[MUSIC PLAYING]
DAVID J. MALAN: So odds are you use a computer most every day,
but you might not necessarily think of yourself as a computer person.
Something goes wrong, you don't necessarily know how to fix it.
And if you actually want to solve some problem using technology,
the whole world of technology, and computing,
and algorithms and all of that might seem all quite foreign.
So much so that you can't necessarily feel like you're
making an informed decision.
Well, that's just because the technology that we all use around us
is kind of working at this level.
Whereby there's so many people who have come
before us that have invented this level, and this level, and this level,
and this level.
And so what we'll do here is try to start from that ground level up.
And indeed, at the base of all computing, do we really have,
as we'll see, just zeros and ones?
And that's probably something you know, at least, generally speaking.
But it turns out once you go have zeros and ones,
can you very quickly go to text?
Can you go to graphics?
Can you go to videos?
Can you go to spreadsheets?
Can you go to so much more?
But again, that's the world we now inhabit.
But let's now build ourselves up to that point,
so that you actually begin to look around yourselves
and realize, oh, I understand now, what's going on.
And to do that, let's consider this, computational thinking, which really
refers to thinking computationally.
Thinking more methodically, thinking more carefully,
and somehow framing all problems in the world as a sequence of steps.
And those steps are quite simply, input comes in, and output goes out.
And what's in between those inputs and outputs
we'll eventually describe as something called algorithms,
but for now, let's just consider it to be this black box.
We don't know, we don't care what's going on inside there.
All that we care is that we get back a correct output or a solution
to a problem.
But how do we go about representing those inputs and outputs?
If our problem at hand is to analyze a whole bunch of financial data,
a whole bunch of numbers in a spreadsheet,
how can we actually sum all of those numbers
together, or perform some other arithmetic calculation on them?
If we instead have a really big document, a contract of some sort.
And we want to make sure that it is properly spellchecked?
Well, the input there isn't numbers, but is
all of the words and letters in that document.
And the output, we hope, is a document without all
of those little red squigglies.
We want a correctly spelled document.
So those are just some of the problems that you might experience most any day,
but underneath the hood, there's quite a bit of complexity going on.
But that complexity, it turns out, is just
the result of layering, pretty simple ideas, one on top of another.
But in order to get to that point in the discussion,
we need to somehow represent these inputs, these numbers, these letters,
whatever it is that we have at hand.
And to do that, we're going to use binary.
Now odds are you've probably heard that computers indeed only speak
or understand zeros and ones.
But how can that possibly be, when they can do so much today, whether they're
on our desktops, or laptops or even our mobile phones in our pockets,
if all you have at the end of the day is zeros and ones?
How could you possibly count to two, let alone three, or four, or four billion,
let alone representing any number of other types of media
that computers today understand?
Well, odds are you recognize in this word here,
this prefix, bi, bi meaning two, and indeed that hints at the fact
that you only have two letters in your alphabet, so to speak.
Two digits, zero and one.
Now we humans have typically grown up using the decimal system,
dec meaning 10.
And of course we have 0,1, 2,3, 4,5, 6, 7, 8, 9, 10 possible digits
that we can permute and arrange to count even higher than that.
But with just zeros and ones, how do you get to two, how do you get to three?
Well, you're thinking still in base 10, so to speak, base 10 being decimal.
But we want to think now in base 2, so to speak, binary,
where we have just two of these inputs.
And let me propose that if you're like me,
you probably grew up understanding numbers as really just this pattern
of symbols like this, where each symbol was in a column of sorts,
a placeholder, if you will.
And this, if you recall, was generally called the ones place, and then
the tens place, and then with the hundreds place.
And so at the moment, what we really just have on the screen
is a pattern of symbols, a pattern of digits, one, two, three.
But why is this pattern of symbols, one, two, three,
the number you and I think of immediately as 123?
Well, that's because by way of these columns or places do we implicitly,
in our own mind, quickly do a bit of arithmetic?
Of course, if we have one, two, three, that's really like 100 times 1 plus
10 times 2 plus 1 times 3, and of course that gives us 123.
So it turns out that the binary system is actually fundamentally the same.
So if you get decimal, if you've been counting like that since grade school,
you are good to go now with binary, because in fact
and arguably, binary is even simpler, because you don't even
have to keep track of so many digits.
In fact, if we consider this pattern of symbols,
this of course, in our human world, would generally
be immediately recognized by us humans as the number zero.
Of course, it's a little silly to have those
left most zeros, those leading zeros, so to speak,
because they don't really add much mathematically.
But that's OK.
You can put as many zeros to the left of a number in our real world
and it doesn't really affect the value.
Now instead of using one and 10 and 100, let me just use one, two, and four.
Now why those values?
Well before, 1, 10, 100, and again a thousand, 10,000, 100,000, and so
forth.
Those are powers of 10, 10 to the zero is 1, 10 to the one is 10,
10 to the two is 100, and so forth.
So what pattern do you perhaps see here?
These aren't powers of 10 anymore, 1, 2, 4, an 8, 16, 32.
Now you really hear the pattern?
These are instead powers of two.
Two to the zero is one, two to the one is two, two to the two
is four, and so forth.
So that's the only change.
By using base 10, just two digits, zero and one,
can we still count using the same arithmetic system.
So for instance, this of course is still the number zero,
because we have zero fours, and zero twos, and zero ones.
But if we instead change this pattern to 0 0 1,
well what value does this now represent?
In the world of decimal, this would represent the number one.
And that's true in the world of binary, because we have 4 times 0 plus 2 times
0 plus 1 times 1, which of course is just 1.
So how do we get to two?
You might be inclined to just change this zero to a one,
but that would be incorrect.
That's like carrying the one, but then not
wrapping around on the rightmost digit.
Rather, if we want to represent the number two,
we need to come up with a pattern that represents that.
So let me propose instead that we don't just change that zero to a one,
but we also change that one to zero.
Because now we have 4 times 0 plus 2 times
1 plus times 0 which of course is the number we know as 2.
And now you can perhaps grok the pattern at hand.
If we want to count up to the number we know as three,
let's just change that rightmost one to a one.
So that's four times 0 plus 2 times 1 plus 1 times 1.
That of course is three.
Four is perhaps jumping out at you now.
We just change that zero to a one, and then the other two digits to zeros.
And now if we want to count up to five, or to six, or to seven, or to--
dang it, what's supposed to come next?
It would seem that in the binary system, we can only count as high as--
what, seven?
And that's a little strange, because of course, we started at zero,
we counted here up to seven, but surely computers can count higher than that.
And that's fine, just need another digit.
Much like if we wanted to count from 999, in our human decimal world,
up to a thousand, which has four digits.
Similarly here, do we need another digit?
We need an eights place, and so we could represent
the number we know as eight with one, zero, zero, zero.
But the key here is that we needed another digit.
To put it more mechanically, we needed more hardware.
We needed another physical place, at least in this depiction,
to actually store that one.
And now we begin already to hint at a connection between this digital world
of zeros and ones, and the hardware world in which it's typically
incarnated.
And by that, I mean, we need to decide, ultimately,
how and where to store patterns like this.
And you know the nice thing about binary only having two values,
is that that effectively, means you have just two states.
And you might think of these two states as something being on,
or something being off.
And maybe we might think of something being on as a one, and something being
off as a zero.
And so just with two states, can we have these two extremes.
And maybe it's not on or off, maybe it's true, or false, or black, or white,
or red, or blue.
It doesn't even matter what we call the two states,
the key is that we have two of them.
And you know In the physical world, the simplest thing
I can think of to turn on and off, is perhaps something like this.
Just a light bulb.
This here's a little desk lamp, and in fact,
if I go ahead and clip this on here, and let
me plug this into an electrical outlet so that I'm
getting some additional input if you will, electricity or electrons,
this lamp right now I claim is representing the number zero.
It's completely off, and I'm just going to agree, to agree with you, that this
shall represent zero.
But you know what, if I want to represent a one in the physical world,
just going to turn it on.
And so now we have something that's on, or true, we'll call that a one.
Zero, one.
Of course, we're not really counting all that high just yet.
If I want to count higher than zero to one, to something higher,
I'm going to need another light bulb.
I'm going to need more hardware, more memory, if you will,
in the world of computers.
So let's actually now give myself a second zero or one.
Plug this thing in here.
Give it some electricity as well, which again, is really
one of the few physical resources we have in a computer,
whether it's coming from a power cord or a battery.
And so now, if I want to represent the number zero, here we are.
But I'm doing it really with zero zero.
This now is going to be one, this now is going to be two, and this, of course,
is going to be three.
And let's not stop there.
Why have two desk lamps when you can have three desk lamps.
If we instead make a little more room for here, and we
want to instead now count not just as high as three.
Let me go ahead and plug-in this third and final desk lamp,
and go ahead and turn this on.
And of course, we instantly go from three to--
not quite instantly because you have the hands--
to the number four.
And so forth.
So what's now the connection between the digital
and the sort of analog world of light bulbs,
and the physical world of computers?
Well inside of a computer is a whole bunch of tiny, tiny little bulbs,
so to speak.
In fact, you can think of what's inside of your computer is exactly this.
So many little light bulbs that are either on or off.
And thanks to those little light bulbs, can your computer represent numbers?
Zero, one, two, three, four, as many--
as high of a number as it wants so long as it has enough little lights.
Now of course I'm oversimplifying.
It's not actually lights that you have inside of your computer, rather,
you have just the switches.
These switches being called in the computing world transistors,
as you may have heard.
Indeed, one of the things that companies like Intel are increasingly doing,
is packing more and more transistors into their CPUs, central processing
units.
The brains of computers.
And with those additional transistors, can they store even more values,
or equivalently can they count even higher?
