Original subtitles

in this python algorithms course you

will learn to work with recursion data

structures greedy algorithms dynamic

programming and more

you don't need any previous experience

and algorithms to follow along

algorithms help us solve problems

efficiently and employers pay good money

for good problem solvers

your instructor for this course is joy

brock

she has an amazing talent for breaking

down complex topics using humor and

visualizations

[Music]

hi

can i ask you a question raise your hand

if you love being bossed around

i'm not seeing any hands

i know that i don't like to be bossed

around

now you might be wondering why i would

ask a question like that and what does

that have to do with computer algorithms

but that's just it

computers love to be lost around in fact

a computer can't do anything on its own

without being told what to do and that

my friends is what an algorithm is

an algorithm is a specific set of

instructions that we write to the

computer to tell it what to do after all

that's why you're here right to learn

how to tell that old ball and chain

who's boss you know the one that keeps

us perpetually locked in a prison of

screen time so without further ado

welcome to the introduction to

algorithms course i am so glad you're

here we are going to have an insane

amount of fun so grab your beverage of

choice grab a notepad for notes and be

prepared to learn the five basic

algorithms that every programmer should

know

by official definition an algorithm can

be defined as being the current term of

choice for a problem solving procedure

it is commonly used nowadays for the set

of rules a machine especially a computer

follows to achieve a particular goal it

does not always apply to computer

mediated activity however the term may

is accurately be used for the steps

followed in making a pizza or solving a

rubik's cube as for computer powered

data analysis

algorithms are often paired with words

specifying the activity for which a set

of rules have been designed

a search algorithm for example is a

procedure that determines what kind of

information is retrieved from a large

mass of data an encryption algorithm is

a set of rules by which information or

messages are encoded so that

unauthorized persons cannot read them

this comes from the merriam-webster

dictionary what does algorithm mean

mathematically speaking algorithms have

been around for nearly 900 years pretty

amazing isn't it in fact algorithms are

all around us every day and have been

our whole lives in this course we're

going to dive into algorithms that are

finite or determine sequences of

explicit instructions that are used to

solve problems and or perform

computations these computations are

generally to perform calculations or to

manipulate data using natural language

processing natural language processing

or nlp for short is what tells the

computer how to comprehend and interpret

written text

it's important to distinguish the

difference between using well-defined

formal languages to calculate functions

versus algorithms used in artificial

intelligence or machine learning

those algorithms are where automated

deductions are made using mathematical

and logical tests

or where predictions are made based on

patterns learned from experience for

example the other day when i was looking

at the front page of google on their

news i thought you know this algorithm

better be so good that i can read about

the news before it even happens

those types of algorithms are for

another video on another day

the objective here in this course is to

give you the tools you'll need to

undertake the journey of using a

well-defined formal language in this

case python to calculate functions

within a defined amount of space and

time that may be asked of you for

interviews or tests

in other words the goal is for you to be

able to

find differences in algorithmic

approaches based on computational

efficiency

this means that how we write our program

is very important in regards to how much

space it can use up on our computer or

how quickly it can run

and it all begins with the foundational

principles that every algorithm has

in input

well-defined instructions

execution of those instructions step by

step leading to an output that ends in

termination of the program

now there are many different algorithms

that accomplish

these basics when it comes to

programming and as much as i love

spending time with you all

it would be so difficult to cover every

single one so for the sake of brevity

we'll be covering the following

simple recursive algorithms

algorithms within data structures

divide and conquer

greedy algorithms

and last but certainly not least dynamic

programming

for this course you'll need some

understanding of the python language

like writing functions variables

conditionals for loops and basic syntax

an ide or integrated development

environment an ide is the virtual place

to write our programs

now if you don't have a downloaded

platform like pycharm jupiter notebook

or visual studio code you can just use

any online python ide through a google

search

if you would like to download your own

personal ide such as pycharm vsc or

jupiter you can definitely do that as

each one of those platforms are free to

install

go ahead and pause the video now if you

need to get situated otherwise let's get

this party started enroll into our very

first lesson on simple recursive

algorithms

in our first lesson we're going to learn

what is recursion and we're going to

examine three different problems ranging

from easy to mind-breaking

we're going to look at the differences

in computational efficiency and see

which approach is best

is implementing recursion the best way

let's find out

[Music]

if the easiest way to define recursion

is to say it's where a function calls

itself then what's the hard way to

define it in all truth it's easy to

throw a statement out hand on hip that

goes oh it's a function that calls

itself

sure okay good for you but what does

that really mean i think oftentimes when

it comes to programming it's easy to

take for granted this idea that we

should automatically know what it all

means

i don't know about you but i wasn't born

with the ability to comprehend various

methods to solve problems let alone be

able to decipher which which method is

better in the first place you know

we all need to learn these concepts

layer upon layer step by step so let's

start at the very first layer and build

up from there now when we write our

program recursively it's important to

understand that this is in contrast to

writing it iteratively or through

iteration when we do that we're

generally using for loops to iterate

over an object instead of making a

function call back within the program

you'll see what i mean when we get into

our ides and and start to write the code

recursion

is it originates from a latin verb that

means to run back and this is what the

program we're going to write together is

going to do it will run back to itself

over and over as many times as it needs

until the program terminates so does

this mean that we can't use for loops

with recursion no of course you can use

them in fact they will be needed and

you'll see how that all works in just a

moment in order to get started with our

very first lesson involving factorials

and the algorithmic method that we are

going to use for better efficiency we

will calculate the factorial of a number

using the leaf method or last in first

out and it all begins with a mental

picture of a bookshelf that doesn't have

any books on it yet to perform a lethal

operation on your hypothetical bookshelf

you will grab some books and begin

placing them on the shelf let's say you

ordered some britannica encyclopedias

wait why would you do that

why did any of us do that oh right it

was our grandparents that saw to it to

make sure that we had our own modern day

google machine thanks grandma your

legacy lives on at the donation center

okay that was a wonderful trip down

memory lane wasn't it okay so back to

lefo or lifo if that's what you call it

so you put your encyclopedias on the

shelf a to z in a last in first out

scenario what was the last book you you

sat on the shelf z what's the first book

that's going to come off z

so last in first out and this would be

important to remember in our number

oriented recursive function coming up

that the last number that goes in will

be the first number that gets called out

or computed

now if you're a little bit confused it's

okay you won't be for long we're going

to see this in action right now and talk

about the computationally efficient way

to solve as we write our first recursive

program to calculate the factorial of a

number

for a quick recap on factorials say you

take the number 5 and multiply it by

itself minus 1 all the way down until

you hit one

your answer will equal 120. remember

factorials only involve positive

integers so once you hit one the program

terminates

now let's get to work

we're going to look at two ways that we

can write a program that will calculate

the factorial of a number this is going

to be a wonderful foray into differences

regarding complexity and program

efficiency in our first method we're

going to tackle this iteratively or like

i mentioned before through iteration

this means that our program will cycle

through or iterate each and every step

we tell it to

first we define our function with one

parameter the number we want to

calculate in this case we'll just call

it n

then we're going to create a variable

and start it at one positive numbers

remember no zeros here next we construct

a for loop to iterate over that variable

or object if you like

and do something to it in this case

we're going to multiply it by every

number in a range starting at 2 and

adding in our ending at our number plus

one this gives us a range of four

numbers two three four five now why not

six since n or five plus one equals six

remember we're using python so if we

loop to a range of two

comma 6 in parenthesis our indices will

always be minus 1 because python starts

at 0. so an index position of 6 will

give us our last number for our

calculation 5. then we take our object

called fact and multiply it by every

number in our range because we can use

the asterisk and equal sign to condense

our code a little fact equals fact time

asterisk i that assignment operator is

reassigning our variable each time it

cycles through our loop

when the cycling or iteration is

finished we simply return our finished

fact variable if you want to see the

results of each

iteration you can just place a print

statement within the for loop and run

the program

for a recursive method it's going to be

already typed out while i talk

this is where our last in first out

comes into play remember when we talked

about that this is in contrast to our

iterative method that starts at the top

are fact variables starting at 1 and

then iterates through our range going up

until our number is reached in the

algorithmic method of recursion

when we go to call the function if our

number isn't 1 it moves to our else

statement where in our else statement we

have a temp variable that runs back to

our function call and creates a pocket

of space if you will for each number

minus one

it will run back all the way down until

it hits one our last n if you will once

that happens the return statement and

our else statement starts working its

magic and then we start heading back up

our last in which was one becomes our

first out

our first number to get computed

ta-da

[Applause]

now you might be tempted to think that

the recursive method is top dog here but

alas you'd be wrong our iterative method

is twice as fast

for what we're trying to accomplish

which is

just a basic calculation of a factorial

what's important what's important to be

able to understand is a how recursion

works by using a simple program such as

momentous did and b how how and why one

approach can be better than another

in this case recursion is taking up more

space and time by running back to the

function call to create those pockets

for our numbers

versus iteration where it's simply

multiplying each number in place and not

creating anything new

it's simply iterating over each number

reassigning our variable each time and

producing a final output

the objective for our basic exercise is

to just understand and see recursive

algorithms in action so that as you gain

experience you'll be able to exercise

different modes of expression or

expressivity in your writing

and in some cases use your readability

check out this example

on the upside being able to write

code in three lines instead of seven or

eight can be a huge deal depending on

the circumstance however on the flip

side may decrease the performance of the

code let's like this example that we

just coded together

it's just not as efficient as iteration

in this case so if you're ready let's

move up to a more advanced recursive

algorithm involving permutations

a permutation is when you take a set of

elements and find all the different

variations that can be made with it for

instance if we take a b and c we can get

six variations out of these three

letters mathematically speaking we can

find the number of variations by

calculating the factorial of the

elements just like we did a few minutes

ago

for a hands-on approach to understanding

a level up on recursion let's take a

peek under the hood and see how this

will work in a program shelby

[Music]

