Order Notation
Big "Oh"
Let f(n) and g(n) be two real valued functions over the integers n ≥ 1. (Sometimes, it
may be advantageous to allow n to be real. But, for our purposes, n usually will be the
"size" of a problem instance which is normally an integer.)
Definition:
f(n) is said to be "order" g(n) if and only if there exists positive constants
c and N for which f(n) ≤ cg(n) for n ≥ N.
What this says is that, as n gets large, the function g(n) is an upper bound on the value
of f(n). This is handy when we don't have a "nice" expression for f(n), but do for g(n).
That is, we can use g(n) in calculations and know we are not underestimating the running
time of whatever f(n) is measuring. Of course, we don't want to overestimate, either. So,
our goal is to find the simplest and "tightest" g(n), that is, the best possible bound.
Notice that all the "c" does in the definition is to allow us to "talk" about (and use)
g(n) in it's simplest form, i. e., with a leading coefficient of one, recognizing we may
need to multiply it by a constant for some of the functions it bounds.
Sometimes you will see "f(n) is O(g(n))," to be read as "f(n) is 'bigOh' of g(n)". Or,
f(n)
∈
O(g(n)), which is read "f(n) is in (or, an element of) O(g(n))."
{Many times you will see writers mistakenly write "f(n) = O(g(n))" and claim to
mean the same thing. This isn't mathematically correct. In mathematics, ideally, symbols
and operators have just one meaning – not like some programming languages which
allow symbols to mean different things depending upon the context (overloading of
operators). For example, mathematically A = B and B = A mean the
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 Fall '10
 Dutton
 Calculus, Derivative, Coefficient

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