Learn how to represent a directed unweighted graph using an adjacency matrix. We cover why a two dimensional matrix is really just a one dimensional array under the hood, making cell access and updates extremely fast. For directed graphs the starting node is the row and the ending node is the column. A zero means no connection and a one means an edge exists. Self connections are not implied by default. Adding or removing an edge is constant time. Directed graphs do not produce a symmetric matrix like undirected ones. This is one practical way your program can store and work with a graph. If you have been following the previous videos on undirected graphs this builds directly on that. The next video will cover directed weighted graphs with an adjacency matrix. Thanks for watching and I hope you picked up something useful.

00:00 Intro to Directed Unweighted Graphs
00:01 Matrix Stored as Contiguous 1D Array
03:09 Building the Adjacency Matrix
04:25 Filling Outgoing Edges Node by Node
09:43 Directed Graphs Are Not Symmetric
10:36 Adding a New Edge
11:08 Adding Self-Connections
11:53 Removing an Edge
13:04 Adding Bidirectional Edges
15:28 Next Video and Outro

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Hello there. Let’s talk about representing a directed, unweighted graph inside your

machine using an adjacency matrix. It’s going to be fun.

Okay, sorry about that. Anyway, so I hope you’ve been watching my previous videos.

We covered already how to do an undirected weighted graph and also unweighted graph

using an adjacent

and also unweighted graph using an adjacency matrix we’re going to basically represent a graph

inside of the machine like this is one of the ways that your machine can do it we’re not going to

talk about code exactly but you know kind of allude to code but this is this is a really good

way to you know get started in practicing the concept okay first thing i want to say i’ve said

this in my previous videos i’m sorry if you’ve been watching all of them is that this matrix

this two-dimensional matrix it’s really just a one-dimensional array think about this we have

Thank you for watching.

array think about this we have some numbers in an array I’m going to put six numbers in there

we could just as easily represent this as a two-dimensional matrix a two-dimensional array

right it’s the same amount of numbers what’s really happening under the hood when you have

a two-dimensional array is your computer just has it as a one-dimensional array so the question is

why is that useful oh hang on okay anyway so what i wanted to say is that you see

So what I wanted to say is that you see a two-dimensional matrix,

but really what’s happening inside your computer is a one-dimensional matrix.

I’m just going to make a one-dimensional matrix.

I’m going to say we’re putting numbers in the matrix.

I’m just going to put like, you know, six hashes or pound signs or hashtags in the modern era.

So a one-dimensional array could be represented as a two-dimensional matrix

if we just reformat it a little bit.

last part of that first array on a second line that’s what’s happening in the computer

even if you’re working with a two-dimensional array in your computer it’s still just a

one-dimensional array under the hood which means it’s incredibly fast to access one particular

index not necessarily to scooch things remove things and add things but if i wanted to go to

index three and a one-dimensional array if you know time complexities already that’s really fast

That’s constant time. We’ll do time complexities in another video, but um if

but um if i wanted to access you know a certain row in a certain column intuitively you think

well i gotta like scan down and scan to the right uh so maybe it’s slower but it’s not because it’s

still just going to be math to find exactly the right offset in uh your two-dimensional matrix

because it’s really a one-dimensional array all of the items in this array i didn’t say this in

previous videos all the items in this array are contiguously allocated so all we really have to

do is compute an offset from some

compute an offset from some starting memory location so it’s incredibly fast

to read a cell in this matrix and to modify a cell in this matrix you’ll see in a second

that’s well you’ll see in several minutes why that’s one useful feature of a of an adjacency

matrix okay so adjacency matrix uh suppose you’re presented with this matrix you see

i’m sorry with this diagram you see right here this is a graph it’s got direction on all the

edges i really hope i didn’t forget any particular edge i think i’ve got arrows on all of them

think i’ve got arrows on all of them we’ve got nodes and edges and i just

need to figure out how can i put that into a matrix so my computer

my program understands the graph and it it has a representation of the graph

so with directed graphs the start row uh sorry the start node and the end node

actually matter a lot in my other videos where we talked about

undirected graphs we could just easily swap the row and the column

in this matrix and it didn’t really make a difference because the whole matrix

was going to end up being a mirror anyway.

