CSC165 Week 9Larry Zhang, November 4, 2014

today’s outline

➔ exercises of big-Oh proofs

➔ prove big-Oh using limit techniques

Recap definition of O(n²)

Beyond breakpoint B, f(n) is upper-bounded by cn², where c is some wisely chosen constant multiplier.

cn²

f(n)

B

Notation of O(n²) being a set of functions

functions that take in a natural number and return a non-negative real number

beyond breakpoint B, f(n) is upper-bounded by cn²

set of all red functions which satisfy the green.

Prove

thoughts: it’s all about picking c and B➔ tip 1: c should probably be larger than 3 (the

constant factor of the highest-order term)

➔ tip 2: see what happens when n = 1

➔ if n = 1

◆ 3n² + 2n = 3 + 2 = 5 = 5n²

◆ so pick c = 5, with B = 1 is probably good

◆ double check for n=2,3,4…, yeah it’s all good

Proof# 3n²+2n ≥ 0 when n is natural number

# 5 is a positive real

# 1 is a natural number

# generic natural number

# the antecedent

# 2nx1 ≤ 2nxn since 1 ≤ n

# c = 5

# transitivity of ≤

# introduce antecedent

# introduce ∀

# introduce ∃# by definition

what if we add a constant?

thoughts: it’s all about picking c and B➔ tip 1: c should probably be larger than 3 (the

constant factor of the highest-order term)➔ tip 2: see what happens when n = 1➔ if n = 1

◆ 3n² + 2n + 5 = 3 + 2 + 5 = 10 = 10n²◆ so pick c = 10, with B = 1 is probably good◆ double check for n=2,3,4…, yeah it’s all good

Prove

Prove

thoughts: ➔ assume n ≥ 1

➔ upper-bound the left side by overestimating

➔ lower-bound the right side by underestimating

➔ choose a c that connects the two bounds

large

small

Proof # non-negative for n ≥ 0

# -5n⁴ ≤ 0

# n³ ≤ n⁶ since n ≥ 1

# c = 9/2

# 0 ≤ n² # -n⁸ ≤ -n⁵

# n ≥ B, the antecedent

how about disproving?

Prove

➔ first, negate it

➔ then, prove the negation

Prove

thoughts

Proof:

# definition of max

# definition of max and ceiling

# multiply both sides by n² > 0

# conjunction introduction

# introduce ∃

# intro ∀

# definition of big-Oh

summary

➔ the ways we talked about of picking c and B, the main point of them is that we know how to show the bounding relation when picking in these ways.

➔ these are not the only ways, you can be more flexible when you are more familiar with these types of proofs.

➔ In general, larger c or larger B don’t hurt.

so far all functions we talked about are polynomials

➔ between polynomials, it is fairly easy to figure out who is big-Oh of whom

➔ simply look at the highest-degree term

➔ f(n) is in O(g(n)) exactly when the high-degree of f(n) is no larger than that of g(n)

polynomials

how about non-polynomials?

but how do we prove it?➔ we need to use a math

tool, which we didn’t like a lot but studied anyway

Intuition:

➔ if the ratio approaches infinity

when n becomes larger and larger, that

means f(n) grows faster than g(n)

f(n)

g(n)

give me any big number

more precisely

I can find a breakpoint

beyond which

the ratio is bigger than that big number

This is sort-of related to the definition of big-Oh…

How to use limit to prove big-Oh

General procedure

1. prove using “some calculus”

2. translate the limit into its definition with c

and n’, i.e.,

3. relate this definition to the definition of

big-Oh, i.e., means

Step 1. prove using “some calculus”

“some calculus”: L’Hopital’s rule

derivatives

l’hopital

l’hopital again!

http://mathworld.wolfram.com/LHospitalsRule.html

Step 2. translate the limit into its definition

we have proven

then by definition of limit

give me any big number

I can find a breakpoint

beyond which

the ratio is bigger than that big number

Step 3. relate it to the definition of big-Oh

wanna pick a wise n

we have this awesome n’, all n ≥ n’ satisfy 2^n > cn²

to satisfy both

pick n = max(n’, B)

what we have

Proof: # generic c and B

# applying L’hopital’s rule twice, (show the steps, omitted here)

# definition of limit

# definition of max

# definition of max

# definition the limit

# conjunction introduction

# intro ∃

# intro ∀

# negation of definition of big-Oh

next week

➔ more proofs on O, Ω, Θ

➔ (maybe) computability