can see it is clear from the graph when one outperforms the other.
For instance n^3/10 > 5×n^2 solves for n>50
http://www.wolframalpha.com/input/?i=n%5E3%2F10+%3E+5%C3%97n2
Consider asking this kind of questions on https://math.stackexchange.com/
Hope this helps
I'm stuck proving or disproving this statement:
If f ≠ ω(g), then f = O(g)
Intuitively, I think that the statement is false, however, I can't figure out a valid counterexample.
My thought is that we know that f is not bounded from below by a function of g, but that tells us nothing about an upper bound.
Any thoughts? Hints in the right direction?
As a hint, this statement is false. Think about two functions that oscillate back and forth, where each function overtakes the other over and over again. That would make f ≠ ω(g), because f is repeatedly dominated by g, and would make f ≠ O(g) because f repeatedly dominates g.
You'll need to find concrete choices of f and g that make this work and formally establish that f ≠ ω(g) and f ≠ O(g) to formalize this, and I'll leave that as an exercise.
Hope this helps!
I got a practice exam question here asking if the following is true/false.
Let f , g, and h be functions from the natural numbers to the positive
real numbers. Then if g is an element of Big Omega( f ) and g is an element of O(h), and f is an element of O(h) then g is an element of big Theta (h)
I got false for this but it is kind of confusing me now because I don't exactly know what Big omega(f) is.
Can someone clarify if my answer to this question is correct / if not, where I went wrong (and explain if possible please).
Thanks.
Check the link I mentioned in the comment. g is an element of big Theta (h) <=> g is bounded both above and below by h, which is not the case. From your post it's only can be deduced that g is bounded above by h. So "false" is correct answer.
I have a question regarding big O notation.
If g(n)=O(f(n)) and h(n)=O(f(n)) is g(n)=O(h(n))?
Is this allways true, sometimes true or allways false?
Thanks
In words: if g is bounded by f and h is bounded by f, is g bounded by h?
From this we can see that the conclusion doesn't follow from the premises. You can construct a counterexample by choosing f, g, and h in such a way that the premises hold but the conclusion does not.