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Algorithms

CSCI 235, Fall 2015Lecture 4

Asymptotic Analysis II

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Last time...

Say that f is: Notation ApproximatelyLooselylittle omega g f=(g) f is way bigger than g f>gomega g f=(g) f is at least as big as g f>=gtheta g f=(g) f is about the same as g f=goh g f=O(g) f is at most as big as g f<=glittle oh g f=o(g) f is way smaller than g f<g

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Relationships between O,o,,,

(g)(g)

(g)

(g)

(g)

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(g) ⊂Ω(g)

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o(g) ⊂O(g)

is a subset of

biggerf

smallerf

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and

if f = (g) then f = (g)

Why?

Definition of (n):

Definition of (n):

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(g(n))= { f (n) | ∀ positive constants c, ∃ n0 > 0

such that ∀n ≥ n0 0 ≤ cg(n)< f (n)}

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(g(n)) = { f (n) | ∃ positive constants c, n0

such that ∀n ≥ n0 0 ≤ cg(n) ≤ f (n)}

If f=(g) is it necessarily true that f=(g)?

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O and o

if f = o(g) then f = (g)

Why?

Definition of (n):

Definition of (n):

If f=(g) is it necessarily true that f=(g)?

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o(g(n))= { f (n) | ∀ positive constants c, ∃ n0 > 0

such that ∀n ≥ n0 0 ≤ f (n)< cg(n)}

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O(g(n)) = { f (n) | ∃ positive constants c, n0

such that ∀n ≥ n0 0 ≤ f (n) ≤ cg(n)}

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is a subset of and of O

(g) union (g) is a subset of (g)

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(g)∪Θ(g) ⊂Ω(g)

(g) union (g) is a subset of (g)

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o(g)∪Θ(g) ⊂O(g)

(g)

(g)(g)

(g)

(g)

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is the intersection of O and

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(g) = Ω(g) ∩O(g)

In other words:f=(g) if and only if f=O(g) and f= (g)

why?

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O(g(n)) = { f (n) | ∃ positive constants c, n0

such that ∀n ≥ n0 0 ≤ f (n) ≤ cg(n)}

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(g(n)) = { f (n) | ∃ positive constants c, n0

such that ∀n ≥ n0 0 ≤ cg(n) ≤ f (n)}

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(g(n)) = { f (n) | ∃ positive constants c1, c2, n0

such that ∀n ≥ n0 0 ≤ c1g(n) ≤ f (n) ≤ c2g(n)}

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Symmetric relationships

f = (g) if and only if g = o(f)why?

f = (g) if and only if g = O(f)why?

f = (g) if and only if g = (f)why?

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Example 1What is the relationship between f and g?

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f (n) = 5 −1

2n

g(n) =1

What happens if you change the coefficients?

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Example 2

What is the relationship between f and g?

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f (n) = n −1

2n

g(n) =1

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Example 3

Can we have a function that is in O(g) but not o(g) or (g)?

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f (n) = { n if n is odd1 if n is even

Consider:

a) g(n) = 1

b) g(n) = n

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Helpful hints

1) Not every pair of functions is comparable

2) It may be easier to test for o(g) and (g). Try these first and then try O, and .

3) Sometimes you can deduce several relationships from the knowledge of only 1. For example: if a function is o(g) it is also O(g), but never (g), (g) or (g).

4) When in doubt, graph the functions.