the underlying basis of how most

functions for permutations work is to

take off the first element and then

create two subsets of numbers that get

swapped like this

then the original first element gets put

back onto the front of each subset then

we do this all over again this time

instead of popping out the first element

we pop out the second one and create two

new subsets and so on and so forth

the algorithm behind permutations can

really make your head spin so take a

deep breath

find your center and jump in

to

a puddle

let's not jump into the ocean just yet

it's good to start small and by small

let's recursively swap just two letters

of a string this is a little bit harder

to do than a list because strings are

immutable but

this is a good launch point for us to

build up from here also we'll start with

recursion this time and then afterwards

we'll look at a different method for

comparison

to write a recursive function that will

swap just two letters we will need to

create a pocket for where our strings

need to go remember again strings are

immutable and if this were a list we

could simply slice it and create a

variable swap but that's a no go for

strings we're moving into a little bit

steeper territory here but stay with me

okay you can do this

let's create our function and call it

permute with two parameters one called

string and for the string of two letters

that we want to swap and an empty pocket

to put our new string in this is our

parameter default defaulted to an empty

string then we create an if statement

for an edge case in the event that the

string is empty and if it's empty we

just return our empty pocket now we'll

start with looping through the range of

the length of our string and within our

loop we'll start with creating three

variables that will contain one each

letter on our string two we'll need to

take off the front of the string that

we're going to call front and then three

a variable called back that will slice

off the back of our string if you study

this algorithm you'll see some

programmers call it head and tail

current and the rust first next or front

and back i like the terms front and back

as it gives me a better mental picture

but you can use whatever terms you're

more comfortable with

now those first three variables are

fairly obvious but this is where it

might get a little hazy we need to

create a fourth variable that will put

our front and back together and we'll

need this variable of combined elements

to plug into our function call

this is what creates spaces to in our

pocket to be able to move our letters

around i know this may sound a little

complicated at the moment but you'll see

shortly that we're going to need each

and every letter of our string even

however many that will eventually come

to be to have the space it needs to go

into the right place in our pocket now

whether that's going to lead up to

eventually being three letters or four

keep in mind our strengths can't be too

long because of two things the rate of

exponential growth and the fact that

recursion can get exhaustive very fast

the more letters you have recursively

that's a lot of running back okay we

have all the variables we need right now

our letter a front which is just an

empty string right now our back which is

the last set of our string b

and are together that combines our front

and back now for our algorithm this is

where the magic happens this is the true

ham in our sandwich we call our function

premium and then incorporate two

parameters are together

and our letter added to our pocket hence

the addition operator

this is the heartbeat of our algorithm

it needs to run back and put our letters

into our pocket once in reverse and then

the second time in a copy of its

original form because it's still

considered a permutation watch how this

works in the visualizer

i think that if you get really get a

really good handle on this concept and

you have a solid foundation

it becomes so much easier to see how

this works with additional letters if

you can see it mentally this makes all

the difference in the world

it's a neat little program that only has

11 lines of code which is really nice

however this program isn't as efficient

as a program that uses iteration and

swapping again

that program has nearly double the lines

of code i'll spare you the direction of

watching me type it all out and just

show you here

this would be the code that executes

just like the beginning

[Music]

that you saw on the earlier slides where

it pops off the first letter and then

swaps the it pops out the first letter

swaps the last two and then

pops the first letter back on then it

rotates to the second letter

swaps those two

and then puts it back on

check the output here and you'll see

what i mean now if you needed to know

all the permutations of a string and

don't really have time to write the code

or the patience you could use the inner

tools permutations module it's really

easy

but

that would be primarily for you

just to know for yourself or for a quick

look up when it comes to interviews or

testing companies would generally prefer

that you know the mechanics behind the

method for solving whether that's using

recursion or swapping

as with any program the caveat must

always be made can this program be

written differently

as always yes however you will see very

little difference when it comes to

recursion of most programs like this

very generally speaking they will call

the function in a very similar way as it

is written here by taking the front and

back together and incorporating it in

some way

within the function call

along with the letter

hopefully you were able to distinctly

see how recursion works in this program

even just for two letters

it can work great on small strings but

it will get computationally expensive

the longer the string now if you're not

too overwhelmed hopefully not we've only

done two

we're going to move into the mind

breaker section like it should be the

mind blowing section

the end cleans

problem

this computational problem is about

placing eight chess queens on an eight

by eight board so that no two queens are

able to attack each other according to

wikipedia the problem finding all

solutions to the eight queens problem

can be quite computationally expensive

there are 4 billion possible

arrangements of the 8 queens on an 8x8

board but only 92 solutions wikipedia

also goes on to say that the use of

shortcuts and various rules of thumb

that avoid brute force techniques can

greatly reduce the number of possible

combinations which i'm sure is always a

good thing but what is brute force

though

well that's would be brute force would

be looking at every single possible

arrangement 4 billion of them to find

the 92 solutions how long do you think

that would take

probably take decades for our computers

to run a program like that but guess

what the use of permutations can reduce

the possibilities from 4 billion down to

just 40 000. that's a big reduction

now we're not are we going to use

permutations on the end queen's problem

today no way jose

but it's good to know that this problem

is often used in examples for various

programming techniques such as solving

it by using recursive algorithms it's

also used

as an example in constraint programming

which is a paradigm for solving

combinatorial problems that draws on a

wide range of techniques from artificial

intelligence computer science and

operations research

to get a better idea of what that would

be we can use a real world example like

employee scheduling say for instance a

company operates continuously like a

factory they need to create weekly

schedules for their employees and if

that company runs three eight hour

shifts per day in a science three of its

four employees different shifts each day

while giving a fourth one's a day off

even in such a small case like that the

number of possible schedules is huge

each day there are factorially 24

possible employee assignments for four

people

so the number of possible weekly

schedules is 247 which calculates over

4.5 billion

constraint programming methods keep

track of which solutions remain feasible

and makes it a powerful tool for solving

real world

scheduling problems who knew queens on a

chessboard combined with computer

programming could help solve real world

problems like that

so what did we learn in our very first

lesson in our algorithms course

we learned simple recursion and saw it

in action that when a function calls

itself it creates new spaces to store

information and sometimes this may not

be an efficient solution to a problem

it's definitely more elegant with fewer

lines of code

that can come in handy depending on the

problem but it's not always cost

effective

but it is important to see and

understand the difference

factorials and permutations are used in

a lot of computational mathematical and

real world applications

learning how to write programs rather

recursively or iteratively to find the

best method

in this area can go a long way and may

lead to unknown opportunities for you

down the road as a programmer

we learned about constraint programming

where recursion plays a role for solving

mind-bending problems

not only within board games such as

chess but real life issues that

companies can face

as a programmer that works to gain the

tools needed to understand these types

of methods to solve problems can make

them a valuable asset to any company

[Music]