And so now, even though we began the discussion down here
with just zeros and ones in the abstract,
now we've made it a little more physical, in so far
as we can represent those zeros and ones with something physical,
drawing electricity to power these kinds of light bulbs.
Of course, we can now miniaturize that and consider
these light bulbs to really be, what we'll call in the computer world,
transistors.
But at the end of the day, even though we
have now a way of representing information,
the zeros and ones, even though we have a way now
of representing even bigger numbers by using even more transistors inside
of our computer, we're still talking only about numbers.
And with my computer I want to do more than just use something
like Microsoft Excel.
I want to do more than just do math on my data.
I want to actually store things like letters, and words.
Let alone, colors, and images, and videos, and more.
So we need to abstract higher.
So again zeros and ones, we know how to represent it, now let's
build on top of the zeros and ones and start to represent
something more interesting.
Something more alphabetical, if you will.
And so this gives us ASCII, the American Standard
Code for Information Interchange, or more modernly
referred to as a superset of Unicode.
Turns out, human some time ago, decided on a mapping
between letters and numbers.
Somewhat arbitrarily, but in a way that's
convenient when it comes to actually programming, things like this.
And this is to say that humans time ago decided,
you know what, we are going to represent the letter A using the decimal number
65.
Now what does that mean?
Well, this means that if your computer is to store the letter A, or the letter
B, or the letter C, it simply has to store ultimately
the number 65, or 66, or 67.
What does it mean to store number like 65, 66, or 67?
Well that just means to store some pattern of bits, some pattern of bulbs
as we did a moment ago, and turn them on and off in such a way
that you're representing in binary, the decimal number 65, 66, 67.
So if you think about something like Microsoft Word or Google documents,
or any program in which you type text, what you're really
doing by typing on the keyboard, is sending some pattern of signals
that's telling the computer to store not just the letter ABC per se,
but really to store the number 65 or 66 or 67,
or really, to store the pattern of zeros and ones
that ultimately represent those same values.
So again, this spirit of layering binary now
becomes ASCII, or Unicode, something higher still.
And so with this can we represent messages?
For instance, if I wanted to represent a message like this, a familiar message,
there say, I might store these three values.
72, 73, and 33.
Now what is that?
Well turns out if I look back at this pattern here, where 65 is A,
and in tens 72 is H, and 73 is I, well what are we really representing here?
It would seem we're representing the pattern H I,
and then you wouldn't know this from that chart,
but if you look at another online reference
you'll see a 33 is an exclamation point.
So now we have the word Hi or the exclamation Hi.
But that's in the context of a text editor, or word processor like word.
Suppose, instead, you were using not a text
editor, but something like Photoshop, or MS Paint,
or some other graphical program.
Well instead, you might have that same pattern of numbers,
or really bits at the end of the day, but in this context,
something like Photoshop, these numbers are
meant to be values between 0 and 255.
Turns out, long story short, that if you have eight zeros or ones,
where zero or one, you know what, let's just
start calling them by their formal name bits for binary digits.
If you have eight bits, you can count from 0 all the way up to 255.
And the quick math there is 2 to the 8, means
you have 256 possible permutations of zeros and ones.
So therefore, if you start counting at zero you can count as high up as 255.
And so in the context of a graphics program, you can think of 72 in so far
as it's not quite halfway between zero and 255, it's a medium amount of red
and a medium amount of green.
Not too much blue.
Where zero means none of this color, and 255 means a lot of this color.
So if you had a pattern of bits, and in-turn numbers, stored inside
of your computer for the purposes of a graphics program,
it's really like telling the computer give me a medium amount of red,
give me a minimum amount of green and just a little bit of blue,
and combine those like paint, or like frequencies of light.,
until you get the summation of those, really,
are the combination really of those, which
is this murky shade here of yellow.
So again, using the same patterns of bits,
can we represent either letters and words and paragraphs,
or in another context all together, could that same pattern of bits
still represent numbers, but be interpreted not as letters and words
and paragraphs, but as colors.
If you treat each trio of 8 bits as representing some amount of red,
some amount of green, some amount of blue,
otherwise known as, if you've heard the term, RGB, for red, green, blue.
So this principle of starting simple, and gradually making
things more and more and more complicated,
is really a principle of abstraction.
Because at this point in the story, when we're
talking about words and paragraphs and essays and documents,
or in another context we're talking about images, or maybe even
movies and more, we no longer really need to care about,
or even need to know about, or understand,
what's underneath the hood those zeros and ones,
because we've abstracted away from that lower level.
And this principle of abstraction, layering idea on idea
on idea on idea such that you no longer need
to worry about how the lower level ideas are implemented
is nice, because it allows us humans to focus only
on the problems we really care about, which
in theory are those top most problems.
The ones that are immediately at hand, that we're building solutions to,
on the shoulders of, computer scientists, and engineers,
and just colleagues that have come before us.
And we don't have to get lost in the weeds,
so to speak, of the earlier complexity.
And so this is a powerful problem solving technique,
and indeed it's a principle that we'll see applied ultimately
in the world of programming as well.
Now let's try something.
Speaking of abstractions, let me encourage you
at this point to take out a piece of paper if you have one there.
And surely, if you don't have one right there,
surely you could pause this video and actually go dig one up.
Grab a pencil too, or pen.
Yeah, Ill wait.
Now you could just pause this, I can't wait all that long.
Let's assume at this point in the story you've got that piece of paper,
and a pencil or a pen, and let's play a little game.
I of course can't really see what you're doing.
But I'm going to hope that you either do or don't do what I do,
because either way it will be instructive.
And I'm going to leverage some abstractions here,
for better or for worse.
I want you to go ahead, and please don't be one of those people like me
who's just following along pretending like all right,
I have the piece of paper, I have the pencil, this is what I would be doing.
Actually do this.
This will be kind of fun for one of us in the end, if this works out.
Go ahead now, on that piece of paper, with your pen or pencil, and go ahead
and just draw a circle.
OK.
All right.
And below that circle draw a square.
All set.
All right, and below that square draw a triangle.
Now odds are, you know exactly what I meant when I said draw a circle,
and perhaps you did a little something like this.
Or a little more perfect than my circle there.
And then beneath that I said draw a square.
And you just intuitively know what a square
is, so you might have drawn a square like this.
And then the third thing I said was draw a triangle.
And so you might have drawn a triangle that looks like this.
But immediately, ant that looks curiously like part of a jack-o-lantern
now.
But curiously, there's some ambiguity there.
Right, a circle is kind of a circle, but I didn't specify the radius
or diameter, so maybe yours is bigger, maybe yours a smaller,
maybe it's over here on your paper, maybe it's over there on your paper.
So already, these abstractions are useful in that you immediately
knew what to draw, but you didn't really know how to draw it.
Similarly, this square, I don't know why I drew it a little smaller here,
but it's indeed smaller, and it's barely a square.
But it's meant to be a square.
But there's a gap between it and the circle, but I didn't really specify.
So there, too, the abstraction is valuable
in that you immediately knew how to draw a square, but not where to draw it.
Or in what size to draw it.
And lastly, the triangle, perhaps the most,
the juiciest opportunity for ambiguity, I
didn't tell you how to orient that triangle.
Maybe you, instead, did something a little different,
where your triangle wasn't drawn like that,
with the pointy part at the bottom.
Maybe you, instead, did something like this.
Or maybe it was somewhere else on the paper altogether.
So now at this point in the story, you could have followed my instructions
exactly as I described, but we could still somehow come up
with different solutions.
And in fact, if I can now spoil what the actual task at hand was,
this was the picture I was describing.
This picture here has a circle, below which
is a square, below which is a triangle.
But that leaves out some key details.
And the curious thing here, is that even though abstraction is a useful
mechanism, once you start to move away from those implementation details,
if you will, you very quickly realize that I don't really know what
you're telling me to do, necessarily.
And the challenge is, that computers, as complicated or intimidating as they
might seem to you, they're really not all that bright.
Right.
They can only do what they are explicitly told to do.
And so if you, the human, or you eventually perhaps,
the programmer, don't actually specify absolutely
precisely what you want the computer--
or the human in this case-- to do, you might not
get the output, or the correctness of a solution, that you're hoping for.
In fact, let's try one other, and let's see
what the other extreme might feel like.
So on that same piece of paper, maybe on the flip side
or another sheet of paper, let's play this game just one more time.
And this one's going to be a little harder.
I'm going to try to tell you exactly what to do.
So lesson learned, that was too abstract.
Let's now drill down on the implementation details.
OK.
So let me think like a computer might, or I think a computer might,
go ahead and put your pen or pencil down on the piece of paper
toward the top middle of the page.
Now from that point, move say Southwest to halfway down the piece of paper.
So really at a 45 degree downward angle.
And then go ahead and move south east, or a different 45 degree angle,
toward the bottom of the page.
So you've kind of sort of made the left half of a triangle,
or not really that, 2/3 of a triangle, two sides of a triangle.
But stay where you are.
At this point in the story your pen is probably below your original dot.
But you've drawn two lines that form an angle.
From where your pen now is, draw a line Northeast,
or in an upward and to the right 45 degree
angle, to the same height as your previous line.
And then double back and go say Northwest, or a different 45 degree
angle, still back to the original point.
And at this point in the story, you have really either nothing at all,
or you have a diamond.
And therein lies a curiosity too.
I could have just said diamond, draw a diamond, or draw a kite.
Which would be an equivalent shape.
But that too lends itself to same ambiguity, so I drill down deeper.
But my God, it's taken us just a minute or more just to get to this point.
And we're not done yet.
We're not drawing a diamond or a kite.
Now you have a diamond or a kite, with a top vertex or point, a bottom one,
a left and a right.
