Comparing Functions

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Comparing Functions

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Notes on Notation

• N 0,1,2,3, ...
• N 1,2,3,4, ...
• R Set of Reals
• R Set of Positive Reals
• R R U 0

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Comparing f(n) and g(n)

• Let f be a function from N to R.
• O(f) (Big O of f) is the set of all functions g
  from N to R such that
• 1. There exists a real number cgt0
• 2. AND there exists an n0 in N
• Such that g(n) cf(n) whenever n ³ n0

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Notation and Pronunciation
Proper Notation g Î O(f)
g is oh of f
Also Seen g O(f)
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Big Omega
Let f be a function from N to R. W(f) (Big W of
f) is the set of all functions g from N to R such
that 1. There exists a real number cgt0 2. AND
there exists an n0 in N Such that g(n) ³ cf(n)
whenever n ³ n0
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Big Theta
Q(f) O(f) Ç W(f)
g is of Order f
g is Order f
g Î Q(f)
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Little o and Little Omega
o(f) O(f) - Q(f)
w(f) W(f) - Q(f)
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English Interpretations

• O(f) - Functions that grow no faster than f
• W(f) - Functions that grow no slower than f
• Q(f) - Functions that grow at the same rate as f
• o(f) - Functions that grow slower than f
• w(f) - Functions that grow faster than f

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Limit Formulas
if
for some
if
or
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More Limit Formulas
if
for some
if
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The Last Limit Formula
if
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Properties

• Transitivity
• if f Î O(g) and g Î O(h) then f Î O(h)
• Same holds for Q , W, o, and w
• Anti Symmetry (Sort of ...)
• f Î O(g) if and only if g Î W(f)
• May replace O with o and W with w

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Some More Properties

• Symmetry
• if f Î Q(f), then g Î Q(f)
• Reflexivity
• f Î O(f)
• Also true for Q and W
• Not True for o and w

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And Even More Properties

• Big Theta is an equivalence relation
• f R g if and only if f Î Q(g)
• O(fg) O(max(f,g))
• Also true for Q and W