welcome to level two

working with data

that is structured i like to think of

data structures like something that

holds money say for instance you have on

your person a big wad of cash consisting

of a lot of different bills what could

you do with it well you could store it

sort it search it add it spend it

delete it

count it so with data structures they're

simply containers or collections of data

they provide a means of managing small

or large amounts of values for uses and

such things like

databases web traffic monitoring user

registrations and or indexing within a

contiguous memory location like an array

they can range from as simple as lists

to linked lists stacks cues hash tables

trees and heaps

these are all examples of data

structures and since algorithms are

steps to solving problems what kind of

problem-solving techniques are common

with data structures well just like

watts of cache data often needs storage

sorting counting retrieving searching

spending all those fun things now arrays

can be a really basic container for

information but don't let that fool you

many different algorithms can be used on

one-dimensional arrays but what about

two-dimensional arrays what about 32

dimensions since that's the maximum

amount can you imagine

arrays are among the oldest and most

important data structures and are used

for almost every program for a basic

level of algorithms with data structures

we'll go easy and stick to a linear or

one-dimensional array

a one-dimensional array is just an array

stored in an unbroken block of memory to

access the next value in an array we

just move sequentially to the next

memory address it's best to see this

visually so check these slides out for a

better idea

in our basic section we're going to

cover

linear search binary search

bubble sort

and insertion sort

we'll be covering a little bit more for

the basic level on data structures

because it's a really big topic actually

entire books are written about them let

alone design techniques and algorithms

i'll do my best not to put you to sleep

hopefully it'll be enough to watch your

appetite where array is our concern as

this is a huge topic as well and as much

as python is a wonderful program with

tons of features like built-ins and

libraries and modules that can help

programmers code faster it's still

really important to understand how these

features work algorithmically and the

concepts underneath it all

let's get started

a linear search could be equated to

pulling that a lot of cash out of your

wallet

wait does anyone carry cash anymore no

okay well let's pretend that you have a

wallet full of really big bills i mean

go big or go home right and you need a

500 bill from your stack

why well you're at the mall of course

and you want to buy that really cool

pair of socks that you saw in the window

and you might be thinking 500 for a pair

of socks and i would say yup that's

inflation for you but this is a fantasy

remember besides inflation slomation you

got a wallet full of big bills and in

this fantasy let's go for a big ticket

item

well maybe not sucks but maybe a t-shirt

in a linear or sequential search we

could technically transpose transposo

those terms you would have to go through

each bill to find that 500 same with an

array of say six random integers

if we were wanting to find the eight we

we would have to manually check each

element in order to return the position

of the eight

algorithmically it would look like this

and can you imagine if we had an array

containing a large amount of numbers it

would be very very time consuming and

inefficient to use a linear search

because it would be comparing each

integer one by one

now can it be used for searching sure

but it's only when the numbers are

relatively few and relatively sorted but

hey let's code this out anyway we'll

define our function with two parameters

our array and the element that we're

searching for then we'll write a for

loop to iterate over the length of our

array and if there is an element there

that's the same as our element then just

return where it is in the array when we

print it out we get index position 4 as

that is where our 10 is

a more efficient way to search is to

move up to a binary search and the clue

is in the name binary as in by meaning

something that has two parts

in this search algorithm we would create

two pointers and repeatedly divide the

search interval by half naturally this

would reduce our search space so that

we're not searching the whole array but

only the divided parts

the catches at the binary search

is that it is usually mostly

generally

done on sorted arrays that looks like

this

and this way the algorithm checks

whether the middle element is equal to

the target that we're searching for in

every iteration if the target value

matches the element then its position in

the array is returned if the if the

target value is less than the element

the search continues in the lower half

of the array the target value is greater

than it searches in the upper half of

the array ultimately the algorithm

doesn't need to search the half our

target isn't in with each iteration

now let's look at some code

because our program is only searching

half the array it makes it a much better

search

let's look at the iterative method first

in our function we can create four

parameters our array a start point an

endpoint

and the target that we're searching for

we'll create a while loop to iterate

over our array from start to end and

then create our midpoint our variable

called mid is the magic in the sauce we

only want to traverse the half that

contains our element feel free to print

out the mid when you run the whole

program for yourself to get a good idea

of this

next we can implement our if statements

we'll only need three

check if our target is in the upper half

then reassign our start to begin moving

to the right of the array

if our target is lower than our mid we

reassign our end to begin moving to the

left of the array if our target matches

our mid then just return it easy peasy

then we just return our start or we can

return something like negative one to

print out some strings to make it a

little fancy

but either way you prefer to have your

results printed we have to be sure to

declare our start and end in the

function call our start is zero from the

beginning of the array and our end is

the last element

our length of array minus one

for our recursive function i won't spend

a lot of time here because the

difference between our iterative program

and the recursive program is very

minimal runtime and memory space is very

very close meaning they're both really

efficient

either way it's good to see both

programs

basically all we're doing is just

calling back to the original function

within our else statements

feel free to pause the video for a

moment and take a closer look

okay now that we have some super basic

searches under our belt let's move into

a deeper section of sorting because well

sorting is fun

who doesn't like to sort money piles of

50s here piles of honey is there

holy crap i sound like mr krabs

all right all right let's mix the money

references let's get to sorting our

boring old elements

as for sorting there are a vast array of

techniques

that are available and here's a little

quick list

actually

it's not that quick as there are many to

choose from

bubble sort is the simplest sorting

algorithm that works by repeatedly

swapping the adjacent elements if they

are in the wrong order for instance the

largest integer bubbles up to the top or

iterates its way to the end of the array

through a comparison of each element

although this isn't a very efficient

algorithm perhaps the worst it's still

an important concept to understand the

only real perk the bubble sort has

compared to other algorithms is its

built-in ability to detect whether or

not an array is sorted in the first

place in a best case scenario the list

is already sorted or nearly sorted and

this is in contrast to other algorithms

even those with better than average

performance whose entire sorting process

caused more causes more inversions

reversals reversals or swaps to happen

an optimized algorithm for the bubble

sort is to find the nth largest element

and put it at the end of the array there

is an unoptimized version that you can

check out for comparison purposes if you

want to look that up on your own you're

more than welcome to do that but the

optimized version is done so that the

inner loop can avoid having to look all

the way at the end when it's running the

last iteration it's just saving time and

energy by not having to do that although

bubble sort is one of the simplest

sorting algorithms to understand and

implement its performance means that

it's that its efficiency decreases

dramatically on arrays of larger amounts

of elements and it's even been argued

that it shouldn't even be taught anymore

as it's outdated and is even hazardous

to some cpus

yikes i bet you didn't realize that

you'd be entering dangerous territory

here i mean who knew conan could be

fraught with such tenuous escapades such

as this

how sexy

let's code it

our function will take just one

parameter our array

and then i implemented an iteration

count just for informational purposes

next we create a for loop to iterate

over our indices in the array going to

the right combined with a nested loop

inside that iterates over our indices

going left

or reverse that we're going to call j

i strongly encourage you to write this

program for yourself and within our

nested loop print out j just to see it

in action

we have our iteration counter when this

prints you'll see why i added this now

we just tell it to swap elements if they

are lower than the one to the left in

our if statement if our element to the

left is greater then swap it

then we just return our newly sorted

array so let's print this so you can see

our iteration count

with 36 iterations that's pretty

efficient

check this out now i know i said that it

wasn't going to look at the unoptimized

but guys check this out double the

iterations for two functions and it's

double the code

i liked our first program

simple shorter faster

unstable

oh

that was dangerous i kind of like it but

hey if dangerous not your thing you

thrive in a more stable and secure

environment that's fine too we can

totally move on to the insertion sort

algorithm insertion sort is another

simple sorting algorithm that builds the

final array one item at a time

it's still inefficient on large lists

like the bubble sore but it can have

some advantages such as simple

implementation meaning a reduced amount

of code and performs well in small data

sets and relatively sorted arrays and it

doesn't require much space meaning that

it can be done in place therefore not

using much computer memory in order to

perform well the concept of the

insertion sort is to split the array

into two sections in order to compare

its elements we can consider one side

sorted and the other side unsorted our

algorithm takes one element at a time

from the unsorted side and compares it

to our elements on the sorted side

if our element is larger or smaller it

places it where it needs to be on our

sorted side each generation takes an

element from the unsorted side and

places it into our sorted side all the

way till there are no more elements to

compare to

and in the end we are left with a sorted

array

for our program our parameter in our

function call will be our array starting

with a for loop that will iterate

starting at our second element we don't

need to start at our first integer

because that's our sorted side

we leave that alone for now we create

two variables our key which is our

current element and a pointer to the

left of our key

we'll implement a while loop to iterate

as long as our index is greater or equal

to zero and our index is greater than

our key

we then program it to reassign the

larger one to the right position and

then move down and put our small element

to the left

we do this all in place without creating

extra space

it's fairly simple and straight to the

point

working with arrays can last for days

you like that no but seriously what

we've touched on is only the tip of the

iceberg as a personal challenge to

yourself try searching and sorting a

two-dimensional array are you up to the

challenge

for our advanced level algorithm section

we're going to dive into linked lists a

linked list is a linear data structure

in which the elements are are not stored

at contiguous memory locations

remember this picture

also remember that arrays are stored at

contiguous memory locations

a wikipedia definition states a linked

list is a linear collection of data

elements whose order is not given by

their physical placement in memory

instead each element points to the next

it's a data structure consisting of a

collection of nodes which together

represent a sequence in its most basic

form each node contains data and a

reference in other words a link to the

next node in the sequence

this structure allows for efficient

insertion or removal of elements from

any position in the sequence during

iteration

more complex variants and additional

links allowing more efficient insertion

or removal of nodes at arbitrary

positions a drawback of linked lists is

that access time is linear and difficult

to pipeline

some say that linked lists are among the

simplest and most common data structures

are they though

as an example

think of a photo storage website if you

click on the website landing page this

would be considered your header

then

when you click on the first image this

would become a node

the arrow next to it that you can click

to see the next picture would be your

pointer

now if you get to the third or fourth

picture and want to go back to the first

picture

and look at that again

those back arrows would mean that you're

on a doubly linked list

a doubly linked list means that you can

traverse the list forward and backward

however for this lesson we're only going

to cover a list that we can go forward

on the singly linked list

now the upside of a linked list over a

conventional array is that the list

elements can be easily inserted or

removed without re allocation or

reorganization of the entire structure

because the data items do not need to be

stored contiguously

linked lists allow insertion and removal

of nodes at any point in the list and

they keep up with a constant number of

operations by storing the link previous

to any link being added or removed

during a list traversal so let's talk

about our four basic algorithms or

common problem solving operations for

singly linked lists

every programmer needs to be able to do

the following traverse a linked list or

be able to travel along the nodes and

pointers

search a list for the header a node or

tail a tail is the last and final node

insert a new header note or tail

and to delete a note or tail a link to

list generally always needs a header

let's create a simple linked list

representing a small family member group

we want to be able to do all the

operations mentioned above this will

require a little bit of object oriented

programming so hopefully you are

somewhat familiar with that but if not

that's okay we'll stay simple and make

it easy peasy

we need our pretty standard class node

and class linked list to start our nodes

will be our family members and our

linked list will represent our whole

family this can be considered a little

backwards but i like to instantiate our

objects first so it's easier to see what

we're dealing with being able to

visualize what we're doing is an

important skill in programming and

design because it helps us think about

the big picture throughout the solution

let's picture our knowns as a family of

mom dad and two kids we'll create our

family head as bob his wife amy our

header will point to amy as the next

node and then our two little nodes max

and jenny they all need to point to each

other as they will be linked together in

our list so we will make sure our

pointers are created which is

object.next

everything looks good and we have all

our members individually as nodes but to

really see our algorithm in action we

need to link all our members together in

our linked list where we can operate on

that list and design them to be really

complex or really simple

linked list problems are often used as

interview and exam questions so knowing

how to create linked lists how to

traverse them how to add and take away

elements is very important once you get

really familiar with those concepts you

can move into learn learning how to

traverse how to reverse a list swap

elements maybe even go on to doubly

linked lists or circular

it's a really big topic to explore and

fun too and if you're not having fun

why are why are you here like why

so back to our family

let's get them created and then set it

up to where we can traverse this family

list

once we have established our class we

just need to implement our functions

that tell our linked list class what to

do

this is where we implement all the

things we need done with our nodes and

the first thing we're going to do is

make our nodes by creating the data and

a pointer

let's create our objects

then create pointers that go to each

family member