Move your pen to the left one, and draw a vertical line down.
Now go back to the diamond or the kite, and on the right vertex or point,
draw similarly a vertical line down.
And now, in that original diameter kite, at the bottom most vertex or point,
draw a vertical line down.
And now, if you followed along with these very precise machine
instructions, you've got three vertical lines that are just kind of dangling.
I don't even want to mislead you with any hand gestures, three vertical lines
that are dangling, go ahead and draw two lines that
connect the ends of those three lines.
Oh my God.
Like it's so complex to do what we just did, and I would put money on the fact
that you did not draw correctly, what I was trying to get you to draw,
which is just a cube.
Right, a cube is a wonderfully simple abstraction.
A shape with which you and I are probably long familiar,
and it's so easy to say draw a cube.
But as we saw before with the circle and the square and the triangle,
just saying draw a cube is ambiguous.
At what angle should it be oriented?
How big should it be?
Where on the piece of paper should it be?
And so I was trying to be so much more precise
this time by having you put your pen down at the top,
go down to the southwest and draw this line,
then go down to the southernmost point here, then another Northeasternly line,
and then a north westerly line.
And that just got us to the kite.
But the takeaway here, is that when it comes
to making a computer do what you want to do,
you can't just speak these abstractions.
You actually have to implement them, or program them, or code them
at least once.
In fact, some of the earliest graphical programs in the world of computing,
were kind of as low level as this.
There was an old programming language called logo,
in fact, that allowed you to program by moving a cursor,
like a turtle of sorts, up and down and left and right on the screen.
And putting either down or up a marker of sorts,
and that you could draw shapes like this by just moving around on the screen.
But to draw things like this clearly, as in our verbal example here,
you have to be so darn precise.
And it just gets so tedious so quickly.
It certainly would take all of the fun out of using a computer,
or all the fun out of using programming a computer, if you
had to do this every time.
But that's where there's an ingredient here to be leveraged.
One of the things that a computer scientist, and a programmer,
and engineers, more generally, very often do,
is they absolutely implement these kinds of low level details once.
They go through those very methodical, if mundane, steps
of getting something just right.
And then they save the instructions they wrote.
They save the programs they wrote, if you will,
so that they can reuse them later.
And the fancy words for these things will eventually see
are called libraries.
Or functions.
Or other names still.
So once one human in the world has implemented a program, if you will,
with which to draw a cube, similarly can we stand on his or her shoulders
and reuse that same routine.
And hopefully, they were clever enough to allow us to parameterize it.
To customize it, by maybe changing the angle and the size and the depth
and so forth.
So it doesn't just have to be that one cube.
And so here we have a wonderfully powerful problem solving technique.
Abstraction.
Which allows us to say what we mean, and the rest of the humans in the room
just immediately understand-- at least after some instruction-- what
it is we're talking about.
But with computers being these very little literal devices,
we can only talk at those levels of abstraction
once we've actually built up software, implemented solutions to get us
to that point in the conversation.
And this is why, at first glance, using a phone in your pocket
or a computer on your desk might actually seem super, super complicated.
There's so many moving parts.
And absolutely there are.
Windows and Mac OS are literally the result of millions
of lines of programming code these days, having been written over the years.
And so of course it's to be expected that you might be a little daunted
or overwhelmed by the apparent complexity.
But one of the goals here for this lesson,
is to really help you appreciate that beneath all that complexity,
is a simpler idea.
And then an even simpler idea.
And then a very simple idea, and so forth.
And so once you sort of bottom out and understand those first principles,
zeros and ones, binary on top of which might be Ascii or Unicode, on top
of which might be some other encoding still,
can you resume the current conversation and understand
that what might have looked completely complex at first glance,
is really just the result of assembling, if you will,
a whole bunch of pretty simple puzzle pieces.
So at this point in the story, we now have a way of representing information.
But now let's just stipulate.
We know how to represent information.
At the end of the day, Yeah it's binary, And at the end of the day
you can think about it as decimal, and maybe you're using Ascii in Unicode,
or maybe you're using graphics, or whatever
is going on underneath the hood.
All we need to know and care about now is that you can do it.
And we don't really have to think too much more at that level.
Now we can resume our look at the overarching model
at hand, which is problem solving.
We now have a way to represent our inputs and outputs.
What then is inside that black box?
Well the buzzword du jour these days is perhaps
algorithms, where even if you don't necessarily know what it is
or how to make one, or how to use one in the context of a computer program,
algorithms seem to be increasingly the solution to all of our problems.
Well an algorithm isn't all that complex, fancy
though the word might sound, it's really just step
by step instructions for solving a problem.
And those steps can be in English, or they can be in something we'll call
pseudo code, sort of code like but it's not really an actual language,
or can be in Java, or C, or c++, or JavaScript,
or any number of other programming languages.
Algorithms are, again, just sets of instructions for solving a problem.
Well what's one such problem we might want to solve?
Well in the real world we might have--
the real old world shall we say--
we might have a problem that once upon a time looked like this.
A phone book with a whole lot of pieces of paper inside of it,
on which were a whole bunch of names and a whole bunch of telephone numbers.
And the phone book was alphabetized from A to Z typically.
And maybe there were some other sections like the yellow pages, or apparently
the red pages, whatever this is here.
But we'll just assume that these are the white pages,
so to speak, with just a whole bunch of humans names and numbers.
So suppose I want to find one such human,
a human whom we'll call Mike Smith.
How do you go about finding Mike Smith in this phone book?
Well I could, somewhat stupidly, but arguably quite correctly,
start at the beginning of the book.
And I see here instructions for calling 911.
If I move on to page two, I now see someone's names and numbers.
But these people's names all start with A.
And so I continue going through the A section.
And the A section.
And then eventually I get to the B section.
And the B section.
And then the Cs, and the Ds and the Es and Fs and so forth.
And eventually, tediously but correctly, I'll get to the Ss in the phone book.
And if I see Mike Smith here, I can now pick up the phone and call Mike Smith.
Now no one out there, if you've been still use this technology,
is going to have looked up Mike Smith in that way.
You're going to fly through this phone book far faster than I, right,
you learned probably in grade school why count by ones
when you can count by twos?
So two, four, six, eight, ten, twelve.
Sounds faster, is faster.
It's going to get me to Mike faster, but is it going to get to me to Mike
correctly?
I'm going twice as fast by doing two pages at a time.
So I'm going to have flip half as many times.
But there's actually a bug, so to speak, a potential mistake here.
What is that?
Well in my cleverness to get to Mike's name
twice as fast, what if I go ever so slightly too
far, just because by bad luck, Mike is sandwiched between two of the pages
that I so cleverly was skimming two at a time?
Of course I'm looking at this side, maybe this side,
but maybe Mike is in the middle.
And so it turns out with that algorithm, that twosie approach,
am I going to have to have a little bit of a check at the end?
Such that if I hit like the T section, and see a name starting with T,
I better wait a minute, let me double back,
I might need to go back at most one page to make sure
that I didn't actually miss Mike Smith.
So at the end of the day, it's not correct
if you naively just do two pages at a time,
but it is correct if you do two pages at a time,
with one final reversal by a page, just to make sure
once you go past SMITH in the phone book,
that you didn't accidentally miss Mike just one page prior.
So it's still super fast, you just need that one little check.
But still, no one out there, is going to look up Mike Smith one page at
a time, two pages at a time.
You're going to open in the middle of the damn phone book,
look down and see, oh, not in the S section yet, I'm in the M section.
And so what you intuitively know how to do since growing up perhaps,
is that you know Mike is not in this half of the phone book.
Mike is clearly in this half of the phone book.
And so at this point you can figuratively-- but in our case
literally-- tear the problem in half.
Throw half of the problem away, and be left with,
we single use phone book, and half at that, but half the size of the problem.
So if we had 1,000 pages originally, now maybe I've got only 500 pages.
And so I can repeat this intuition.
Jump roughly to the middle.
Darn, I'm a little too far now, I'm in the T section.
Again, let's tear half the problem away, throw it away as well,
and now be left with a 250 page problem, which can now be 125 page problem.
Which is getting easier and easier and easier, until we repeat,
repeat, repeat, repeat.
Until theoretically, we're left with just one page in the end.
Maybe Mike's on it, maybe Mike's not, but if he's in the phone book,
he will be on this page.
So how efficient was that.
Well if that phone book had 1,000 pages, in my first algorithm
it might have taken me what, like 700 plus steps to get to Mike?
Or in the worst case 1,000 steps, right, it's alphabetical.
But maybe it's not Smith, maybe it's someone
with the last name that starts with Z. In the worst case in that first phone
book, maybe it would take me 1,000 steps maximally to find Mike Smith.
Pretty slow.
What about that second algorithm where I was going two pages at a time?
Well, with that algorithm, it's going to take me like 500 steps
maximally to find Mike Smith.
That's twice as fast, that's pretty good.
It's not nearly as amazing as the algorithm we settled on.
The intuitive one, arguably, where I divided and conquered.
I have the problem again and again and again.
Because if I start at 1,000, and I go to 500, then 250, then 125 and so forth,
rounding as needed, I actually get to one page much faster.
Put another way, how many times can you divide 1,000 in half
before you're left with just the number one?
Well if you do the math, either in one direction or the other,
you'll see that it's roughly ten.
In fact, if you want to pause even and grab a calculator, or a piece of paper,
or pencil, or just think through it in your head, you can start with 1,000
and go to 500, 125, and so forth.
And you'll eventually hit one, after just 10--
give or take, depending on how you round--
steps.
That's pretty powerful.
But not that big of a deal.
Right.
Ten still is like a lot of page turns.
But what about an even bigger problem.
The types of problems that Google, and Microsoft, and Facebook,
and Oracle, and really big companies deal with that
have lots and lots of data.
