Introduction to Algorithms(2nd edition)

by Cormen, Leiserson, Rivest & Stein

Chapter 3: Growth of Functions

(slides enhanced by N. Adlai A. DePano)

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

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

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)

The Θ-Notation

f

c1 ⋅ g

n0

c2 ⋅ g

Θ(g(n)) = { f(n) : ∃c1, c

2 > 0, n

0 > 0 s.t. ∀n ≥ n

0:

c1 · g(n) ≤ f(n) ≤ c

2 ⋅ g(n) }

The O-Notation

f

c ⋅ g

n0

O(g(n)) = { f(n) : ∃c > 0, n0 > 0 s.t. ∀n ≥ n

0: f(n) ≤ c ⋅ g(n) }

The Ω-Notation

Ω(g(n)) = { f(n) : ∃c > 0, n0 > 0 s.t. ∀n ≥ n

0: f(n) ≥ c ⋅ g(n) }

fc ⋅ g

n0

The o-Notationo(g(n)) = { f(n) : ∀c > 0 ∃n

0 > 0 s.t. ∀n ≥ n

0: f(n) ≤ c ⋅ g(n) }

fc

1 ⋅ g

n1

c2 ⋅ g

c3 ⋅ g

n2

n3

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 ≥ n

0: f(n) ≥ c ⋅ g(n) }

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

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

Asymptotic Analysis and Limits

f1(n) = O(g1(n)) and f2(n) = O(g2(n)) ⇒ f1(n) + f2(n) = O(g1(n) + g2(n))

f(n) = O(g(n)) ⇒ f(n) + g(n) = O(g(n))

Comparison of Functions

Standard Notation and Common Functions

MonotonicityA 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) .

Standard Notation and Common Functions

Floors and ceilingsFor 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, x1 < x x x < x+1.

Both functions are monotonically increasing.

Standard Notation and Common Functions

ExponentialsFor all n and a1, the function an is the exponential

function with base a and is monotonically increasing.

LogarithmsTextbook 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).

ai

Important relationshipsFor all real constants a and b such that a>1,

nb = o(an) that is, any exponential function with a base strictly greater than unity grows faster than any polynomial function.

For all real constants a and b such that a>0, lgbn = o(na) that is, any positive polynomial function grows faster than any polylogarithmic function.

Standard Notation and Common Functions

Standard Notation and Common Functions

FactorialsFor all n the function n! or “n factorial” is given

by n! = n (n1) (n 2) (n 3) … 2 1

It can be established that n! = o(nn) n! = (2n) lg(n!) = (nlgn)

Functional iterationThe 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 n=0 f (i)(n) = f(f (i1)(n)) if n>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

Iterated logarithmic functionThe notation lg* n which reads “log star of n” is

defined as lg* n = min {i0 : 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

We consider algorithm A better than algorithm B if T

A(n) = o(T

B(n))

Why is it acceptable to ignore the behavior of algorithms for small inputs?

Why is it acceptable to ignore the constants?

What do we gain by using asymptotic notation?

Asymptotic Running Time of Algorithms

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

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