now that we have our nodes we can now

link them in a list by creating our

linked list class this is where we

create a function to traverse our nodes

in a list and be able to print them all

out

we only need one argument self because

it doesn't need any other value or input

to be to be able to do what we need

which is starting at our header and

printing them all out next by next by

nuts

then we just call family.traversal

and out pops our family members now if

you want to shoot these out in a

particular way such as in a list or

fancy strings that would have to be done

within another function in the linked

list class which unfortunately is not

within the scope of this course that's

more in tune with object oriented

programming a whole other separate topic

the goal here is to focus on being able

to operate or solve problems to creating

simple function calls on linked lists

now our family member list is looking

pretty good but word on the street is

that bob isn't treating amy too well and

he's been running around on her

from the looks of it bob might be

packing his bags and leaving the family

but it's not all bad because dave has

been hanging around a bit and although

he's bob's brother he's a really good

guy

i know i know it sounds really back once

but seriously dave is going to step up

to the plate and he's looking forward to

being a dunkle to the kids

you know you know a dunkle like when

your uncle becomes your dad and it's not

weird at all

so for our list we need to put dave as

the new header position and because this

is a computer and it can't do anything

that we don't tell it to do we need to

implement some functions

let's call it insert new header

and this is going to fall under our

insertion algorithm

we need to know how to insert elements

into the list and not only that we have

to tell it where to insert for davis the

new head we are going to put him at the

front of the list

so for our function apart from self we

need new data or information to be added

to the parameter since we are adding a

new name

our new name is our new node so we will

need our node class

remember the nodes are our family

members then all we need to do is assign

him as the next new node to the

self.head and then assign yourself.head

to the new node let's traverse the list

now

oh boy bob is still in the picture and

you know what amy can't have her new

hubby under the same roof as her ex that

is just not gonna work this is a little

two backwoods so bob needs to

go go

creating our new header simply pushed

all our nodes down so we need to get bob

out of there let's create a function

that deletes a node however we can't

just delete some rando node we have to

tell our computer where bob the node is

in the first place

this means we have to search our family

list so let's create a search search

function with one parameter x

see what i did there pretty clever we

need to know if bob is even in there to

delete so in order to search we need to

start at the beginning of the list

let's create a temp variable assigned to

the self.head and make a while loop that

will traverse as long as that list or

header isn't empty if any node or temp

node isn't equal to x or whatever it is

we're searching for we want to return

true or else return false

then we need to assign our temp to the

next node because in a while loop it

will need to continue until the node is

not is none then it will return our pool

great let's find out if bob is in there

by printing our search function uh oh

it's true bob's still there and he's

giving amy a hard time about leaving

let's help her out and create a delete

function

this is going to be similar to our other

functions where we need to put our data

in the parameter and begin at the head

of our list

we need our if else statements and we

will

and we will say bob isn't there then

tell us and if he is let's kick him out

we do this by changing the pointers from

the previous node and pointing it to the

node after bob so in essence we're

skipping over bob but watch what happens

when we try to try to delete amy

bob will come back

why does this happen

because our list has four nodes

as created here

and those nodes will always be there

unless we go in and manually delete the

object itself

so far we've traversed our list searched

our list inserted a new header deleted a

node and last but not least we're going

to delete our tail

yep that would be poor jenny

she's had enough of these adult

shenanigans and she's out of here

heading to new york to become the

greatest fashion editor that vogue has

ever known she's ambitious okay

we started the head and as long as the

temp dot next next is not none our

pointer is two nodes ahead we make our

temp the next node and set it to none

voila

best of luck jenny

so there you have it some very basic

foundational algorithms for linked lists

which in itself is a complicated subject

but if you get these classical concepts

down solid it becomes easier and easier

to build up from on your own once you're

more comfortable with singly linked

lists try to challenge yourself with

doubly linked lists with pointers that

point back who knows maybe bob will

shape up and amy will go

or point get it

back to bob

for our mind breakers section in data

structures we're going to be looking at

hash tables right off the rib i'm going

to tell you the key phrases or terms you

need to be familiar with when it comes

to hash tables associative arrays

hash functions

key value pairs collision

and chaining

a hash table consists of an array of

pockets or buckets where each stores a

key value pair in order to locate the

pocket where a key value pair should be

stored the key is passed through a

hashing function

this function returns an integer which

is used as the pairs index in the array

of buckets when we want to retrieve a

key value pair we supply the key to the

same hashing function receive its index

and use the index to find it in the

array

associative arrays can be implemented

with many different underlying data

structures and it refers to an array

with strings as an index rather than

storing element values in a strict

linear index order this stores them in

combination with key values multiple

indices are used to access values in a

multi-dimensional array which contains

one or more arrays a non-performant one

can be implemented by simply storing

items in an array and iterating through

the array when searching remember the

wallet with lots of cash

associative arrays and hash tables are

often confused because associative

arrays are so often implemented as hash

tables according to wikipedia

a hash table uses a hash function to

compute an index also called a hash code

into arrays of buckets or slots from

which the desired value can be found

during lookup the key is hashed and the

resulting hash indicates where the

corresponding value is stored

in many situations hash tables turn out

to be on average more efficient than

search trees or any other table lookup

structure for this reason they are

widely used in many kinds of computer

software particularly for associative

arrays database indexes caches and sets

to summarize we can use hash tables to

store retrieve and delete data uniquely

based on their unique key

the most valuable aspect of a hash table

over other abstract data structures is

its speed to perform insertion deletion

and search operations

hash tables can do

all of them in constant time

for those unfamiliar with time

complexity big o notation constant time

is the fastest possible time complexity

hash tables can perform nearly all

methods except list very fast in o b o

one time

hash tables have to support three

functions

insert for key value

get

key

or delete a key

collision happens when there could

possibly be two or more pieces of data

in

data set x that will hash to the same

integer and data set y

this means that whenever two keys have

the same hash value it's considered a

collision

what should our hash table do if it just

wrote the data into the location anyway

we would be losing the object that is

already stored under a different key

ideally we want our hash values to be

evenly distributed among our buckets as

possible to take full advantage of each

bucket and avoid collisions

but generally speaking this is unlikely

so this is where two common methods to

solve are employed separate chaining

using linked lists or binary search

trees for example and local addressing

using linear probing

with some basic concepts of hash tables

under your belt a fun exercise or

challenge that you can try would be

pattern matching on strings using the

robin carp algorithm

yes it's fairly advanced and yes this is

our mind breaker problem but shoot for

the stars people who knows you might

just surprise yourself

what have we learned what have we

learned from lesson two

okay

we touched on a one-dimensional raise as

our data structure of choice at our

basic level

we looked at contiguous versus

non-contiguous different ways to search

linear

binary

and we sort bubble insert and even

talked about an array with more than one

dimension we learned about dunkles

no no no no we learned about link list

by creating a family of sorts to

demonstrate how to traverse search add

and delete headers

nodes tails on linked lists some of the

most common operations used that every

programmer should be familiar with

we broke our heads on hash tables and

hashing functions by going over what

they are and what can happen when

problems arise from colliding data

within a bucket ultimately hash tables

are for storing retrieving and deleting

data based on key value pairs and on

average are more efficient than search

trees or any other table lookup

structure

[Music]

i learned about the divide and conquer

concept from a very early age

it was traumatizing actually you see my

brother and i used to fight all the time

when we'd swim in the pool and finally

one day our mom came and said to us just

divide it in half and you stand once and

you stay on another side

my brother picked the top half

so yeah divide and conquer

i drowned but only momentarily

in programming when we cross paths with

programs that are too complicated to be

solved all at once we can implement

methods to break them down into smaller

pieces and if we solve all the smaller

pieces we are in essence dividing out

the big

solving the small and in the end this

makes our lives so much easier a divide

and conquer algorithm paradigm can solve

a large problem by recursively breaking

it down into smaller sub-problems until

they become simple enough to be solved

directly it could be compared to the

age-old problem of swallowing an

elephant

can't do that so we devour it one bite

at a time

actually it's more of a solution of

three bytes dividing solving and then

merging it all back together

unfortunately since divide and conquer

algorithms are usually implemented

recursively this means they require

additional memory allocation and without

sufficient memory a stack overflow error

can or will occur

this is due to the overhead of the

repeated subroutine calls along with

storing the call stack the state of each

point in the recursion and that can

outweigh advantages of any approach

on the flip side if the dnc solutions

are implemented by a non-recursive

algorithm that stores the partial sub

problems in some explicit data structure

like a stack queue or priority queue it

can allow more freedom in the choice of

the sub-problem that is to be solved

next a feature that is important in some

applications including breath first

recursion and the branch and brown

branch and bound method

for function optimization

second even if a solution to the full

problem is already known like sorting a

brute force comparison of all elements

with one another divide and conquer can

substantially reduce computational cost

lower cost means higher efficiency

third explicitly recursive subdivision

can have benefits to the temporary

memory of the computer not the main hard

drive

which is called a cache

using divide and conquer one can even

design an algorithm to use all levels of

the cache hierarchy in such a way as to

have an algorithm that is optimally

cached oblivious which means operating

independently from its temporary memory

storage

this approach has been successfully

applied to many problems including

matrix multiplication which we're going

to get into in just a few minutes

so let's get into it and start at the

base level algorithm for divide and

conquer the merge sort

conceptually merge sort works as follows

divide the unsorted list into two sub

lists about half the size

sort each of the two sub lists

merge the two sorted sublists back into

one sorted list

invented by john von neumann in 1945 the

merger sort is an efficient general

all-purpose and comparison-based sorting

algorithm an example of merge sort would

be to first divide the list into the

smallest unit one element this becomes a

list itself

then compare each element which is a

list itself remember with the adjacent

list to sort and merge them done

repetitively the end results in the

elements sorted and merged

i understand that can be a little bit

confusing but let's think of it in this

way

say we have two different lists of

numbers list a and list b and then we'll

have an empty list we'll call c

lists a and b are ordered from least to

greatest and what we would do is

repeatedly compare least value min of a

to the least value min of b and remove

the lesser value and then append that

value into c

when one list is exhausted we append the

the remaining items into c they're

already started so we're good there

then we just return list c which is

sorted we're systematically merging a

and b together and appending it into c

but for this lesson and although you can

append or extend on an empty list to see

we're going to swap our elements i just

wanted to share a very simplistic

explanation for that hoping that would

give you an idea of the concept

typically creating more space for an

empty list or temporary list for storage

isn't always the best approach we want

we want to do our best to optimize our

program if we can

let's head into our ides and write our

programs

the best way to see the different

methods of the merge store and their

efficiencies is to compare four programs

that i have here

some things i wrote myself

some things i borrowed

we're going to look at a single array of

10 integers and just sort it by brute

force this means that the program takes

the first number in index position 0 and

compares it to each and every number in

the array

if it's the lowest it gets reassigned at

the first position

then it moves to the next number and

does the same thing it gets compared to

each and every integer and the lowest

number gets reassigned at index position

one and so on and so forth

for this little iterative program it's

168 iterations or cycles to get an

output and that's just for 10 elements

keep that in mind as we look at the

differences within the merge sort

our first real program for the merge

sort is about as simple as we can get it

we're starting with an array that is

already split in half and sorted the

program we're looking at right now won't

work properly if our two halves are not

already sorted

why

because ultimately this is our divide

and conquer algorithm we are dividing

our array into two parts and conquering

both halves separately by sorting them

and then merging them back together

remember just a few minutes ago when we

talked about list a list b and empty

list c

this program is comparing each number at

each index position determining which is

lower and then popping it out and then

appending it into our empty list c

and

then just in case either list is longer

than the other that leftover portion

just gets added to c and since it's

already sorted we'll have no problems

there

this is a neat little non-recursive

merge sort for an array that has a small

amount of elements and is already

divided in half however since this isn't

a perfect world and you'll be more than

likely asked to write a program for an

array that isn't divided and sorted

beforehand

we're going to have to rewrite this

so here's our big dilemma do we sort our

halves recursively or do we sort them

iteratively

or do we sort our halves using python's

built-in feature called sort or sorted

let's shoot for sorting our halves

recursively this is what is called the

top-down method

what it does is repeatedly produce

sub-lists half by half by half until

there's only one sub-list or one element

remaining which becomes a list all in

itself a list of one element how many of

you are willing to but it's the optimal

or optimized method

well it's not

once we have our have sorted we continue

on very much like before we compare each

element and append it into our empty

list

now if recursion isn't the optimal

method then it must be iteration right

well yes

mostly this is the bottom up method

think of the top down just in reverse it

starts at one element then two elements

then four remember that very first brute

force method of sorting a list we can do

that after we divide the array into two

parts and then merge it all back

together hmm

let's see what happens when we run it in

something like leeco that'll really put

it to the test

[Music]