Suppose, for instance, that I'm searching through not a phone
book, but a database.
A big program that stores lots and lots of data.
And suppose the data that's being stored is still names and numbers.
How much time might it take to find someone like Mike Smith,
in a database that's got like 4 billion names and numbers in it somehow?
Well, four billion names and numbers.
Well if we use that first algorithm, might
take as many as four billion steps to find
Mike Smith in a really big database.
In a really big computer program, that's not too smart.
But if I instead use the twosie approach,
flipping through two database records at once,
maybe it's not 4 billion operations, maybe it's just 2 billion.
That's good.
That's half as many operations.
But what if I use this super clever intuition
that I kind of grew up with here, with that divide and conquer algorithm.
Well, I can start with 4 billion database records, go to 2 billion,
then 1 billion, then 500 million, 250 million, 125 million, and so forth.
I'm getting to one much, much faster.
In fact, I can only divide the number 4 billion
in half, roughly 32 times total.
Again, depending kind of sort of on how you round, but 32 times.
32 is so much smaller than 2 billion steps.
And 32 is certainly smaller than the original starting point,
4 billion steps.
So this is a really powerful problem solving technique
to divide and conquer.
And here, too, even though what computers might seem to be doing these
days is super complicated and sophisticated--
and it is in many ways--
but some of the ideas that those computers and the programmers who
program them are leveraging, are actually pretty familiar to us already.
Inside of this black box might not be something super fancy, but just
a clever adaptation of some of your grade school
human intuition to the context of a computer program.
Now it's one thing to talk about algorithms, especially
if we're just spit balling it verbally.
But computers, of course, need us to be more precise.
And they need us to state our thoughts more methodically.
So what does this mean?
Well, let me propose that we write some code, or really pseudocode,
for this same algorithm, where we're looking for someone like Mike
Smith in a phone book.
And it's pseudocode because it's not going to be Java, or c++,
or JavaScript, or anything else.
It's going to be English like syntax, that's kind of sort of like code.
And before long, we'll see some actual code as well.
But step zero, and just to be playful here,
I'm not going to start counting at one, I'm going to start counting at zero,
just because with any number of bits, or digits,
the lowest number that I could count with is of course zero.
Pick up phone book is the first thing that I did.
One, open to the middle of the phone book
is the second thing I did in that third and final algorithm.
Look at the names was the next thing I did, looking down at that phone book.
And then if Smith is among names, and notice,
this is semantically different from those first three steps,
because this expression starts with if.
So this is kind of like the proverbial fork in the road,
if Smith is among the names, let's do this.
What do we want to do?
Call Mike, that's great.
Otherwise, or else if Mike is earlier in the book,
let's go a different direction instead.
Let's instead open to the middle of the left half of the book, all right.
So that would be the left half that I threw away earlier,
and then go back to step two.
Because once I have opened to the middle of the left half of the book,
I don't have to actually dramatically tear it.
I now need to look for Mike's name again, as per step two.
Else if Mike is later in the book, I actually
want to open to the middle of the right half of the book,
and then go back to step two as well.
Now you might think that's everything to this program.
But there's actually a remaining step.
Indeed I've got room left on the screen for a remaining step or two,
but there are more than three possibilities.
One of them is that Mike is among the names, one of them
is that Mike is to the left, the other is that Mike is to the right.
What's the fourth?
If I don't consider the fourth, and indeed if in a program
I don't implement the fourth, my program might crash.
My computer might hang.
My computer might behave in an unpredictable way,
because if the programmer wasn't so precise as to anticipate
something that might happen, who knows what the computer might do.
And indeed, that's often why your own computer might
have a little spinning beachball, or icon,
or it might crash outright or freeze.
It's because something unanticipated happened.
So let's be precise.
There's a fourth and final scenario I can think of,
which perhaps on your mind too, just quit.
Because in the fourth scenario, if Mike's not among the names,
and he's not to the left, and he's not to the right in the book,
he must not be there.
And so let's avoid just hanging infinitely somehow or other,
by actually proactively deciding to quit.
But now let's tease apart what some of these terms actually are now.
So in yellow are some things that really just look like verbs, or actions.
And we're going to call those statements, or more specifically
functions, or procedures would be a reasonable synonym too.
And each of these yellow terms is really a call
to action for the computer to just do something unconditionally.
But sometimes, we want that computer to do something conditionally,
as evinced by these yellow terms now.
If, and else if, and else and else, kind of paints the picture of a four-way
fork in the road.
Where each of these branches, or conditions,
leads us to take different action.
And you'll see that I've indented lines four and six and seven and nine and ten
and twelve, because they are meant to happen
only if lines three and five and eight and eleven, or eleven,
actually applies.
So those indentation kind of captures the logic of this program.
Lastly, or second to last there is this.
This is what we call a little more fancily, a Boolean expression.
These yellow phrases here are kind of like questions.
There are either yes no, or true false, or one and zero
or any number of other binary terms.
But these are questions we're asking.
The answer to which is going to be true or false.
Smith is among the names, true or false?
Smith is earlier in the book, true or false?
Smith is later in the book, true or false?
And so these Boolean expressions, named after someone
who the last name of Bool, long ago, is a way
of having conditions take you in a different direction based
on whether something is true or false.
Three such examples there.
And then, lastly, there's these two lines.
On seven and ten there's this expression go back
to step two, which is to induce a bit of cyclicity, right.
You can sort of think about it visually if you go from step seven,
or ten for that matter, back up to step two.
This might happen again, and again, and again,
if Mike is still to the left half, the left half,
the left, as you're whittling down the phone book.
And so we're going to induce, and induce, and induce potentially,
this cycling or looping behavior.
So these lines here now in yellow represent
what a computer program might call a loop.
Now these same English phrases we'll eventually see can be translated
into Java, and c++, and JavaScript, and Python, and Ruby,
and other languages still.
But the key takeaway for us here today is
one, the precision with which we specified these steps, two,
the fact that there's this ordering, what happens after the other.
The fact that there is these conditions, some only
happen if something is true, and the fact that some of them
can happen again, and again, and again, based on some kind of looping.
But is this good?
Like this is correct, and that of course, was one of our goals
with the phone book, initially, was let's get it correct,
and better still, let's get it correct and efficient.
correct and fast.
But how fast now is this?
In fact, how do I put a number, of really a formula,
on the performance of the algorithm so that I
can claim that I am a good programmer, or I am a good problem solver?
I've not only solved your problem correctly, but really, really well.
Well let me propose that we analyze these three functions.
Kind of in the abstract.
We don't need actual arithmetic expressions here, per se.
We'll just do things variably as follows.
So here's a nice little Cartesian plane, with an x-axis of size of problem,
and a y-axis of time to solve.
And the farther out you go on size of problem, from left to right,
means bigger, bigger, bigger, bigger, bigger, bigger, bigger bigger problem.
And by bigger problem I mean more and more pages.
More and more input, whatever the problem actually is.
And then time to solve, is not very much time, lots of time.
So on the y-axis too, the higher you go, the more
time it takes to solve a problem, or really the slower
it is to solve the problem.
So let's now draw this red line here as a depiction of the first algorithms
performance, or running time.
Whereby, there is a one to one relationship between size and time.
A one to one relationship between number of pages,
and maybe number of page turns, or seconds, or whatever
your unit of measure happens to be.
So that if I have a phone book of this size, this dot here on the red line
is how many seconds, or page turns it takes me to find it.
And there's a linear relationship there, as implied by this variable--
as we'll call it-- as in algebra.
n for number of pages, for instance.
It's linear in so far as Verizon, if the phone company like Verizon
increases the number of pages in the phone book just buy one next year,
a few more people move into town so they add one more page to the phone book.
That might take one additional page turn to find someone like Mike,
or anyone else in that phone book if Verizon just adds a page.
Or if they doubled the number of pages, it
might double the amount of time it takes to find someone.
There's a linear relationship between the two.
Now with the second algorithm, where I was flying through the phone book
two pages at a time, there's still a linear relationship,
but it's not quite as bad so to speak.
For instance, if I have this many pages in my phone book,
my first algorithm might take this many seconds in the first algorithm,
but because I'm flying through the phone book two at a time,
it will take me half as much time.
Half as many page turns with that second algorithm.
So we'll describe it algebraically as n over two.
Where n, again, is just the number of pages.
So there is a relationship between those two lines and those two formulas.
But that third algorithm.
Super, super fancy I'll wager.
And the shape of its line is fundamentally different.
If you remember your logarithms, you'll see here
this green curved line, otherwise depicted here as log of n,
or really log base 2 of n, but we'll just keep
it abstracted away as just log of n.
This is a fundamentally different shape.
And it still goes forever up and up and up and up.
Even though it looks like it's flattening out,
it's not perfectly flattening out, it's just growing and growing very slowly.
And that's key, because what's powerful about that third and final algorithm,
is that Verizon, for instance, doubles the number of pages in the phone book
next year, because a whole lot of people move into town, or maybe two towns
merge or whatever.
Then how many more steps will it take me to find someone
like Mike Smith in that phone book?
Well just one additional page turn.
One additional tear of the phone book.
Because with each tear can I take a bite out of that problem.
That's 50% of it.
I can literally tear it in half.
Meanwhile, Verizon doubled the number of pages in the phone book
with my first algorithm, as per this straight linear relationship, aka n,
it would double the amount of time necessary, or double the number of page
turns necessary, to find Mike Smith in that case.
So again, in the case of the first algorithm,
would double the amount of time taken.
The case of the third algorithm would increase by one step.
One step.
So if that phone book went ridiculously from four billion to eight billion,
no big deal.
But that third algorithm, just need one additional page turn.
One additional tear of the phone book.
