1. Introduction to Algorithms
(3rd edition)
by Cormen, Leiserson, Rivest & Stein
Chapter 3: Growth of Functions
(slides enhanced by N. Adlai A. DePano)

2. Overview
 Order of growth of functions provides a
simple characterization of efficiency
 Allows for comparison of relative
performance between alternative
algorithms
 Concerned with asymptotic efficiency of
algorithms
 Best asymptotic efficiency usually is best
choice except for smaller inputs
 Several standard methods to simplify
asymptotic analysis of algorithms

3. Asymptotic Notation
 Applies to functions whose domains are the
set of natural numbers:
N = {0,1,2,…}
 If time resource T(n) is being analyzed, the
function’s range is usually the set of non-
negative real numbers:
T(n)  R+
 If space resource S(n) is being analyzed, the
function’s range is usually also the set of
natural numbers:
S(n)  N

4. Asymptotic Notation
 Depending on the textbook,
asymptotic categories may be
expressed in terms of --
a. set membership (our textbook):
functions belong to a family of functions
that exhibit some property; or
b. function property (other textbooks):
functions exhibit the property
 Caveat: we will formally use (a) and
informally use (b)

5. The Θ-Notation
f
c1 ⋅ g
n0
c2 ⋅ g
Θ(g(n)) = { f(n) : ∃c1, c2 > 0, n0 > 0 s.t. ∀n ≥ n0:
c1 · g(n) ≤ f(n) ≤ c2 ⋅ g(n) }

6. The Ω-Notation
Ω(g(n)) = { f(n) : ∃c > 0, n0 > 0 s.t. ∀n ≥ n0: f(n) ≥ c ⋅ g(n) }
f
c ⋅ g
n0

7. The O-Notation
f
c ⋅ g
n0
O(g(n)) = { f(n) : ∃c > 0, n0 > 0 s.t. ∀n ≥ n0: f(n) ≤ c ⋅ g(n) }

8. The o-Notation
o(g(n)) = { f(n) : ∀c > 0 ∃n0 > 0 s.t. ∀n ≥ n0: f(n) ≤ c ⋅ g(n) }
f
c1 ⋅ g
n1
c2 ⋅ g
c3 ⋅ g
n2 n3

9. The ω-Notation
f
c1 ⋅ g
n1
c2 ⋅ g
c3 ⋅ g
n2
n3
ω(g(n)) = { f(n) : ∀c > 0 ∃n0 > 0 s.t. ∀n ≥ n0: f(n) ≥ c ⋅ g(n) }

10.  f(n) = O(g(n)) and
g(n) = O(h(n)) ⇒ f(n) = O(h(n))
 f(n) = Ω(g(n)) and
g(n) = Ω(h(n)) ⇒ f(n) = Ω(h(n))
 f(n) = Θ(g(n)) and
g(n) = Θ(h(n)) ⇒ f(n) = Θ(h(n))
 f(n) = O(f(n))
f(n) = Ω(f(n))
f(n) = Θ(f(n))
Comparison of Functions
Reflexivity
Transitivity

11.  f (n) = Θ(g(n))  g(n) = Θ(f (n))
 f (n) = O(g(n))  g(n) = Ω(f (n))
 f (n) = o(g(n))  g(n) = ω(f (n))
 f (n) = O(g(n)) and
f (n) = Ω(g(n))  f (n) = Θ(g(n))
Comparison of Functions
Transpose
Symmetry
Symmetry
Theorem 3.1

12. Standard Notation and
Common Functions
 Monotonicity
A function f(n) is monotonically
increasing if m  n implies f(m)  f(n) .
A function f(n) is monotonically
decreasing if m  n implies f(m)  f(n) .
A function f(n) is strictly increasing
if m < n implies f(m) < f(n) .
A function f(n) is strictly decreasing
if m < n implies f(m) > f(n) .

13. Standard Notation and
Common Functions
 Floors and ceilings
For any real number x, the greatest integer
less than or equal to x is denoted by x.
For any real number x, the least integer
greater than or equal to x is denoted by
x.
For all real numbers x,
x1 < x  x  x < x+1.
Both functions are monotonically
increasing.

14. Standard Notation and
Common Functions
 Exponentials
For all n and a1, the function an is the exponential
function with base a and is monotonically
increasing.
 Logarithms
Textbook adopts the following convention
lg n = log2n (binary logarithm),
ln n = logen (natural logarithm),
lgk n = (lg n)k (exponentiation),
lg lg n = lg(lg n) (composition),
lg n + k = (lg n)+k (precedence of lg).

15. Standard Notation and
Common Functions
 Factorials
For all n the function n! or “n factorial” is
given by
n! = n  (n1)  (n  2)  (n  3)  …  2  1
It can be established that
n! = o(nn)
n! = (2n)
lg(n!) = (nlgn)

16.  Functional iteration
The notation f (i)(n) represents the function f(n)
iteratively applied i times to an initial value of n,
or, recursively
f (i)(n) = n if i=0
f (i)(n) = f(f (i1)(n)) if i>0
Example:
If f (n) = 2n
then f (2)(n) = f (2n) = 2(2n) = 22n
then f (3)(n) = f (f (2)(n)) = 2(22n) = 23n
then f (i)(n) = 2in
Standard Notation and
Common Functions

17.  Iterated logarithmic function
The notation lg* n which reads “log star of n” is
defined as
lg* n = min {i0 : lg(i) n  1
Example:
lg* 2 = 1
lg* 4 = 2
lg* 16 = 3
lg* 65536 = 4
lg* 265536 = 5
Standard Notation and
Common Functions

18.  Asymptotic analysis studies how the
values of functions compare as their
arguments grow without bounds.
 Ignores constants and the behavior of
the function for small arguments.
 Acceptable because all algorithms are
fast for small inputs and growth of
running time is more important than
constant factors.
Things to Remember

19.  Ignoring the usually unimportant details,
we obtain a representation that succinctly
describes the growth of a function as
its argument grows and thus allows us to
make comparisons between algorithms in
terms of their efficiency.
Things to Remember

20. Tips to Help Remember
 May be helpful to make the following
“analogies” (remember, we are comparing
rates of growth of functions)