okay well that didn't go as planned but

it does work it's just bad on really big

arrays

so what is the bust

using python's awesome sort or sorted

feature of course

well splitter array have python sorted

for us and the rest is good to go with

our 10 element array it's only 34 steps

to completion compare that to our brute

force program no dividing no conquering

with almost 170 iterations our bottom up

iterative method is smoking

check it in leak code

talk about optimization

now let's move out into deeper waters

with some matrix multiplication

matrix multiplication is wild y'all

but it is a fundamental concept to

understanding computing among the most

common tools in electrical engineering

and computer science are rectangular

grids of numbers known as matrices the

numbers in a matrix can represent data

and they can also represent mathematical

equations

in many time sensitive engineering

applications multiplying matrices can

give a quick but good approximations of

much more complicated calculations did

you know that one of the areas of

computer science in which matrix

multiplication is particularly useful is

graphics it's crazy to think that a

digital image is basically a matrix with

rows and columns of another matrix

corresponding to rows and columns of

pixels and the numerical entries

correspond to a pixel color value

decoding digital video for instance

requires matrix multiplication and did

you know that mit researchers were able

to build one of the first tips to

implement a high efficiency video coding

standard for ultra high definition tvs

partly because of patterns they

discerned in the matrices

pretty cool huh

so the dimensions are listed as rows and

columns or in many equations n times m

where n represents the number of rows

and n represents the number of columns

be careful that you that you know the

difference between rows and columns and

don't mix them up rows go across columns

go up and down to see how matrices are

multiplied check out this graphic that

shows how it all works we multiply each

element in the row by each element in

the column and then we add them together

in order to code this we'll start with

the naive method it's not too bad to

learn however just know that it can get

very expensive as we increase the size

of our matrices for larger matrix

operations

we wouldn't want to use too many nested

loops because it could really start

bogging down

ideally we would want to use optimized

software packages like numpy

numpy should be the go-to which is

insanely faster and easy too but before

we start cheating we really need to

examine what's going on underneath the

hood like how a program is created that

multiplies matrices in python

because if we can get a really good

handle on this

we'll get a better handle on the mind

breaker

[Music]

okay so what we're doing is basically

creating a calculator in python that

multiplies matrices and you might be

wondering hey what's the deal here how

is this part of dividing and conquering

algorithms we're definitely going to get

into that but this is all preparation

for the mind breaker lesson you'll need

this under your belt first plus it's a

really good foray into matrices which is

one of the most well studied problems in

numerical computing today so yeah let's

dig in

we'll create two matrices x and y and to

store the results of our products we'll

create an empty matrix we'll call result

the most popular method in my humble

opinion is to use nested loops to

iterate over the rows and columns and

implement the results into our matrix of

zeros using an addition operator that

adds our product of the rows

and the columns of x and y

we could do this recursively sure but it

wouldn't really be advantageous the goal

is to help you understand the logic

behind matrix multiplication so we can

move on to our mind breaker lesson

strassen's algorithm for matrix

multiplication

volker strassen is a german

mathematician who made important

contributions to the analysis of

algorithms and has received many awards

over his lifetime he's still alive by

the way

he began his research as a probabilist

his 1964 paper an invariance principle

for the law of the iterated logarithm

defined a functional form of the law of

the iterated logarithm showing a form of

scale scale and variance in random walks

this result now known as strassen's

environments principle or strassen's law

of the iterated logarithm has been

highly cited and led to the 1966

presentation at the international

congress of mathematicians

now if you know what an iterative

logarithm is that shows a form of scale

and variance in random walks or in

variance principles in general well then

please by all means invite us to your

latest book site

but if you're a mere mortal like the

rest of us

hold on to your hats because that was

only the beginning for strassen

three years later strassen shifted his

research efforts toward the analysis of

algorithms with a paper on the gaussian

elimination come on everybody knows what

the gaussian elimination is

and there in that paper he introduced

the first algorithm for performing

matrix matrix multiplication faster than

the o and three time bound that would

result from the naive method

this means that his algorithm is like

the best of the best when it comes to

computing the product of two two by two

matrices there is no formula that uses

fewer than seven multiplications

how he did it he doesn't say maybe

somebody flew into german can call him

up and ask him like what kind of person

do you have to be to discover some great

mathematical algorithm like what are

your hobbies can we talk what do you do

on the weekends i mean at this point at

the time of recording this he's 85 so i

don't know he probably just relaxes his

brain muscles for the most part i mean i

know i would they're probably still

smoking anyway not to knock on this

awesome person and as great as the

algorithm is and the contribution it's

made to matrix multiplication

unfortunately it's too numerically

unstable for some applications

in practice fast matrix multiplication

implementation for dense matrices use

strawson's algorithm for matrix sizes

above a breaking point then they switch

to a simpler method once the subproblem

size reduces to below the breakover see

divide and conquer in any event the

publication of its algorithm did result

in much more research about multi matrix

multiplication that led to faster

approaches it opened the door to bigger

and greater things which is pretty neat

okay now that we've got that all under

our belt let's take a deep dive into

what the algorithm really does just a

few minutes ago we saw the naive method

so you should be somewhat familiar with

it

in two two by two matrices we have eight

multiplications with stress since we can

get this down to seven with basic

mathematical formulas

and i've gotten it all written out so

you can see it all work in a program

let's take a peek

we're still going to keep our 2x2

matrices for simplicity yay

but we're going to look at our famous or

infamous methods of optimization

iteration and recursion for strassen's

algorithm we're dividing our matrices

not by halves but by quarters and then

we'll solve or conquer

each sub-matrix to solve the entire

problem

in our iterative method we're breaking

down the two matrices all the way down

to each individual square or quadrant

we'll name our passes p and simply plug

in our mathematical formula used to

compute our multiplication and addition

on the quadrants

now although we called numpy cheating

we're going to implement it here just

for a little bit of ease by instead

creating a result matrix of zeros we're

going to make a new matrix called c and

number each quadrant of c as c1 c2 c3

and c4 for each quadrant we plug in our

equations for our passes these equations

are our conquered solutions to the

quadrants

lastly we use numpy to implement a new

matrix by using the np dot array feature

and input our four c quadrants

pretty cool

the recursive solution is very similar

only we're recursively splitting the

matrices into our quadrants and

recursively computing our seven passes

what did we learn what did we learn we

learned that the merge store is an

efficient general purpose sort that can

be implemented in several ways including

the top-down and bottom-up approach

we learned that matrix multiplication is

used in simple graphics all the way up

to hd tvs and we wrote our own python

program to calculate two matrices

we looked at strassen's algorithm for

matrix multiplication

and learned that he discovered how to

get the lowest amount of calculations

possible for multiplying two matrices

to this day computational efficiency of

matrix multiplication is still a hot

topic in theoretical computer science

[Music]