Not in the case of the first algorithm.
I would need an additional 4 billion page turns to find who I'm looking for.
OK.
But a phone book is a physical problem.
And searching for phone numbers in a phone book is a physical solution.
But what can computers do electronically?
Like what is the analog in our computer to the problems
we've been solving thus far?
Well what about our own iPhones, or Android phones, or the like,
where you have contacts these days?
Sort of a digital version of a phone book.
And those contacts are typically sorted alphabetically.
But even then, you might know a lot of people, and most of us
probably don't scroll, scroll, scroll, scroll, scroll.
And if you do, you're doing it wrong.
You can instead search by keyword, or by first name, or by last name,
to actually find someone much more efficiently.
And typically their name jumps to the top of the list.
So that's an algorithm.
The Google, and Apple, and other companies
have implemented algorithms for searching, or even sorting
your list of contacts so that they can find people for you quickly,
so that you can just send them a text message or make that call.
But let's simplify further.
Rather than even get into the sorting of names, and alphabetical phrases,
let's keep things simple for a moment.
And just assume that we want to store things like numbers.
A list of numbers for instance.
So if we want to store a list of numbers, how can we do it?
Well let me propose that the screen here, or if you will a piece of paper
in front of you, really just represents your computer's memory.
Now you may know that inside of your computer
there's different types of memory.
Something called the hard disk, or maybe a solid state disk,
or something called ram, or random access memory, something called Rom
or read only memory, something called l-1 cache, l-2 cache,
sometimes l-3 cache and the like.
So there's different types of memory.
But the one we're going to think about right now is ram, random access memory.
And this is the type of memory that you might
have a gigabyte of, four gigabytes of, 16 gigabytes of, depending on how fancy
your laptop or desktop computer is, or your phone for that matter.
So inside of your computer, or phone, is a whole bunch of memory.
A whole bunch of RAM.
And if you have, for instance, 4 gigabytes of RAM,
that is four billion bytes.
Because what that means is giga means billion,
so four giga means for billion, four gigabytes means four billion bytes.
Mega, meanwhile means million, give or take.
And in fact, earlier when we talked about rgb,
And four megabytes then would be four million bytes.
So 4 gigabytes is even bigger.
And if you want to count higher still, four terabytes
would be a huge amount of memory.
So if you have four gigabytes of memory, that's
four billion bytes of memory, and a byte meanwhile, is just eight bits.
So why talk in terms of bytes?
Well it's just easier to talk in terms of eight bits,
rather than individual bits.
Because with one bit, you can't really do all that much.
You can only count from zero to one.
So with 8 bits at least we can count a bit higher and express a bit more.
so if we have 4 billion bytes of memory inside of our computer,
you know it stands to reason that we could number each of those bytes.
And maybe the top most byte up here represents byte number zero,
and maybe the bottom most byte over here represents the number four billion,
give or take.
In fact it's not exactly 4 billion, it's a little more than that.
But for our purposes, let's just assume that if this screen here
represents all of my computer's memory, and you know for that matter,
let's think of my memory as being divisible into rows and columns,
just to kind of keep things orderly.
This is not actually what it looks like underneath the hood.
And it's definitely not as wavy as I making it out to be here.
But you can think of it really, as rows and columns,
that you can address each of the cells in this table.
So kind of sort of like a spreadsheet with a whole lot of room for values.
So this might be byte zero one, two, three, dot dot dot if you will.
Four billion or so in the bottom right hand corner.
But where these things are laid out doesn't really matter for our purposes
right now.
So that's how you might address memory.
But how might you use memory?
So suppose I want to store a whole bunch of values
inside of my computer's memory.
I don't really care where it is for the moment.
And therefore, I don't care what those addresses are.
But let's suppose that I do care that the numbers are close by to each other.
They are contiguous in memory.
Back to back to back to back.
And we'll see that that has some advantages for efficiency.
So suppose that I want to store one number in my computer's memory,
and suppose that this memory and this memory and this memory
here, those are all being used already by other programs,
or other things going on in my program.
And so the first available slot, for instance, say is this one here.
And the number I want to store is the number one.
And the next number that I want to store is going to be the number 50.
And the next number I want to store is going to be the number 51.
And the next one 61.
And then the next one 109.
And then the next one 121.
And so forth.
Which is to say, if I'm storing not people's phone numbers or names,
just more simply for the sake of discussion,
all I care about for whatever reason is storing a list of numbers.
I might store them in my computer's memory in exactly this way.
Each of these squares represents a byte, or eight bits.
We know that we can abstract above bits, and actually
represent things like decimal numbers.
So I'm just stipulating that inside of these boxes
really is eight bits, or some pattern of eight zeros and ones.
But that's too low level.
We're going to abstract away from that and just talk in terms, for now,
in decimal digits.
So this thing here might be how I store something
underneath the hood in my computer.
And what's nice about this, is that it's super, super easy to find values.
In fact, let me just highlight this range of values, and slap a name on it.
This boxed in region here in a computer program,
would very often be called an array.
It's a list of sorts, of values, but what's key
here is that this list of values is back to back to back
to back all contiguous in memory.
When that is the case, when you have this property inside of a computer,
you can do things pretty darned fast.
Because there's a mathematical relationship among the locations of all
of these values.
For instance, if you are the computer, and you
find yourself looking at the number 50, how do you
find the next number in the list?
Well, you just add one to your location.
Not one to your value, one to your location.
How do you go from the number 51 to 61?
You just add one to your location.
Again, because if this is byte zero, one. two, three, and I
don't know where we are here.
Maybe this is byte 100, 101, 102, 103.
Because all of these values are back to back to back to back,
you can very quickly access them just by looking to the left,
looking to the right, by a fixed number of bits.
This is advantageous, because you can jump around.
In fact, this is where the name random access memory comes from.
You have random access, not in the sense that you just go anywhere randomly,
but you can go anywhere you want in the same amount of time.
Even though this byte four billion looks really far away from everything else,
doesn't matter.
Because arithmetically the computer can say, give me byte number four billion.
And that's a constant time operation, it's not linear.
Doesn't take more and more steps the farther away it is, because all of this
is electricity and electronic, not moving parts.
You can just instantly jump to any of the cells in your computer's memory
like that.
The catch is, me the human, can look at these values and just see,
oh I see a whole bunch of values.
Six values inside of that array of memory there.
But a computer can't do that.
In fact a computer can really only see one value at a time.
And so when I say the computer can go get value four billion, it can do that.
But it can only go get that one value.
If it wants another, it has to go get that value and another value.
And indeed, inside of a computer, inside of a CPU that a company like Intel
might make, there's an instruction that's
often called load, which refers to exactly this process.
Go load a value, maybe one byte, maybe four bytes, maybe eight bytes,
from memory, and bring it back to me.
And put it in something called a register, typically.
But that load operation is kind of like, if you've ever seen those fancy soda
machines where it doesn't just drop your soda on the ground,
instead there's that fancy little robotic arm that for whatever
reason goes left and then goes right and then goes up and then goes down,
and then finally decides where your soda is.
Then it moves it over the dispenser and drops it out.
Well that kind of robotic arm is kind of like a physical incarnation
of a computer, in that it can't just take a look back and figure out
where the coca-cola is, it has to go to that particular location
and actually drop it for you.
But computers, when there aren't moving parts, can just jump there instantly.
You don't have to wait for that robotic arm to move.
And so in this case, the computer can get one of these values instantly,
but it doesn't know what value is there until it arrives.
Right It's kind of like being--
another analogy might be a whole bunch of lockers
in a train station or the like.
Where the only way you can see what's inside of a locker is by renting it
or by putting in a key and opening it up and seeing what's inside.
Similarly, can you think of there being tiny little doors,
occluding these numbers so that the computer has to open it up
before it sees what's there.
Speaking of doors, let's suppose that the computer's memory actually
has some doors in front of it.
Depicted here, is just these simple rectangles.
And behind each of these doors is some random numbers,
among which is one number I do know and care about.
In fact, I want to find myself the number 50,
but I don't know where it is.
So I'm going to take a pretty naive, but correct approach, of just starting
from the left.
Just as we did with a phone book, starting from the first page.
And I'm going to go ahead and knock on, and then open the first door.
And I see the number 15, that of course is not the number 50.
So I'm going to proceed further.
To go and knock on and touch the second door, and I see 23.
Still not the number I'm looking for, so I knock and touch again.
16.
Knock and touch again.
Still no number, but I have found the meaning of life.
And let's try one more door here and touch the number 50 behind this door.
So here too is a very linear algorithm, but these numbers
are kind of all over the place, right.
It starts at 15, gets a little bigger, then gets smaller, then smaller still,
then bigger then bigger again.
So it doesn't seem that these numbers are sorted.
And in fact, if the analog then in my phone might be,
my contacts might be just randomly ordered.
Maybe ordered in the order in which I inputted
them, which probably isn't ideal when I want to find someone.
I don't really care in what order they were put in,
I care that I find them quickly.
And so maybe I should do a little work up front,
or maybe Apple or Google should do a little work upfront,
and actually maintain my contacts in sorted order.
Which indeed they do.
Or in this case here, why don't I at least maintain my numbers
in sorted order.
So let me go ahead and reset all of the doors, and start again.
This time with an assumption.
This time with a feature, if you will, that the numbers are now
sorted behind these doors.
So they go from smallest to biggest, and there might be some gaps in the middle,
but in so far as they're now sorted, I can
leverage some of that old school phone book intuition and divide
and conquer this problem.
Let me go ahead into the middle of the problem now, not the left,
and let me go ahead and knock on and open the middle door.
And there's 16.
Now 16 is kind of a small number, but I now
know that none of the numbers to the left
are going to be the one I'm looking for.