was going to end up being a mirror anyway but that’s not going to happen this time so what i’m

going to say for the way that i like to do things is the starting node is the row and the ending

node is the column so this is also an unweighted diagram an unweighted graph so we don’t really

need to store floats or infinities or anything like we did in the other video we can just say

that no connection means no connection is a zero and yes connection is a one of course you could

You could still do that with floats if you want to,

of course you could still do that with floats if you want to but whatever you could have a a matrix

of booleans if you really wanted to like true and false and true and false but you know for this

diagram we’re going to say 0 and 1. so i’m going to start with node number one and node number one

therefore is going to be the the source node and i’m going to see where does where does it go out

to what edges are outbound from the number one so the start node is the one that means the row is

one that i’m working with and the destination nodes or the end nodes or the sync nodes those

Those will be the calls.

nodes are the sync nodes those will be the columns so i’m not really going to look at how six is

coming into one because i’m only interested right now in what uh what one starts uh in terms of an

edge you know what edges come out of one so one forget about the six and the two those are incoming

edges i’m just going to look at the outgoing edges so one goes out to three and four and five so

by default in our diagrams at least at this point we’re going to say

our diagrams at least at this point we’re going to say that self connections are not implied

meaning by default if i don’t see an actual edge on the diagram going from one back into one then

i’m going to say one is not connected to itself so i’m gonna put a zero to say there’s no connection

uh one going to uh the two that’s not happening because two is actually going into one

so i’m going to put another zero then uh there uh one going to three that is happening

uh one going to three that is happening so i’m gonna put a one so one does have an outbound

connection or an edge to three so i’m gonna put a one in that row and column

one going to four uh yes so i’m gonna put a one one going to five uh yes so i’m gonna put a one

there one going to six uh no so i’m gonna put a zero there i should have put you know

edges that go back and forth uh to make this more interesting but you can probably imagine

imagine well i’ll add that maybe later in this video so one going to seven uh nope someone put

a zero so that’s the first row we represented all of the outbound edges from one to some other node

and we used a zero to say it doesn’t exist and a one to say it does exist so then we just move on

to the two so if i’m focusing on the two node then i’m on the two row and its outgoing connections

are going to be represented in the columns.

in the columns so does two go to one yes I’m going to put a one there does two go to itself

self-connections are not implied at this point and then uh hang on a second

does two go to three um nope because three goes into two and not the other way around

uh we’ll add those other edges in a second does two go to four nope does two go to five

nope there’s two

nope does two go to six um i don’t see that so i’m going to say nope and then does two go to seven

uh yes i’m going to put a one so now we’re done with the two um this is a directed graph by the

way so like i said about my previous videos we will not see a mirror at least we won’t necessarily

see a mirror we might accidentally make a mirror depending on what the graph is but we’re not going

to see a mirror. So we have to do this whole graph with no shortcuts. So the

shortcuts so the three uh does it go out to the one that’s a no does it go out to the two

um where the heck is the three oh yeah it goes out to the two

uh no implied self-connections three goes to the four nope three goes to the five yes

3 goes to the 6

nope

3 goes to the 7

uh three goes to the seven nope okay so now we’re done with the three

and then we’ll say now does four go to the one i’m looking at the four where the heck’s the four

i’m losing the four oh four doesn’t go anywhere except for the seven so i can actually just put

like zeros in this whole thing you know just to save a little time oh god there we go but the

four does go to the seventh. I’m going to put a one right there. Okay.

the sevens i’m gonna put a one right there okay so uh the five um it has outbound connections i’m

gonna i’m gonna be cheap here i’m gonna be cheesy the five goes to six and seven so i’m just gonna

put zeros everywhere else so whoops in fact i probably could have just copied that whole row

of zeros and then just overwrote the last two but i guess so the five goes to the six that’s a yes

The 5 goes to the 7. That’s a yes also.