apparently if you google greediest

person a real live person does pop up

can you imagine like googling the worst

person in the world and your face gets

blasted on the screen

now i'm not going to show who this

person is because i don't know who he is

and it's not really important just kind

of funny but it does tie in with our

algorithm

what is greed to you like what is a

greedy approach

say your beautiful old granny baked a

pie for your family and uncle bob shows

up you know amy's ex-husband

and he takes the biggest piece of pie

for himself that'd be pretty greedy

wouldn't it by him eating the largest

portion that sure would make the pie

easier to finish wouldn't it it wouldn't

be great for everyone else or their

rumbling tummies but by taking the

biggest portion and leaving just little

portions to take care of

that's

sort of the concept behind the algorithm

the paradigm behind the green concept is

that it builds up a solution piece by

piece always choosing the next piece

that offers the mo the obvious and most

immediate benefit

by using several iterations and by

obtaining the best result at a certain

iteration the results should be computed

in other words it follows the

problem-solving method of making the

locally optimum choice at each stage

with the hope of finding the global

optimum something to keep in mind though

is that they do not consistently find

the globally optimum solution because

generally speaking they don't operate

exhaustively on all the data

nevertheless they are useful because

they are quick and often give good

approximations to the optimum if a

greedy algorithm can be proven to yield

the global optimum for a given problem

class it typically becomes the method of

choice

a really fun problem that we can lead

off with is called assign mice to holes

what we need to do is put every mouse to

its nearest hole to minimize the time

how can we do that and what's the

optimal approach

what would you think about sorting the

positions of the mice and holes first

that would be pretty greedy of us

wouldn't it

this allows us to put a particular mouse

closest to the corresponding hole in the

holes list we can then find the maximum

difference between the mice and the

corresponding hole position to determine

how much time it will take to get there

now does being greedy always equate to

maximum of something regarding problem

solutions not necessarily don't let that

word fool you

for this problem we're finding the

maximum difference hint subtraction to

find the minimum amount of time it takes

for our mice to travel

as a quick side note another problem

that i would encourage you to learn on

your own a triple dog dare you is the

kids with greatest candies problem using

the greedy approach

this is considered an easy problem so

don't worry too much the problem is

similar to this one by asking for the

distribution of the minimum number of

candies that can be given to a kid

and then seeing which kid will contain

the maximum number of candies

so again the greedy approach can be

tricky we have to carefully decipher

word usages to make sure we're solving

the problem accurately

but with that being said the assigned

mice to holes problem will help set you

on some solid ground and make it a bit

easier to work on other challenges now

if you do decide to tackle the candy

problem and i do highly encourage you to

do that i'll give you a little hint

the brute force method would be a one by

one distribution of candies to each

child until the condition satisfies

and yet the greedy way would be to

traverse the array just twice from

beginning to end and back again by

greedily determining the minimum number

of candies required by each child

as a personal challenge to yourself work

out how that could be written it'll be

fun let's head over to our ides and slam

dunk some mice into holes

okay we need to get our mice into holes

and we have to do it quickly but first

things first we have to make sure that

there are an equal amount of holes to

our mice this is our base

now

we initialize our greedy approach by

sorting the mice first then we create a

variable to store our maximum difference

and loop through the length of our mice

in our if statement we create the

condition that if the position of the

mouse subtracted from the position of

the hole is greater than the maximum

difference

our max diff variable gets reassigned to

hold that difference

lastly we just return the largest

difference our last mouse gets to the

hole in x amount of time

by sorting our two lists first this puts

us at an advantage to finding the

minimum time faster

our code is concise short and along with

python's sort feature we are really

rocking it out

next up is kind of a doozy the

fractional knapsack a pretty well known

problem for our greedy algorithm

if you were a little bored with our mice

and no holes you won't be with the

knapsack

for a step up on the greedy method let's

take a peek into what's called the

fractional knapsack now this is intense

fractional knapsack freshness people

okay so this problem on the surface

seems really complicated but it's only

as complicated as you want to make it

you know what i mean so let's not make

it complicated okay so just so what you

can do is just envision me it's the

apocalypse it's the end of the world

wait maybe it's just a monday food is

scarce people are bartering water is

practically non-existent so i need to

conserve every bit of energy i have to

walk to the market and try to trade what

i have to stay alive okay now i have

five objects i can take our knapsack or

bag is our container for holding our

objects think of our array as a storage

device

and each object is worth something some

are worth more than others and each

object weighs something some objects way

more than others the key to this problem

is to make the right decision on which

objects are going to bring me the most

profit i mean i gotta know i have

animals to feed

that haven't become somebody else's

dinner you know while at the same time

they can't be too heavy that i can't

carry in my container because this is

crucial i might have to make a decision

based on its object the weight of the

object not necessarily its profit if i

want to make it back alive from zombies

r us

now the real kick in the head on solving

this

lies in that little teeny tiny word

fractional this means that my objects

can be divided if they are divisible

say one of my objects contains three

chapsticks i can divide that object and

bring only two chapsticks if it will

make a difference who knows cat fat

might be a real hot commodity remember

in that book of eli movie where he

killed that cat

well it was for food but he also made

chapstick out of it

not saying my chapstick would be that

it'd probably be made from beeswax or

something

don't forget i'm risking my brains to go

to the market to feed my animals in the

first place so no cancel be harmed here

no siree bob okay wow that was a really

long

way off the beaten path if let's get

back to the problem so the weight is

divisible and it would help to best fit

how many profitable objects in my

backpack then we should do that why we

need to fill to the brim in our fixed

size knapsack the most valuable items

that we are able to carry

for the steps of our algorithm keep in

mind we are given weights and values of

n items and we need to put these items

in a knapsack of capacity w to get the

maximum total value in the knapsack by

the way in the o1 knapsack problem a

whole different problem is one where

we're not allowed to break items down or

divide them using fractions

we either take the whole item or don't

take it but in this problem fractions

can be used

a really efficient solution to this

problem is to use the greedy approach as

opposed to brute force remember brute

force is calculating every possible

solution and is typically

computationally expensive meaning it

takes a lot of runtime and memory space

the basic idea is to calculate the ratio

of value slash weight for each item and

sort the item on the basis of this ratio

then take the item with the highest

ratio greedy remember and add them until

we can't add the next item as a whole

and at the end add the next item as much

as we can

this will generally always be the

optimal solution to this problem

you'll see what i mean when we get into

our ides and start coding it so let's

get to work

okay

we'll define our functions with three

parameters

we need our values our weights and the

capacity of the bag

now we have our created variables here

but i implemented print calls to show

you what's going on

we have our items a range of zero to

four equaling five items

the ratios are values divided by weight

using floor division to eliminate floats

list comprehension is good to use when

you can

then we'll sort our ratios in decreasing

order

lastly the ratios index positions are

printed here to show the to show you the

order that it will be computed in

i'm going to

take out the third variable here it as

it won't be needed

i just wanted to show you how it matches

up with its index position

our ratios being sorted is taken care of

in the items.sort line

we need a counter to keep a running

tally of our max value

we will create a list that is the length

of our values we'll call fractions that

will contain all zeros

i'll tell you why we're doing this at

the end of the program now it's time to

run our for loop we'll iterate over our

items and if the weight of our item is

less than capacity it will add a one to

our fractions list of zeros at its index

position then we'll add that value to

our max value and subtract the weight

from our capacity now when all those

conditions satisfy and we hit our else

we take the fractional difference of our

last item and add it to our max value

then we return our maximum value which

is 500 now here's the key you might be

wondering this

we have an item that weighs

a 10 and it's worth 90.

at a capacity of 150 we could put 15

of these items into our bag for a value

of 15 times 90 equaling 1350.

why can't we just do that

we can't do that because what we're

looking at today is what is called the

bounded knapsack problem or bkp for

short

the scenario i just painted a 15

15 of the same items would be allowing

for copies of that item called the

called the ukp type short for unbounded

knapsack problem and that's a whole

other ball of wax kids

this is why we set up the fractions list

containing nothing but zeros

this is important because it's putting

in our highest values first

and then taking a fraction of the last

item from maximum capacity in our bag

each item counts as one in our fractions

list until every zero is turned into a

one but our very last item to change

from zero to one is that special one

that gets the remaining capacity divided

by the weight

that figure gets multiplied to our value

and then added to our max value in the

case of our program that ends up being

28 and a half percent and what's 28 and

a half percent of 140

is the value of 40 that puts us up to

500. sorting our ratios in decreasing

order gives us the optimal or best

approach this is considered greedy as

you are placing items with the largest

ratios into our bag first

now you can sort by price and you can

sort by weight as those are considered

methods and they will get you close to

500

but it's just not 500. and we should

always strive for the best method

algorithmically

hopefully you now understand a pretty

well-known problem not only in logistics

but other fields as well like

manufacturing and finance it's all about

optimization

what is the optimal way to get the items

of highest value into a bag that you can

reasonably carry

and this greedy concept can be carried

out in other reasonable scenarios as

well

with the goal of calculating solutions

whether that be approximations or actual

figures

let's move on to our last section with

good old fibonacci

in our mind breaker episode we're going

to talk about another well-known

mathematician leonardo alpisa aka

fibonacci considered to be the most

talented western mathematician of the

middle ages

according to wikipedia he wrote the

libre abachi the book of calculation and

is it is a historic 13th century latin

manuscript on arithmetic which includes

the posing and solving of a problem

involving the population growth of

rabbits based on idealized assumptions

the solution

generate generation by generation of

rabbit offspring was a sequence of

numbers later known as the famous

fibonacci numbers or fibonacci sequence

in a little side compartment of

mathematics however what we are going to

talk about today is the greedy algorithm

for egyptian fractions first described

by fibonacci for transforming rational

numbers into egyptian fractions

an egyptian fraction is a representation

of an irreducible fraction as a sum of

distinct unit fractions and this means

that every positive fraction can be

represented as a sum of unique unit

fractions a fraction is a unit fraction

if the numerator is 1 and the

denominator is a positive integer for

example 1 3

is a unit fraction to take a closer look

we'll reduce a fraction of 7 14 in a

pizza example the egyptian way

[Applause]

[Applause]

[Music]

in order to generate egyptian fractions

we will utilize the greedy algorithm

because although possible to use brute

force search algorithms to find it it

would be extremely inefficient

remember 3d algorithms are for

optimizing our problems we always want

to know and understand the optimal

solution as a side note although

fibonacci's greedy algorithm isn't the

only way to efficiently solve our

problem he's our focus for today

let's look at a program

so yeah i got this program here

we're importing some tools

scroll down some

we got three functions going strong a

main a validate a converting function we

got some wrappers nestled functions we

got some yields which means it's a

generator not even gonna touch that

this is a lot of code

heading into nearly 50 lines

is it good

yes

hard to write

depends but if you're looking for

kindergarten style programming you come

to the right place

i totally dumbed it down to show you the

mechanics not that you're dumb i love

simple make it plain am i right

so we have this pie remember that bob

took a big old chunk of let's change

that to a pizza since it kind of goes

with our theme of leonardo of pizza

or fibonacci

so to get this real simple we'll write a

function that takes our numerator and

our denominator

we'll implement an empty list for our

denominators we won't need one for our

numerators because we already know that

they're going to be ones

then we write our while loop that as

long as our numerator isn't zero we can

create a variable that stores our

quotient using the math dot seal that

will round it up

this is important for when we have a

number that's a float lower than one

next we reassign our numerator to be the

product of the quotient and the

difference of our denominator subtracted

from our numerator then we reassign our

denominator to be multiplied by our

quotient or x

for our fraction of 7 12 this is going

to put a 2 and a 12 into our egypt list

and those two numbers are our lowest

common denominators or lcd for short

all that's left to do now is put a 1

over those lcds as those are our

egyptian fractions and that can be

accomplished a few ways but basically i

just implemented some good old-fashioned

string formatting by iterating over our

egypt list of denominators and adding

them to an empty string

once we do that we just return it the

caveat to this really simple program is

that it probably doesn't work for really

complex fractions or when the numerator

is larger than the denominator

but this is a good exercise in

understanding how to break down a

fraction the egyptian way ultimately all

of this runs more along the line of

recreational mathematics however there

are still open problems involving

egyptian fractions that have yet to be

solved or proved

who knows maybe you can be the next

mathematical genius to solve one of them

so what did we learn about greedy

algorithms we learned optimization of

solutions to problems using the greedy

algorithm provided in our example with

the mice in the holes it's like a

delicious big pie where you take the

biggest slice out first

this makes the pie easier to finish we

learned that chapstick made from cats is

a hot commodity in the apocalypse

kidding

we walk through the fractional nap stack

problem where we find the largest value

of an item in a way to maximize the

total value within the knapsack we want

to avoid brute force here because

finding all the possible subsets would

take too much time we learned that

fibonacci was the first to describe an

algorithm for transforming rational

numbers into egyptian fractions and it

is where we really take the largest unit

fraction we can and then and then repeat

that on the remainder although this is

purely mathematics it's still a good

representation of the greedy algorithm

[Music]