Because I'm still looking for 50.
So I can sort of throw that half of the problem
away, or really just turn a blind eye to it.
And now go to the right half of the problem,
identify the middle door here, knock on and open this one, there again is 42.
But now, again, I can apply some of that intuition
and know that 50 is bigger than 42, so must be
that 50 is right here at the last door.
And in this case, have I gotten away with opening fewer doors?
It's not a frightening number of fewer doors, it's still relatively small,
but that's only because we could fit some seven doors or so on the screen
here.
But if the number of doors were four billion, or if the number of doors
were even larger, surely with this divide and conquer approach,
could I find my number super fast.
But there's kind of a trade off here, right.
Like yes, I can now find numbers faster, or equivalently
I can find people presumably in my address book in my phone
faster, if they are maintained in sorted order, but what price did I pay?
In fact, a theme in computer science, and in programming specifically,
is this theme of trade offs.
Yes, you can speed up searches, but you're going to have to pay a price.
What is that price in this context?
Well it took someone some amount of work to sort those contacts, right.
Before class I had actually come up with the sorted order of those doors,
and then put the right numbers behind the right doors.
There was a non-zero amount of work.
Not a huge amount of work, but it's non-zero.
So really, by speeding things up during search,
I have to first incur an up front, one time cost, of sorting things
before I even begin to search.
Or equivalently, someone at Google, or someone at Microsoft,
or someone at Facebook, or any of these big companies with big data,
if they want to use searching efficiently,
they need to pay a price upfront of sorting the data first.
And indeed, that's what Android and iOS and other devices
too, must be doing underneath the hood.
Someone at some point in time, probably long ago at this point,
implemented a sorting routine.
But how quickly can we sort numbers?
How quickly can we actually achieve that result of sorted values
so that we can then do search more efficiently?
So for this, let's try something a little visual,
a little old school, in fact.
We're here in this beautiful theater.
And in fact we have a whole bunch of music stands from the musicians.
Let's go ahead and use a few music stands,
each of which will represent a location in my computer's memory,
and then I'm just going to have some numbers that I
want to put inside of those locations.
And let's see how expensive it is for me to actually sort those values.
So I have here in my hand the numbers one through eight,
and I have here eight memory locations as represented here by music stands.
And let me go ahead and just put these numbers in pretty much random order.
I've got my little cheat sheet there up on the screen,
so that we reset each time to the same location.
And I'm just kind of putting these in some random order.
Where four's going to be over here, and the numbers
are going to sometimes get bigger, sometimes
get smaller from left to right.
The key, though, is that they're not actually sorted.
And so I start now with these values as such.
All right.
So suppose I wanted to find now, maybe the number 50.
Well 50 is not among these numbers, so we better think a little smaller.
Suppose I want to find myself the number seven.
Well again, as in the case of a computer, or the doors,
I can only find seven not just by kind of cheating like a human
and say, oh there it is over there.
I have to get to that point more methodically.
And define the number seven linearly, using linear search, so to speak.
I have to check is this it, nope.
Is this it, nope.
Is this it?
Nope.
Is this it?
Nope.
Is this it?
Nope.
Is this it?
Nope.
Is this it?
Yes.
And now I found it, coincidentally, some seven steps later.
But seven could have been anywhere, but in the worst case,
it's all the way over here.
Now could I use binary search on this?
Let's actually slap a label on that algorithm we've leveraged with a phone
book.
Dividing and conquering.
If you do it half and half and half and half, looking for some value,
it's actually more technically called binary search.
Bi meaning two, and so it's two because you're splitting
the problem in half each time.
Could I use binary search here?
Well there's one minor problem, which is that I
don't have an odd number of music stands, which
means the middle element is like between eight and one.
But that's fine.
We can deal with that arithmetically by just rounding up or down,
depending on how we want to implement it ultimately.
But even then, that's not the biggest problem.
If I look at eight here, and I'm looking for seven, well
based on my previous divide and conquer strategy,
I should be looking to the left because seven is smaller than eight.
Of course I'm not going to find it to the left of eight,
because it's not there, it's over there.
And that's because the numbers aren't sorted.
So as in the case with my hypothetical address book,
if I want to be able to search this efficiently,
or search a database sufficiently, or search most anything efficiently,
I need to know something about the data.
And ideally that it's sorted.
So how do I get to the point of sorting this data?
Well, you know what, I could take a fairly localized approach.
Like four and two, I'm looking at them right here.
And technically, I'm looking at them one at a time.
I see four, I see two, and I know they're out of order.
Well, as the computer, I could do this.
I could, using my load and store instruction, essentially,
or a move instruction, do that.
And now I've improved the sorting of this list.
Now four and six, those look OK.
Six and eight, those look OK.
One and eight, Mm-mm, not OK.
So let me just make a quick fix.
Let me just swap one and eight.
It's not perfect yet, but it's better.
Eight and three, let me make a quick fix.
Well, that's better.
Eight and seven, a quick fix, that's better.
Eight and five, quick fix, that's better.
So this is great.
Eight is all the way to the right as intended,
and one is not yet all the way to the left.
So I haven't solved the problem, but I have improved it.
I've gotten a step closer to the correct solution of sorted numbers,
because some of the values moved at least one step closer
to their intended location.
But of course, I've got to do this again.
So two and four their good.
Four and six they're good.
Oh six and one, now that's out of order.
And so I can transpose these two.
So this algorithm has a name, whereby I am transposing pairwise elements that
are out of order, such that the biggest numbers are bubbling their way up
to the top. eight already made its way there, seven just made its way there.
And so if I keep doing this, these bigger values
are going to continue to bubble their way up.
Two and four those or fine.
Four and one, ooh notice, four is starting to bubble to the right.
Four and three, it's continuing to bubble.
Four and six is OK.
Six and five, mm-mm.
Six bubbled its way up, and now I got a good one there.
I really fixed a lot of those values.
I just need to do this once more.
One and two, fixed that.
Two and three, three and four, four and five, six and seven, seven and eight.
Let me do another spot check just to make sure.
One and two, three and four, five,six, seven, eight.
I'm good.
But I did a lot of walking, and I did a lot of transpositions.
And it turns out this algorithm is correct,
does have a formal name called bubble sort.
But it's not very efficient.
In fact, if you actually do the math out, this takes as many as like n times
n steps in the worst case.
So if you have n elements, that's eight in this case.
N times n, that's 64.
It's not actually 64, but it starts to approach 64 asymptotically,
so to speak.
So if we did a formal analysis, long story short,
it would start to feel like something quadratic, so to speak.
Not a fast algorithm.
But there's other ones.
So let's actually try out another algorithm.
Let me go ahead and restore the original, seemingly random order here.
And indeed it is random in so far as I'm just
putting it in some non-sorted order that I thought through initially.
One's going to go here, three is going to go here, seven and five.
So now we're back to the original unsorted order.
You know what, I can do another algorithm altogether here.
Let me not look at them pairwise, let me just go even simpler.
Let me just find the smallest number, and deal with it.
All right, four is pretty small indeed it's the smallest I've seen thus far,
I'm going to hang on to this.
Oh wait a minute, two a smaller.
Let me hang on to that.
Six not smaller.
One, eight not smaller.
One is smaller, let me go ahead and hang on to this.
Three, not smaller.
Seven, five.
OK so here's the smallest element.
I found it I've selected the smallest element.
You know what, this belongs all the way to the left.
I'm going to put it there.
I'm going to do that.
Of course, this is cheating.
Like this music stand, even though it's kind of wide,
should only be storing one value.
There's only enough room for one byte for instance of information.
So where do I put four?
Like four can't stay on the same music stand.
You know I can't cheat and do that.
So I could move eight over, move six over, move two over,
that's a lot of work.
Or I could just, fine, four was already out of order in the first place.
Let's just put four over here.
It's not made the problem any fundamentally worse,
because they started in random order.
I randomly put four there.
The key is, one is in really good shape now.
Let me repeat this process of selecting the smallest element.
Let me look for the next smallest.
Two is pretty small.
Small, smallest, yep.
OK I found it.
Two is the smallest.
Let's put him co-incidentally right where he goes.
So that was kind of a waste of time, but verification that it's correct.
Let's keep looking.
Six is pretty small now.
Eight is not, oh four is smaller.
Let's grab four.
Oh Three is even better.
Let's grab that.
Seven, five.
OK 3 needs to be put in its place.
I don't want to touch one and two, because those are good.
Six, sorry you don't belong there.
Let's go ahead and evict you and put six over there.
So if I repeat this process of just continually iterating, or looping,
if you will, through the list, looking for the smallest element,
and evicting whatever is in the place that it belongs,
notice that I've sorted it seemingly faster.
But I actually got a little lucky there.
It turns out that this algorithm, which also has a name called selection sort,
is also quadratic in nature.
If you've got n elements, it's going to take roughly n times n total
steps in order to achieve that.
Not in every case.
But in the worst case, so to speak.
So it turns out this is just two ways now to sort elements.
And there are dozens, if not more, other ways to actually sort elements,
with each one getting maybe a little better, maybe a little worse,
maybe a little better depending on the inputs.
So sometimes it really depends.
But these are all examples then of algorithms.
And algorithms have performance associated with them.
Algorithms have a running time associated with them.
And one of the things that theoretical computer scientists do,
is not only design algorithms like these,
or really even way more sophisticated algorithms than these,
but also analyze them.
So that we can make arguments, mathematical arguments, sometimes
that state as a proof, this algorithm is correct.
No matter what, this will sort your elements correctly,
if implemented correctly in a computer, and this
is how much time it will take in general in order to achieve that.
But the takeaway here, is that it was a lot of work, right.
