# CS Fundamentals: Scalability and Memory

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• SCALABILITY AND MEMORYCS FUNDAMENTALS SERIES

http://bit.ly/1TPJCe6

• HOW DO YOU MEASURE AN ALGORITHM?

• ???

• CLOCK TIME?

• CLOCK TIME?

• DEPENDS ON WHOS COUNTING.

• ALSO, TOO FLAKY EVEN ON THE SAME

MACHINE.

• THE NUMBER OF LINES?

• THE NUMBER OF

LINES?

• THIS IS TWO LINES, BUT A WHOLE LOT OF STUPID.

• THE NUMBER OF CPU CYCLES?

• THE NUMBER OF

CPU CYCLES?

• DEPENDS ON THE RUNTIME.

• ALL THESE METHODS SUCK.

NONE OF THEM CAPTURE WHAT WE ACTUALLY CARE ABOUT.

• ENTER BIG O!

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ASYMPTOTIC ANALYSIS

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ASYMPTOTIC ANALYSIS

Big O is about asymptotic analysis

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ASYMPTOTIC ANALYSIS

Big O is about asymptotic analysis

In other words, its about how an algorithm scales when the numbers get huge

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ASYMPTOTIC ANALYSIS

Big O is about asymptotic analysis

In other words, its about how an algorithm scales when the numbers get huge

You can also describe this as the rate of growth

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ASYMPTOTIC ANALYSIS

Big O is about asymptotic analysis

In other words, its about how an algorithm scales when the numbers get huge

You can also describe this as the rate of growth

How fast do the numbers become unmanageable?

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ASYMPTOTIC ANALYSIS

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ASYMPTOTIC ANALYSIS

Another way to think about this is:

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ASYMPTOTIC ANALYSIS

Another way to think about this is:

What happens when your input size is 10,000,000? Will your program be able to resolve?

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ASYMPTOTIC ANALYSIS

Another way to think about this is:

What happens when your input size is 10,000,000? Will your program be able to resolve?

Its about scalability, not necessarily speed

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

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PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

In computer science, it means essentially means

• TEXT

PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

In computer science, it means essentially means

the asymptotic rate of growth

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PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

In computer science, it means essentially means

the asymptotic rate of growth In other words, how does the running time of this function

scale with the input size when the numbers get big?

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PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

In computer science, it means essentially means

the asymptotic rate of growth In other words, how does the running time of this function

scale with the input size when the numbers get big?

Big O notation looks like this:

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PRINCIPLES OF BIG O

Big O is a kind of mathematical notation

In computer science, it means essentially means

the asymptotic rate of growth In other words, how does the running time of this function

scale with the input size when the numbers get big?

Big O notation looks like this:

O(n) O(nlog(n)) O(n2)

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O

n here refers to the input size

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PRINCIPLES OF BIG O

n here refers to the input size

Can be the size of an array, the length of a string, the number of bits in a number, etc.

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PRINCIPLES OF BIG O

n here refers to the input size

Can be the size of an array, the length of a string, the number of bits in a number, etc.

O(n) means the algorithm scales linearly with the input

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PRINCIPLES OF BIG O

n here refers to the input size

Can be the size of an array, the length of a string, the number of bits in a number, etc.

O(n) means the algorithm scales linearly with the input

Think like a line (y = x)

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O

Scaling linearly can mean 1:1 (one iteration per extra input), but it doesnt necessarily

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PRINCIPLES OF BIG O

Scaling linearly can mean 1:1 (one iteration per extra input), but it doesnt necessarily

It can simply mean k:1 where k is constant, like 3:1 or 5:1 (i.e., a constant amount of time per extra input)

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O In Big O, we strip out any coefficients or smaller factors.

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PRINCIPLES OF BIG O In Big O, we strip out any coefficients or smaller factors.

The fastest-growing factor wins. This is also known as the dominant factor.

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PRINCIPLES OF BIG O In Big O, we strip out any coefficients or smaller factors.

The fastest-growing factor wins. This is also known as the dominant factor.

Just think, when the numbers get huge, what dwarfs everything else?

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PRINCIPLES OF BIG O In Big O, we strip out any coefficients or smaller factors.

The fastest-growing factor wins. This is also known as the dominant factor.

Just think, when the numbers get huge, what dwarfs everything else?

O(5n) => O(n)

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PRINCIPLES OF BIG O In Big O, we strip out any coefficients or smaller factors.

The fastest-growing factor wins. This is also known as the dominant factor.

Just think, when the numbers get huge, what dwarfs everything else?

O(5n) => O(n)

O(n - 10) also => O(n)

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O

O(k) where k is any constant reduces to O(1).

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PRINCIPLES OF BIG O

O(k) where k is any constant reduces to O(1).

O(200) = O(1)

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PRINCIPLES OF BIG O

O(k) where k is any constant reduces to O(1).

O(200) = O(1)

Where there are multiple factors of growth, the most dominant one wins.

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PRINCIPLES OF BIG O

O(k) where k is any constant reduces to O(1).

O(200) = O(1)

Where there are multiple factors of growth, the most dominant one wins.

O(n4 + n2 + 40n) = O(n4)

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PRINCIPLES OF BIG O

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PRINCIPLES OF BIG O

If there are two inputs (say youre trying to find all the common substrings of two strings), then you use two variables in your Big O notation => O(n * m)

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PRINCIPLES OF BIG O

If there are two inputs (say youre trying to find all the common substrings of two strings), then you use two variables in your Big O notation => O(n * m)

Doesnt matter if one variable probably dwarfs the other. You always include both.

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PRINCIPLES OF BIG O

If there are two inputs (say youre trying to find all the common substrings of two strings), then you use two variables in your Big O notation => O(n * m)

Doesnt matter if one variable probably dwarfs the other. You always include both.

O(n + m) => this is considered linear

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PRINCIPLES OF BIG O

If there are two inputs (say youre trying to find all the common substrings of two strings), then you use two variables in your Big O notation => O(n * m)

Doesnt matter if one variable probably dwarfs the other. You always include both.

O(n + m) => this is considered linear

O(2n + log(m)) => this is considered exponential

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COMPREHENSION TEST

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

O(3n + 5)

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

O(3n + 5)

O(n + 1/5n2)

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

O(3n + 5)

O(n + 1/5n2)

O(log(n) + 5000)

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

O(3n + 5)

O(n + 1/5n2)

O(log(n) + 5000)

O(2m3 + 50 + n)

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COMPREHENSION TEST

Convert each of these to their appropriate Big O form!

O(3n + 5)

O(n + 1/5n2)

O(log(n) + 5000)

O(2m3 + 50 + n)

O(nlog(m) + 2m2 + nm)

• What should n be for this function?

• For each character in the string

Unshift them into an arrayAnd then join the array together.

Lets break it down.

Make an empty array.

• For each character in the string

Unshift them into an arrayAnd then join the array together.

Initialize an empty array => O(1)

Then, split the string into an array of characters => O(n)

Then for each character => O(n)

Unshift into an array => O(n)

Then join the characters into a string => O(n)

Well see later why this is.

Make an empty array.

• For each character in the string

Unshift them into an arrayAnd then join the array together.

Initialize an empty array => O(1)

Then, split the string into an array of characters => O(n)

Then for each character =>

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