Concrete mathematics : a foundation for computer science

434 ASYMPTOTICS
inefficient “because its running time is O(n2)!’ But a running time of 0 (nL)
does not imply that the rrrnning time is not also O(n). There’s another
notation, Big Omega, for lower bounds:
f(n) = fl(gin)) W If(n)1 3 Clg(nil for some C > 0. (9.17)
We have f(n) = fI(g(n)) if and only if g(n) = O(f(n)). A sorting algorithm
whose running time is n( n’ ) is inefficient compared with one whose running
time is 0 (n log n) , if n is large enough.
Finally there’s Big Theta, which specifies an exact order of growth: Since 0 and 0 are
f(n) = O(g(n))
‘In) = o(g(n)) w and f(n) = n(g(n)) .
umercase Greek
It%ers, the 0 in
(9.18) O-notation must
be a capital Greek
We have f(n) = @(g(n)) if 0.
This is essentially the relation f(n.) + g(n) of (9.8). We also have
(9.19)
f(n) - 4(n) W f(n) = s(n) +0(64(n)). (9.20)
Many authors use ‘0’ in asymptotic formulas, but a more explicit ‘0’ expression
is almost always preferable. For example, the average running time
of a computer method called. “bubblesort” depends on the asymptotic value
of the sum P(n) = ,YF=, kn k kl/n’. . . Elementary asymptotic methods suffice
to prove that P(n) N m?!, which means that the ratio P(n)/a approaches
1 as n t co. However, the true behavior of P(n) is best understood
by considering the d@eerence, P(n) - J7cn/2, not the ratio:
n ( P(nllJ7m72 1 P(n) - &G
1 0.798 -0.253
10 0.878 -0.484
20 0.904 -0.538
30 0.918 -0.561
40 0.927 -0.575
50 0.934 -0.585
The numerical evidence in the middle column is not very compelling; it certainly
is far from a dramatic proof that P(n)/- approaches 1 rapidly,