[Music]

what do you think when you hear the word

dynamic

well if you're a programmer or an

aspiring programmer

you may not know what to think

what does being dynamic really mean

for once in my life it would be awesome

to be described as dynamic because when

we talk about a dynamic person

it means energetic

spirited

active

lively

zestful vital

vigorous

maybe this is why when it comes to

computer science students can become

intimidated with the subject of dynamic

programming

ultimately the word dynamic means a

process or system characterized by

constant change activity or progress

with its roots in greek to the word

dynamous meaning power

hint dynamite

so it's no wonder we subconsciously shy

away from it because it just sounds like

a handful it dangerous pretty wild how

words and their meanings really deep

down can make us feel a certain way you

know

but i'm here to help you out and put you

at ease and tell you a little secret

they introduce dynamic programming to

children

okay maybe not children children but

like upper elementary middle schoolers

kids like that so if they can learn

about it so can you

but first we have to unplug our

preconditioned mentality if you will to

think and believe that anything in

computer science is too hard

it's not it's all just one big treasure

hunt to find information and knowledge

that best suits your intellectual filter

not all people teach all things the same

way

not all people learn information or

absorb knowledge the same way so let's

take a deep breath and step back to look

at the different dynamics that are out

there

that's a long list

for us there should only be two that

should really mean something

that's dynamic programming languages and

dynamic programming as a methodology you

can have programming languages that are

either dynamic or static python is

dynamic by the way and then you have

programming methodologies that are

concerned with analyzing a problem by

developing algorithms based on modern

programming techniques

in the world of dynamic programming

dp is used when the solution to a

problem can be viewed as a result of a

sequence of decisions with the intended

outcome of reducing run time and any

implementation that can make a solution

more efficient is considered optimizing

it

in an early lesson we saw several

problems that can be viewed in this way

i.e the knapsack problem

essentially the difference between the

greedy method and dp is that in the

greedy method only one decision sequence

is ever generated and in dp many

decision sequences may be generated

however sequences containing sub-optimal

sub-sequences cannot be optimal and as

far as possible will not be generated

and this is the key that helps make a

reduced run time optimal substructure

means that optimal solutions of

subproblems can be used to find the

optimal solutions of the overall problem

in dp this is called the principle of

optimality the principle of optimality

as relayed by richard bellman who

introduced dynamic programming in 1953

states that an optimal policy has the

property that whatever the initial state

and initial decision are the remaining

decisions must constitute an optimal

policy with regard to the state

resulting from the first decision

so what does that mean

in general we can solve a problem with

optimal substructure by doing three

things

breaking the problem down into smaller

sub problems

solving the sub problems recursively

and then using the optimal solutions to

construct an optimal solution for the

original problem

but there's an issue with this that we

need to be aware of and that is

sometimes when we do this we can get

overlapping subproblems we want to avoid

that

for instance in the fibonacci sequence

f3 equals f1 plus f2 and f4 equals f2

plus f3 computing each number involves

computing f2 because both f3 and f4 are

needed to compute f5 a naive approach to

computing f5 may end up computing f2

twice or more

in order to avoid this we instead save

or store the solutions to problems we

have already solved in dynamic

programming there are three basic

methods

we have recursion

top down with memoization and bottom up

with tabulation

and although we could have chosen the

well-worn path of the fibonacci sequence

to do this

oh no

i'm encouraging you to step out and

think bigger have confidence

you are smart and you can do big things

so

let's get ugly with it in our first

lesson

what are ugly numbers and why do we call

them ugly and what do we do with them

to answer the first question i saw this

and i think it will give you a big hint

okay

that's funny and i wish i could have

been the one to think it up but alas i

was too slow to the game

darn

i stumbled upon a pretty funny thread

about ugly numbers and i stole that so

shh don't tell on me

but the long and short of it is this

to check if a number is ugly in the

first place

we divide the number by the greatest

divisible powers of two three and five

and if the number becomes one

then it's an ugly number otherwise it's

not

we're employing our power of prime

factorization

we learned that way back in the fifth

grade so that tells me that you probably

don't remember

but it'll all come back to you today

the sequence of one to six eight nine 10

12 15 shows the first 11 ugly numbers by

convention one is included

all right now that you got that under

your belt let's tackle the problem of

finding the nth ugly number

can we find the hundredth ugly number

would you believe that if we crafted a

program using brute force it would be

over a thousand iterations

for the hundredth number

i think i broke my visualizer trying to

run it

and we don't want to break things or

make our computers smoke out the back or

anything like that so it's time to

incorporate our spirited energized

vigorous programming that's like

dinomite but not a lot of dynamite

we'll start small and just find the 15th

ugly number how does that sound

since we're going big and not going home

we're gonna shoot for the bottom up

method for a dynamic approach

if we use a pretty simple iterative with

some recursion solution we're still

looking at nearly 600 steps or

iterations in the program

to get our results as compared to just a

little over a hundred steps for the

bottom-up dynamic programming method

talk about optimization

if we use simple iteration and

continuously loop through all the

numbers to find whether each number is

ugly or not it can really bog down when

we get to larger numbers

with dynamic programming instead of

starting from the head we start from the

bottom we're also using a table to help

us store the values for easy access if

we look at the ugly number sequence 1 to

6 8 9 10 12 15 we can see that it can be

further divided into sub sequences of 2

3 and 5. and these subsequences

themselves are ugly numbers therefore we

take every ugly number from the above

sequences and choose the smallest ugly

number in every step to make sure that

we don't miss any number in between

let's head into our ides

okay so if we're looking for the nth

ugly number in a sequence we have to

write a function that will tell us if

our numbers

can be evenly divided in the first place

meaning no remainder

x modulo y equaling 0 gives us what we

need to prove that if there is a

remainder we'll just write it to return

our original number for our second step

before we can find the nth ugly number

position we have to check if it's an

ugly number we can do this recursively

by going back to our first function the

successive division

what this does is plug in the number

we're checking and successively or

successfully if you want to think of it

that way dividing it by two three and

five if our number is successfully

divided by two three and five with no

remainder we have an ugly number on our

hands let's check a six

it's true six is ugly

okay so we have what we need to find our

nth number six is fairly easy since

every number before it is ugly so the

sixth ugly number is

six but what's the 15th number well

let's write a program to find out

essentially what we're doing is merely

creating a counter coupled with

recursively calling back our ugly check

function we'll create two variables one

for i to iterate up our sequence and one

for our counter and then we just

incorporate a while loop to say as long

as our number is greater than our

counter we'll add one to i heading up

our number sequence and if that number

within our while loop is ugly add a 1 to

our counter

then we just return i

so what's our 15th ugly number

it's 24.