I had the luxury with that phone book, long ago,
of just opening it up to the middle and looking for Mike Smith,
and da da da da da, found Mike Smith pretty darn efficiently,
because Verizon, or whoever made the phone book,
actually did all of the upfront work for me.
So here's a question then.
Suppose that you have a whole bunch of random values,
and you want to search for something specific among those values.
Mike Smith, number 50, whatever it is you're looking for,
should you sort those values first before searching?
Should you sort the values first?
Well, I would wager you should sort them if you're
going to be searching over them pretty darn often.
Right.
Because it might be painful, might be boring,
might be tedious to actually sort those values either manually,
or algorithmically with a computer, and a computer program.
But it'll probably pay off over time.
So there's this notion of amortization of the cost over time, whereby yeah, it
might take you a decent amount of steps.
Quadratic, seconds, minutes, whatever you want to measure it in.
But, it's going to pay off in the long run.
Of
Course, if you add data, you're going to need
to keep that new data sorted as well.
But again, it's this trade off.
However, consider a scenario where you just have a whole bunch of data,
it's in random order for whatever reason,
and you just want to find Mike Smith.
Or you just want to find the number 50.
And after that, you do not care about the data.
You're never going to look up someone else.
You're never going to look up another value.
Maybe then, in that case, you should just blindly
go through it brute force, so to speak.
Left or right, in some sense, or linearly, as opposed
to trying to do anything clever like we did with the phone book.
Because it's faster to just brute force your way through the solution,
than doing some upfront sophistication just to improve the subsequent result.
So again, it's a trade off.
It's a cost benefit analysis that you need to decide, ultimately,
on what is most important.
Which resource, time, space, money, people, is most important to you.
So in all of these examples thus far have we
assumed that the data is contiguous.
Back to back to back with other values.
And indeed, that's exactly what the music stands represented,
were these fixed locations in memory.
Bytes in your computer's ram, so to speak, that could have values.
But you had to physically move those values from one location to another
if you wanted to sort the values therein.
you couldn't just kind of insert a new music stand,
and push the others aside, because that's simply not allowed.
That's not the metaphor in question.
Those music stands were fixed just as the locations
in memory of your computer are as well.
And as it turns out, just as we drew a picture before wherein we had some
sequence of back to back values in memory, turns out that in programming,
actually using languages that aren't just pseudo code, but actual code,
Java and c++ and others.
Still, there is very often something called an array, or a list.
A data structure that gives you the appearance of having values
back to back to back to back.
And in some languages, these values are indeed
next to each other in terms of their bits inside of your computer's memory.
But there's a problem with a world like this.
Because if you put in a whole bunch of values up front, as I did earlier,
and you want to actually insert some new value,
where do you actually put that value?
For instance, earlier we had one and 50 and 51 and 61 and 109 and 121.
Suppose I want to put the number 55 in this sequence of numbers.
And better still, I want to keep the whole list sorted.
Well dammit.
There's really not room for 55 in there.
I know where it should go numerically, it should go right here, of course,
between the 51 and the 61.
But there's no room.
Or I could put it arbitrarily over here, but then it's not sorted.
I could put it, well I can't put it there
because the dot dot dot remember from earlier, suggested
that something else was there.
Something else from your program or computer
was already using that location.
Now I could put it maybe below, but that's kind of weird.
The whole point here is this contiguousness.
And indeed, with the music stands a moment ago,
was it advantageous for me to be able to take one step
and be at another music stand?
One more step be at another music stand.
I didn't have to walk all around stage looking for my values,
because they were indeed back to back to back
and that made things very efficient and predictably close to one another.
So what are my options?
Like surely a computer can do this.
It's not the case that computers are so dumb that once you write your values
you can't solve any new problems.
So what are my options?
I could put the 55 here at the end, but then I'm
going to have to reshuffle all of the numbers.
Or maybe I ask the computer for more memory altogether,
and you know what, I re do this, and I say let me
put one here, 50 here, 51 here, 55 here, then 61, 109, and 121.
In other words, why don't I just use different memory?
Different bytes in my computer's memory, but that feels kind of lame,
because now I need twice as much memory, at least temporarily.
So I can kind of copy the old values into the new,
but leave room for that new value.
So that seems a little time consuming, and tedious certainly.
And indeed, it would be for a computer as well.
But maybe there's another solution altogether.
And here too, we're experiencing yet another trade off.
The contiguousness of my memory has thus far been an advantage.
Because again, each of my music stands just one step away.
In the case of actual ram, each number is just one byte away,
and that lends itself to again, random access, ergo random access memory.
But it's kind of painting me into a corner,
because I'm finding that now I can no longer really
add new values without incurring significant cost.
And by significant cost I mean having to duplicate the entire array of memory
into a new location, leaving one additional space for that new value,
which is going to take as many steps as there are values in the original array.
So you know what, maybe I take an even older school approach,
and maybe I do something proactive.
A little defensive.
I'll put one there, and 50 here, and 51 here, and 61 and then 109,
and then 121 and so forth.
And wow, this was clever of me, because now I
left myself little gaps in my values.
So that I still get the predictability of every value is now two steps away,
but now I can fit 55 in here.
Well, clever as you might initially think that, now we've
created kind of a problem.
Because most values are two bytes away from each other.
But now there's this weirdness where 55 is only one byte away
from its neighbors.
So that's really just complicating things now.
I can no longer predictably take just one step at a time.
Sometimes I take two, sometimes I take one,
and that just feels like it's devolving into a very confusing bug prone
situation already.
And in fact, if you ever saw, or maybe grew up with the basic programming
language, back in the day, you didn't number your lines
of code zero, one, two, three, four.
If you numbered of them at all, you instead
did something like line ten, line 20, line 30, line 40.
Which wonderfully left your room for like nine more additions later on,
without having to renumber everything.
Of course now, programs just number or lines automatically for us.
And they're not even strictly necessary, those line numbers.
So we've tried this, and it's just, it's not really a good solution.
Better would be I'll claim some dynamism.
What if instead I consider my computer's memory
still to be this grid of locations.
But let me do this.
Let me propose that the first number.
I'm going to put in, maybe is here, and just to keep things prettier I'm not
going to draw the whole grid again.
But let's assume that one byte I'm using is right there,
and I put the number one in there.
And then eventually, I decide to add that second number.
And I go ahead and add the number 50.
And the number 50 just so happens to end up, oh I don't know, over here.
And it's farther away in memory.
Maybe enough time has passed that the bunch
of memory in between those two values is being used for some other purpose.
And that's OK.
But the 50 and the one are no longer next to each
other, or therefore pictorially, or really related,
unless I somehow interconnect them.
So for now, let me go ahead and draw an arrow.
And let me kind of weave together my values.
Much like popcorn on a thread, or any number of other metaphors
where you're linking things together.
Like with a chain, for instance, can you make a list like this?
Because my other numbers, suppose that more time passes.
I insert a third value like 51, just happens to be over there, whatever.
It's not contiguous, which is unfortunate.
But that's OK, so long as I somehow link these together like this.
And then maybe that next number here, 61, get lucky and it's pretty close.
So it's over here.
But we still need an arrow to this one here.
And then maybe we have maybe 121 and 109, each of which
is in some different locations.
So we might need this arrow here.
And this arrow there.
So it's kind of all over the place.
And if you ever play the game growing up,
it's kind of like chutes and ladders now,
you kind of have to like follow the chutes to get from one number
to another.
But that's OK.
In fact, it's actually quite beautiful, my handwriting aside.
It's not quite as efficient though.
Because it turns out that if you're at one of the numbers in this list,
and you want to jump to another, you can get to the next number pretty fast.
If I'm at one, I can get to 50 pretty quickly.
I just follow that arrow somehow.
I don't know how it works in memory, and we don't really
need to know how it works.
We can kind of abstract that away for the sake of discussion at the moment.
But I can get to 50 based on this picture alone pretty darn directly.
But I can't really get to like 109 pretty quickly.
If I'm starting at one, I want to get to 109.
previously, I could just take one two, three, four steps away, four bytes away
and I could just take a step that's like four times as big as usual
and I'm instantly there.
But these arrows seem to suggest that the memory could be anywhere.
And so it's almost like following a map.
If I'm at one, all right let me go find 50.
Once I found 50, I can pick up another map,
and now I can follow another arrow to 51, which is maybe over here.
Now I pick up another map.
Oh here's 61.
OK now this leads me to, Ah here is 109.
It wasn't as simple as just one big step that's
four times as big as was possible in the case of contiguous memory.
Now, I kind of have to weave my way through.
That actually kind of hurt.
Weave my way through my computer's memory to get where I'm going.
So it's a trade off, right.
Because now I have perfect dynamism.
No matter where there is available memory in my computer,
over there, over here, over here, over here, I can use it.
And I can just kind of stitch together this data structure,
the structure for my data.
I have eliminated the gotcha that my data structure is a fixed size.
So now if I want to add another value, suppose I want to add 55.
OK, suppose that 55 happens to have some available space here.
You know what I can do here, I can just get rid of this arrow,
and then I can go ahead and stitch this in like this.
No big deal.
I don't have to touch any of the other elements.
But think back to the array.
The array being the rectangular example where
everybody was back to back to back.
That I had to make room for things, or duplicate all of it.
Had to do more work here, just to allocate space for your 55,
and then go ahead and just update two of these arrows,
or pointers, or references as that might be called in different languages.
So that's pretty powerful.
But again I've lost the random access, which
means I can't do things like binary search anymore.
Which we're predicated on simple arithmetic.
Divide by two, divide by two, divide by two, Maybe round,
but divide by two each time.
And these arrows.
Kind of abstracting maybe too generously here.
These arrows are going to cost me some memory too.
Now I've drawn them prettily as arrows on the screen,
but those actually themselves are some kind of value.
