Concrete mathematics : a foundation for computer science

A loerarchy?
7
that includes entries like this:
9.1 A HIERARCHY 427
1 -x log logn 4 logn + n’ + nc 4 nlogn 4 cn 4 nn 4 ccn
(Here c and c are arbitrary constants with 0 < E < 1 < c.)
All functions listed here, except 1, go to infinity as n goes to infinity.
Thus when we try to place a new function in this hierarchy, we’re not trying
to determine whether it becomes infinite but rather how fast.
It helps to cultivate an expansive attitude when we’re doing asymptotic
analysis: We should THINK BIG, when imagining a variable that approaches
infinity. For example, the hierarchy says that logn + n”.ooo’; this might
seem wrong if we limit our horizons to teeny-tiny numbers like one googol,
n = 10’O”. For in that case, logn = 100, while no.ooo’ is only loo.” z 1.0233.
But if we go up to a googolplex, n = 1 O”“‘, then logn = 10”’ pales in
comparison with no.ooo’ = 10’Oq6.
Even if e is extremely small (smaller than, say, l/lO'"'oo), the value
of logn will be much smaller than the value of n’, if n is large enough. For
if we set n = 10102k, where k is so large that e 3 10pk, we have logn = 10Zk
but n’ 3 1 O'@. The ratio (logn)/n” therefore approaches zero as n + co.
The hierarchy shown above deals with functions that go to infinity. Often,
however, we’re interested in functions that go to zero, so it’s useful to have
a similar hierarchy for those functions. We get one by taking reciprocals,
because when f(n) and g(n) are never zero we have
1 1
f(n) 4 g(n) H - -
g(n) + f(n) ’
Thus, for example, the following functions (except 1) all go to zero:
1
-+&+- .-&-&+
nlog n
1 1
-+log
n log log n 4 1.
(9.5)
Let’s look at a few other functions to see where they fit in. The number
rr(n) of primes less than or equal to n is known to be approximately n/inn.
Since 1 /ne + 1 /Inn 4 1, multiplying by n tells us that
n”’ + 7r(n) + n .
We can in fact generalize (9.4) by noticing, for example, that
na’ (logn)az(loglogn)a3 4 nB’(logn)PZ(loglogn)83
w (xl,a2,a3) < (b1,62,p3). (9.6)
Here ‘(LX’, 012,013) < (p’, (32, (33)’ means lexicographic order (dictionary order);
in other words, either a’ < p’, or 0~’ = (3’ and CX~ < f12, or a’ = (3’
and a2 = 62 and 0~3 -C 83.