now if we run this program we have a lot

of running back in our recursive calls

for every number in our sequence

and that's a lot of running back with

600 steps or iterations that's a lot

considering with dp we can write a much

more efficient program that cuts all

those iterations by nearly 80 percent

let's check out the difference the first

major thing we can do is eliminate those

three functions within our original

program we won't need those remember

when i said we were going to implement a

table to store information

well that's what we're doing then we'll

start our first position in that table

or in other words a dynamic programming

array to store the ugly numbers next set

pointers for our multiples of 2 3 and 5

and we can do this all in one line which

is pretty cool you'll see how this works

as we get deeper into the program here's

our multiples of two three and five in

their own variables after that we loop

from one to n at each step and we find

the minimum ugly number out of three

multiples and store it in our dp array

we're also increasing the pointer for

the number whose multiple has been added

and replace it by its next multiple so

suppose a multiple of two has been added

to the array we increase twos pointer by

one and multiply the previous multiple

to two to get the next multiple and so

on and so forth for other numbers as

well and this way starting from the base

case we generate all the ugly numbers

and then apply dynamic programming to

store the numbers in the array until we

get the nth ugly number

this approach is way better than the

previous one because instead of checking

all the numbers for being ugly we just

used the previous ugly numbers to

generate new ones

watch how this runs through the program

this really is the better way

very efficient now let's put this

knowledge to good use on a very famous

problem called the traveling salesman

problem

if you haven't heard of this yet you

will

best to get it in your brain bucket now

rather than later

the traveling salesman problem is one of

the most intensely studied and famous

problems in computational mathematics

with its main main really main focus on

optimization

so much research has gone into this

challenge of finding the cheapest

shortest fastest route in the visitation

of locations by this dashingly handsome

or beautiful salesperson

this person starts in their home city

travels around and returns to their

starting point most easily expressed in

a graph

the general form of the tsp traveling

salesperson appears to have been first

studied by mathematicians during the

1930s in vienna and at harvard notably

by karl menger

menger defines the problem considers the

brute force algorithm and observes the

non-optimality of the nearest neighbor

heuristic using the mailman as an

example now the order in which the

person travels isn't cared about as long

as he visits each one

once during his trip

and finishes where he was at the first

in the most cost effective way but

there's a catch

maybe the salesman needs to take a plane

to one of the cities and this will cost

him a pretty penny

maybe he can walk to one of his cities

therefore making it a free trip the cost

associated with the trip describes how

difficult it is to to traverse the

journey or graph for us computer

scientists whether that's money or time

or distance required to complete the

traversal the salesman ideally wants to

keep every cost as low as possible he

wants the cheapest fastest shortest

route wouldn't you the easiest and most

expensive solution

is to simply try all the permutations

and then see which one is cheapest using

brute force the problem with this though

is that for n cities you have n minus 1

factorial possibilities this means that

just for 10 cities there are over 180

000 combinations to try since the start

city is defined there could be

permutations on the remaining nine

and at 20 cities this just becomes

simply infeasible

this is why four is often used in the

majority of problems as you'll quite

often see in the 4x4 matrices keeping it

small makes it much more manageable and

understandable which is good

the awesome news for us is that the

problem is solvable but keep in mind

that there can also be solutions to the

problems that are approximations or

commonly called heuristic algorithms if

that's what is being asked for according

to wikipedia even though the problem is

computationally difficult many

heuristics and exact algorithms are

known so that some instances with tens

of thousands of cities can be solved and

even problems with millions of cities

can be approximated within a fraction of

one percent

wow

so let's get to the nitty gritty and

look at our steps number one consider

city one as the starting and ending

point

number two generate all n

one factorial permutations of cities

three calculate cost of every

permutation and keep track of minimum

cost permutation four return the

permutation with minimum cost

it's not hard to see how this could get

computationally expensive in a very

short amount of time so we will move

into the optimize method right away for

our savings

and our sanity

the goal is to break this problem down

into the smallest pieces possible and

then solve solve solve

let's get into it

because permutations are a big part of

the tsp problem we're going to import

intertools permutations to make our

lives a whole lot easier

it's totally okay for this one

we'll implement a variable that we'll

call v

short for vertices and vertices are the

dots on our graph think of our sales

person traveling to four cities on our

graph a single dot is called a vertex

on our graph and they link or connect to

one another now we can move on to

implementing our function call here with

two parameters our graph that will be a

four by four matrix and s is going to be

our counter that will need to calculate

the sum of our path i'll explain this a

little bit deeper at the end of the

program

we'll create an empty list for our

vertices to be stored in and we'll call

this vertex just to make it plain this

is going to tell us the best route for

our rows in the graph again this is

going to be come real clear at the end

of the program so hold on to yourselves

then we iterate over our range of v and

if the index does not equal us which is

zero we append it to our empty vertex

list this is going to show us where our

salesperson is traveling within the

graph but here's a little catch that we

need to be aware of check out our graph

see those trails of zeros

we have no need to calculate those so

when we append our indices to our vertex

list we're only producing three integers

one two and three

next we create another empty list that

we'll call min path

we need to know the best route so this

will store our lowest combination of

vertices within the graph

then we create a variable that we'll

call next permutation and we'll

implement the inter tools permutations

module here to give us all the

combinations of the vertices for one two

and three

since we have three numbers we know that

the amount of permutations will be six

then we iterate over all those

combinations and create a counter that

we'll call current pathway we'll

reassign our s to a variable we'll call

k and k will represent the row in our

graph

and as we iterate over our indices

within our permutations by using j

this becomes our columns i know this

sounds i know this all sounds really

confusing but stay with me when we add

the k and j of our graph to our current

pathway look what prints out

this is calculating all the paths within

our graph and adding them together

pretty cool isn't it

next we reassign k to j and then use an

addition operator to add the sums to our

graph to the current pathway append them

to our min pass hold them in a variable

called x that's sorted

and simply return the very first number

in our sorted list which is our lowest

number

our shortest route on the graph to see

how this travels

on the graph watch this

this is how it is adding up within our

program

hopefully you can see and understand how

this is all working it sounds so much

more complicated than it really is but

it's only math

we're simply calculating the paths of a

matrix by pulling out all the

combinations or permutations of the

indices and returning the lowest number

or lowest sum of the path it's not too

bad my recommendation to you would be to

create your own graph and then run it in

the program once you do that

map that course on paper or even

whiteboard it to see it visually

for our very last lesson today and not

only for dp but

all together

i'm so sad to leave you

so let's make this session a great one

what we're doing for our mindbreaker

sounds really intimidating but i can

assure you that once we break this down

you're gonna say wow that wasn't even

that bad and then you can brag to all

your friends and then they'll shun you

for your genius

but it'll be totally worth it i i mean

because who needs friends am i right

now your brain won't break too much if

we take each word out and examine it

first we have palindromic which is

religious palindrome matrix we'll keep

it low and slow with a 3x3

and then just paths which we're only

going to go right and down only

since we worked on a matrix in our last

problem this should be old hat hopefully

by now you have a decent handle on

breaking down the problem into smaller

ones called sub problems and solving

them little by little in order to solve

the whole

tips for getting started on the

palindromic matrix path problem would be

to first know how to check whether a

string is a palindrome or not in the

first place

you can use python's python has a little

pip install program called

palindrome that

checks whether a string is pendulum or

not but i do highly recommend you learn

how to do this on your own

because it will help with this problem

the second tip would be knowing how to

traverse a matrix going right and down

only

we're not going to get too crazy for our

final lesson and just have some fun with

it by printing our palindrome strings

we're not going to return counts or find

the longest because that might make some

of us cry

and there will be no crying today any

tears that gets shed must be happy tears

only whether that's joy at finishing

this course or not seeing my face

anymore

tears of accomplishment are fine but no

tears of sadness okay to keep it simple

we'll start at ground zero and start

with an empty three by three matrix and

find all the unique paths

what's really cool about this is that

it's actually just a math problem

what we'll do is create in our three by

three by starting with a row of ones

that we will build up from

our bottom row and far right column will

be ones and when we add their adjacent

numbers together and build up we get a

row of ones one one one at the bottom

and then three two one in the middle and

then six three one at the top

if we return row zero

that will give us six and that tells us

that we have six unique paths in our

matrix

this will come in handy later when we

implement this principle into our matrix

of strings

we can know that we will have six

strings returned to us and all we need

to do is check whether they're

palindromes and print those out

so we're not going to talk much if at

all about the naive approach or brute

force or whatever the difference is in

optimization no

let's part ways on a good note and have

some fun with this

i mean we walked through 14 other

problems with multiple methods in a

relatively short amount of time

and here we are breaking our minds on

palindromic matrix paths so yeah let's

get straight to the optimum let's code

the paths problem first shall we we'll

start by creating our function with two

parameters our row and our column we

already know it's going to be a three by

three then we create our dynamic

programming array of all ones which you

can envision as our bottom row of ones

next we iterate over our rows in a for

loop and then create another dp array

which can be envisioned as an

accumulative next row up in our matrix

our original for loop is going to

iterate three times 0 1 2

for each row in our matrix in a second

for loop we now want to iterate from the

bottom up and reassign our new row

elements by adding one element to the

one next to it going left

we do this twice

while in our final iteration it has

accumulated each row to end with six

three one we then reassign our original

row as the final row and return the very

first element which will be six

our answer to the number of paths now

this isn't super necessary but it will

give you an idea of the path traversal

within a three by three matrix and will

give us a good launch point to moving up

to a matrix consisting of letters we'll

keep it in the realm of just two letters

a

and b and we'll write a very quick

little program to see how many

palindromes of a and b there are

let's import intertools to help us and

then create our rows and columns of

letters

a neat little feature of inter tools is

product with an asterisk on our matrix

this will print out every combination of

letters in our matrix totaling 27. yes

there will be duplicates here but it's

nothing that python's set can't handle

let's create an empty list called p1 and

then iterate over our list of 27 tuples

join them together to make it easy on

the eyes and append them to our empty

list

then all that's left to do is check

which ones are palindromes by making a

new function plugging in our p1 create

another empty list p2 and then iterating

over a non-duplicate set of p1

incorporate an if statement is it a

palindrome yes put it in our empty list

then return it

okay so far we've traversed a matrix in

order to count the number of paths in it

we created a three by three matrix of

just two letters found every possible

combination and created a function to

pull the palindromes from it

we are doing fabulously if you know how

to count the paths of a matrix and find

the palindromes in a matrix you're

totally equipped to take on the

palindromic paths problem

let's get cracking

so

before we start i have to tell you

there's going to be a little bit of a

difference here for this program

it's the order of traversal we're going

to go from the top and right to the

bottom not the bottom up and left to the

top

so keep that in mind as we march forward

we're going to throw in our little

palindrome checker up here first before

we print them out at the end of our

function hopefully by now you're

familiar with that

next we're going to create our main

function with several parameters

a string that will be empty but we're

just going to call it string here

a for our array

our pointers that we'll call i and j

our row and column that we'll call m and

n

now our conditional if statements are

going to traverse the paths of our array

we minus one from m and n because as you

know this is python and we only need to

iterate from indices starting at zero

then one then two for our total size of

rows and columns three by three see the

i plus one and the j plus one within our

recursive callbacks look at these slides

to see the paths

you could also print i and j within the

conditional statements on your own to

see it as well but when we add one to

our pointers we're moving to the next

element then

once the traversal is completed it hits

our else statement and that hits up our

palindrome checker if it's true it'll

print then it moves on to do the next

reversal over and over until it's done

and there you have it

four palindrome strings in our matrix

yay

[Applause]

so what did we learn on our final lesson

we learned about dynamic programming

which is finding the optimal solution to

sub problems that can be used to find

the optimal solutions to the overall

problem

we saw this in our lesson with ugly

numbers the traveling salesman problem

and palindromic matrix paths

in the easy lesson we learned about ugly

numbers what they are and how to write a

program to find the nth ugly number in

the sequence avoiding brute force and

recursion we were able to implement

dynamic programming to optimize the

solution from over one thousand

iterations down to just a little over

one

we looked at the very famous traveling

salesman problem with its main focus in

optimizing a solution to finding the

most efficient route a solution used

across the board for so many companies

in order to keep operational costs low

and lastly we broke down the mind

breaker lesson by studying unique paths

in a matrix

string palindromes within a matrix and

combining those two problems together to

find all the palindromic paths within a

matrix

are you excited i know that i am we

covered so much ground in this course

it's crazy

hopefully you took lots of notes so you

can practice later you should be coding

something every day even if it's just

for a couple minutes 10 to 15. since

every little bit helps now listen to

your mother thank you so much for

joining me today and if you made it all

the way here kudos to you i wish you all

the best in your programming journey bye