And it turns out that those arrows themselves take up space, take up bits.
So I've kind of doubled, let's say, the amount of space
necessary to store this data.
Because I need to somehow store and memory those arrows.
Underneath the hood those arrows are effectively
stored as numbers themselves.
Which is to say, they take up at least a byte or more.
In fact, on most systems, they take up four or eight bytes still.
So it depends on what's important to you.
Do you want to be able to search the data super efficiently
and actually have random access, and do something like binary search
and get that divide and conquer upside?
Or do you want to have dynamism, and be able to grow and shrink
the data structure super fast, but pay a price in terms of memory as well.
And use more of those bits.
It really quite depends.
Now these aren't the only data structures at our disposal.
We can use not just arrays, not just linked lists,
as we'll call that last one, whereby we have
links tying these lists items together.
But we have maybe tree structures too.
In fact, we think back to like your family tree that you
might have made in grade school.
You might have drawn a little something that
might have grandma or grandpa at the top of the tree,
and then their children might be down here,
and then these children might be down here, and so forth.
Well if we generalize away from a fam--
OK ignore that one.
If we generalize away from a family tree,
it turns out that this tree data structure is another kind
of incarnation of that previous idea.
A linked list doesn't have to just be linked from left to right, so to speak,
even if it's kind of swooping all over the screen.
It doesn't have to have a start and an end.
It can be a tree structure, whereby your arrows actually, much
like a branch and a program, can go in different directions.
So each of these edges in this graph actually are directional.
They take you from one place to another.
You can imagine using a data structure like this to store values as well.
For instance, suppose that I wanted to store numbers in this data structure.
Instead of using squares, I'm just doing circles
now just because it's conventional.
Doesn't mean anything else.
Anything special.
These are still just bytes underneath the hood, or ultimately bits.
Now suppose I want to store the number two here, and one here, and three
here, and five here, and six here, and seven here.
This is all very deliberate.
Because notice the pattern that I've kind of adopted.
Four is at the top, in the so-called root of the tree.
What do you notice about all of the values to the left of the four?
All of its left descendants so to speak, to borrow that family tree
nomenclature.
One, and two, and three.
Curiously, all three of those are smaller than the number four.
What about all of its right descendants, down this branch?
Six and five and six and seven.
Those are all bigger than four.
That's kind of neat.
How about here?
Two, this is kind of a mini tree, if you will.
This is a new route, if you ignore everything above it.
Two has two children, left and right.
Notice that the one is less than the two, and the three
is more than the two.
So this is not a coincidence that I drew it like this.
Similarly, if you think of this as a tree, just those three nodes below it,
six is bigger than five, six is less than seven.
So there's a pattern.
So we can actually reclaim some of the capabilities of divide
and conquer by laying out our data, still
with these arrows, these pointers or references if you will.
But not just doing it linearly, or swiveling all over the screen,
but from one start node to an end node.
Where node is a fancy word for these squares or these circles.
What if we instead allow the user to go in two different directions
when searching for data?
And suppose we leverage that same phone book principle
where the smaller values are this way, and the bigger values are that way.
What does that afford us?
Well, this binary search tree, if you will, to give it a fancy name,
actually gives us back a logarithmic running time.
Much like dividing and conquering a phone book.
Because indeed, if I start at the four and I'm looking for seven,
and I immediately realize four is here, which means seven must be to the right
if it's present at all, I can pretty much chop off the half of the problem
that I know seven is not in.
So it's like throwing away the left half of the phone book.
I can ignore almost half of the nodes in the tree
by just snipping off that left branch.
Can't be that easy.
Got to be a trade off.
What it that trade off here?
Well, from the looks of this picture, pretty
though it is, relatively speaking, it looks like now each of my nodes,
each of my values has not just one arrow associated with it potentially,
but as many as two.
And that means we're spending twice as much space as before.
So whereas in our array, we use just one byte per value,
or four bytes per value, or some fixed number of bytes per value,
in my linked list example I had to double
that because I had to also store--
using bytes in some form--
those arrows.
Those pointers, or references as they're called.
But now in this darn tree, I need three times as much data,
because each of my nodes might have a left child so to speak,
or a right child.
So again it's a trade off.
And the prevailing trade off here, thus far, seems to be if you want less time,
you're going to have to spend more space.
Or rather, less time, I'm getting the visual metaphor wrong here.
If you want to spend less time, you're going to have to spend more space.
And if you want to save space, you might have to tolerate a little more time.
But can we do better?
Indeed, can we combine some of the best features of multiple data structures
in order to achieve something better still?
Well, sort of.
It turns out that there exists a very popular data
structure called a hash table.
And a hash table is a data structure that a computer scientist
will use when he or she really wants to store a dynamic amount of data.
It might grow or might shrink.
But they also want to search that data pretty efficiently,
and ideally be able to find data instantly, in constant time,
so to speak.
So you'll often see a hash table implemented, really
with an array something like this, and I've drawn it vertically just
to make the picture prettier.
But again, these are just artists depictions.
Bad artists depictions of what's going on in memory.
It's still just an array of back to back values.
But the values now in the array are not going
to be the numbers I care about, or the words, or whatever.
It's actually going to be arrows, which again we've stipulated take space.
And each of these arrows is going to point
to what we described as a linked list.
So it turns out that a hash table will often
look a little something like this.
A combination of an array, and a linked list
where those linked lists might be not present at all, might be short,
might be long, totally depends.
But we decide where to put our values based on some property of the value.
So for instance, if this array weren't this big, but maybe
had 26 different locations in it.
And suppose that we're no longer storing numbers, we're storing names.
Well every name in the English alphabet starts with an A or B or C
or dot dot dot a Z. One of 26 possibilities.
So if my hash table has an array of size 26,
then maybe, if I'm storing names that start with A,
I could just put them in this linked list here, the first one.
And if they start with B, they'll go in the second one.
And if they start with C, the third.
D the fourth, Z the very last.
And what you'll find in this case is partial optimization.
It's not instantaneous to find someone with a given name.
If there's a lot of people who's names start with A,
you might still have to look through all of the A names linearly
through that list.
But at least you only have to look through 126th
of the possible values in your data structure,
assuming a uniform distribution.
Of course, that's not fair, there's probably fewer names that start with Z,
fewer names that start with Q, maybe a lot
that start with A or D or other such letters.
So your chains might be of variable length,
and maybe that's not the best design.
So maybe we should think a little harder about this.
So this principle of bucketizing things, and putting values in their place,
and using that as a stepping stone to getting
to a complete solution-- in this case of sorted order--
is really useful not just in these increasingly complex examples,
but even in the real world.
So for instance, if you're a fan of playing cards,
you might have a deck of cards at home.
And those cards of course, have different values or numbers
on them, and also different suits.
Clubs and spades and diamonds and hearts.
And you know, sometimes you might want to either shuffle the deck,
or really un-shuffle it and actually sort it.
And if you did want to, sort of obsessively like me,
want to sort your deck of cards, well you
could just go through it one card at a time.
And then try to find or select or maybe bubble up the values,
like our algorithms before.
But most of us probably use a bit of intuition.
So if I see a spade here, I might put it in one pile,
and a heart, I'm going to put that in a different pile.
A diamond, different pile still.
A spade will go up here.
A club now goes over here.
And as I continue this pattern, I'm kind of dividing the problem,
albeit in a different way.
I now have, instead of a 52 card problem,
I'm eventually going to have four 13-card problems.
But how am I getting to that point?
Well, I'm bucketizing things.
Not physical buckets, but there's like four piles here of cards.
But what really am I doing?
I'm really hashing these values.
I am looking at each card as input.
I am passing it through a sort of mental function, a mental black box if you
will, that outputs a value.
Either bucket one, or two, or three, or four, or more specifically spade,
or diamond, or heart, or club.
That's kind of my hash function.
So if you've ever done something like that,
where you're trying to clean up a mess, or you're trying to sort some data,
or organize your cards.
Once you start bucketizing in this way, you are hashing values.
You're taking as input some value, maybe a card,
and producing as output some identifier.
Similarly, with our example of names, if I've
got a whole bunch of names, each of which might start with A through Z,
each time you hand me a name, I'm looking
at the very first letter in that person's name,
and I'm deciding A bucket, Z bucket, D bucket, C bucket or so forth.
C should be over there.
And deciding, based on the input, what the output should be.
So again, here to, this sort of intuitive principle
can very quickly be applied to something much more sophisticated.
Again, compare and contrast the sort of sophistication
of the before and the after.
This is odds are pretty darn intuitive, if not something very familiar to you.
This might still look a bit like Greek.
Certainly pretty arcane versus some simple playing cards.
But to the overarching point, that while computing and technology all around
us might seem especially sophisticated, and complicated,
and well beyond one's understanding, that's
just because it might be presently beyond your familiarity
with some of those building blocks.
Indeed, we started so low level with just zeros and ones,
and then we got to letters, and then maybe words
and paragraphs, and graphics and videos, and eventually
so many more forms of media.
And at the end of the day, it's useful to understand
what's going on underneath the hood, but it's also very empowering.
Because you realize that even once we're at the point
as we are literally right now, of talking it this degree of complexity
and sophistication for how you might store data inside of her computer's
memory, and very efficiently get at it, all of this
was the result of these stepping stones.
These abstractions.
And along the way, in building up these abstractions,
and in solving problems more effectively by standing
on the shoulders of problems solved past,
similarly we have to make some trade offs.
And so at the end of the day, this is what computational thinking is all
about.
This is what computer science and engineering and programming
is all about.
It's solving problems using solutions to problems past,
and along the way making very conscious, calculated, and hopefully
correct decisions, as to what resources and what
objectives are most important to you.