CSC165 Week 9 Larry Zhang, November 4, 2014 todays outline - - PowerPoint PPT Presentation

CSC165 Week 9

Larry 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

• f 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