11/17/10
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21
CS 61B: Lecture 21
Friday, October 15, 2010
ASYMPTOTIC ANALYSIS (continued): More Formalism
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| Omega(f(n)) is the set of all functions T(n) that satisfy: |
| |
| There exist positive constants d and N such that, for all n >= N, |
| T(n) >= d f(n) |
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^^^^^^^^^^ Compare with the definition of Big-Oh: T(n) <= c f(n). ^^^^^^^^^^^
Omega is the reverse of Big-Oh. If T(n) is in O(f(n)), f(n) is in Omega(T(n)).
2n is in Omega(n) BECAUSE n is in O(2n).
n^2 is in Omega(n) BECAUSE n is in O(n^2).
n^2 is in Omega(3 n^2 + n log n) BECAUSE 3 n^2 + n log n is in O(n^2).
Big-Omega gives us a LOWER BOUND on a function, just as Big-Oh gives us an
UPPER BOUND. Big-Oh says, "Your algorithm is at least this good." Big-Omega
says, "Your algorithm is at least this bad."
Recall that Big-Oh notation can be misleading because, for instance,
n is in O(n^8). If we know both a lower bound and an upper bound for
a function, and they’re both the same bound asymptotically (i.e. they differ
only by a constant factor), we can use Big-Theta notation to precisely specify
the function’s asymptotic behavior.
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| Theta(f(n)) is the set of all functions that are in both of |
| |
| O(f(n)) and Omega(f(n)). |
|-----------------------------------------------------------------------------|
But how can a function be sandwiched between f(n) and f(n)?
Easy: we choose different constants (c and d) for the upper bound and lower
bound. For instance, here is a function T(n) in Theta(n):
c f(n) = 10 n
^ /
| / T(n)
| / **
| / * *
| / *** * **
| / * * *
| *** / * * *
| ** ** / * * *
|* ** * * *
* / * * ** *
| / ** *** ˜˜˜
| / ˜˜˜˜˜
| / ˜˜˜˜˜
| / ˜˜˜˜˜
| / ˜˜˜˜˜ d f(n) = 2 n
| / ˜˜˜˜˜
|/ ˜˜˜˜˜
O˜˜------------------------------> n
If we extend this graph infinitely far to the right, and find that T(n) remains
always sandwiched between 2n and 10n, then T(n) is in Theta(n). If T(n) is an
algorithm’s worst-case running time, the algorithm will never exhibit worse
than linear performance, but it can’t be counted on to exhibit better than
linear performance, either.
Theta is symmetric: if f(n) is in Theta(g(n)), then g(n) is in Theta(f(n)).
For instance, n^3 is in Theta(3 n^3 - n^2), and 3 n^3 - n^2 is in Theta(n^3).
n^3 is not in Theta(n), and n is not in Theta(n^3).
Big-Theta notation isn’t potentially misleading in the way Big-Oh notation
can be: n is NOT in Omega(n^8). If your algorithm’s running time is in
Theta(n^8), it IS slow.
However, some functions are not in "Theta" of anything simple. For example,
the function f(n) = n (1 + sin n) is in O(n) and Omega(0), but it’s not in
Theta(n) nor Theta(0). f(n) keeps oscillating back and forth between zero and
ever-larger numbers. We could say that f(n) is in Theta(2n (1 + sin n)), but
that’s not a simplification.
Remember that the choice of O, Omega, or Theta is _independent_ of whether
we’re talking about worst-case running time, best-case running time,
average-case running time, memory use, annual beer consumption as a function of
population, or some other function. The function has to be specified.
"Big-Oh" is NOT a synonym for "worst-case running time," and Omega is not a
synonym for "best-case running time."
