Concrete mathematics : a foundation for computer science

“It is obvious that
the sign = is really
the wrong sign
for such relations,
because it suggests
symmetry, and
there is no such
symmetry. . . .
Once this warning
has been given,
there is, however,
not much harm in
using the sign = ,
and we shall maintain
it, for no other
reason than that it
is customary.”
-N. G. de Bruijn /62]
[Now is a good
time to do warmup
exercises 3 and 4.)
9.2 0 NOTATION 433
And in English, this ‘is’ is one-way. We say that a bird is an animal, but we
don’t say that an animal is a bird; “animal” is a crudification of “bird!’
Fourth, for our purposes it’s natural. If we limited our use of O-notation
to situations where it occupies the whole right side of a formula-as in the
harmonic number approximation H, = O(log n), or as in the description of
a sorting algorithm’s running time T(n) = O(nlogn) -it wouldn’t matter
whether we used ‘=’ or something else. But when we use O-notation in the
middle of an expression, as we usually do in asymptotic calculations, our
intuition is well satisfied if we think of the equals sign as an equality, and if
we think of something like 0 (1 /n) as a very small quantity.
So we’ll continue to use ‘=I, and we’ll continue to regard O(g(n)) as an
incompletely specified function, knowing that we can always fall back on the
set-theoretic definition if we must.
But we ought to mention one more technicality while we’re picking nits
about definitions: If there are several variables in the environment, O-notation
formally represents sets of functions of two or more variables, not just one.
The domain of each function is every variable that is currently “free” to vary.
This concept can be a bit subtle, because a variable might be defined only
in parts of an expression, when it’s controlled by a x or something similar.
For example, let’s look closely at the equation
f (k’ + O(k)) = in3 + O(n2), integer n > 0. (9.16)
k=O
The expression k2 + O(k) on the left stands for the set of all two-variable
functions of the form k2 + f(k,n) such that there exists a constant C with
]f(k, n)l $ Ck for 0 < k 6 n. The sum of this set of functions, for 0 6 k < n,
is the set of all functions g(n) of the form
$(k’+f(k,n)) = ~n3+~n2+~n+f(0,n)+f(l,n)+...+f(n,n),
k=O
where f has the stated property. Since we have
(~n2+~n+f(0,n)+f(l,n)+...+f(n,n)l
< ~n2+/7n2+C.0+C.l +...+C.n
< n2 + Cl,n2 + n)/2 < (C + l)n2 ,
all such functions g(n) belong to the right-hand side of (9.16); therefore (9.16)
is true.
People sometimes abuse O-notation by assuming that it gives an exact
order of growth; they use it as if it specifies a lower bound as well as an
upper bound. For ex.ample, an algorithm to sort n numbers might be called