the seven that’s a yes also and then we’re ready for the six the six has three outbound connections

one three and seven so i’m just gonna you know what i’m just gonna copy a whole row i wish i

had a whole row of zeros and i’m gonna maybe change that to a zero real fast and then i’m

gonna say all right where does where does six actually connect to let’s change those zeros to

ones six connects to the uh the one and the three and the seven

and the 7 okay so 1 goes to 1 3 goes to 1 and 7 goes to 1 okay then the 7 uh has no outbound

connections so this is indeed a row of zeros maybe i could have done that one first it would

have been fun oh well uh notice one more time that there’s no mirror so if we like you know

if you trace your eyes along this identity line right here and you were expecting to see a mirror

you would be sadly disappointed.

a mirror you would be sadly disappointed because what would actually happen is not a mirror let me

do the highlighter here and we’ll say we have like an identity line that goes down the middle

and then i’ll just try to highlight some stuff let’s see the three four five six over here on row

uh one and and those columns let’s go now to uh column one and notice how it’s like not the same

three four five six so that’s not

five six so that’s not a mirror because those numbers don’t match so there’s no mirror in

directed graphs in terms of the matrix anyway let’s see i’m going to get rid of these little

pound signs these hashtags uh the next question is uh let’s do a self-connect how do we add an

edge like if i wanted to add an edge from uh you know one to seven for example actually let me

duplicate this first, and then we’ll work on it.

first and then we’ll work on it if i just wanted to add a brand new edge going from one to seven

uh then i just have to make sure that i look at the row one and then look at the seven column

and then just change that value to a one to say did i do that yeah to say that uh the edge now

exists right so that’s pretty easy to add a brand new edge let’s look at self connects now just in

in case um if i wanted to say that four connects to itself i have to do an arrow

4 connects to itself. I have to do an arrow by the way. Did I do an arrow on the last one?

Yeah, okay. 4, we’re saying now it connects to 4. So before we assumed that there were no

self-connections by default. So all the nodes had no self-connections. But the 4, we just added

a self-connection. So that means right here, that 0 is going to change to a 1. Pretty easy and fast

to add an edge. And by the way, how fast was it to just access a cell and modify a value?

modify a value it’s constant time again big O is going to happen in other videos but uh this

is lightning fast for adding and removing edges okay so then the other thing is um let me see

can I do oh I gotta get rid of that how do we remove an edge oh bro bro what’s happening here

just select that oh how do we remove an edge we talked about this in other videos

we talked about this in other videos for our adjacency matrix but it’s pretty easy we just say

all right suppose i just want to kill this edge going from three to two so now there’s

no connection no edge going from three to two um well i just have to go find the appropriate

part of the matrix and that’s it so row three because three was the start node or the source

node row three and then two is the destination node so i’m just gonna like overwrite that with

the zero and now we have effectively removed an edge remember the computer doesn’t actually

See you next time.

Remember, the computer doesn’t actually see this diagram that you’re looking at on the left.

The computer just has the matrix.

The computer doesn’t really even need to know that the 3 is on top visually.

It just needs to know what the 3 is actually connected to.

At least most applications of graphs.

There are some where it’s more visual, and so there needs to be some coordinates.

But just the easiest and most simple way right here with an adjacency matrix doesn’t need that.

Okay, so then the next thing I want to do is…

is um i want to add edges that kind of like point towards you know two nodes that sort of point

towards each other like you know opposing is this going to work yeah opposing uh edges that’s not a

technical term so suppose for the sake of argument that you’re looking at the three and you say well

the three can go outbound to the two but can the two go outbound to the three yeah totally

Okay, so I now have two edges, each going in a different direction between the 3 and the 2.

And by the way, in your mind, you might be thinking,

hey, isn’t that what undirected graphs are doing?

Yeah, basically, you have just an edge that goes in both directions.

Here we have to draw two separate edges.

Then we just have to update the matrix to match.

So I added an edge going from 3 to 2.

So I’m going to go row 3, and then column 2, and…

Oh, sorry, sorry.

originally which worked from three to two that’s why is that what that’s why

there’s a one there but now I just added an edge from two to three so we look at

row two and then column three and then I update that to match so that zero became

a one and it did become a little bit more like a mirror if we went ahead and

added these like you know opposing edges on every single pair of nodes then the

matrix would start to look like a mirror anyways what else can I talk about we’re

We’re at about 16 minutes.

about we’re at about 16 minutes and i’ve lost my mouse cursor what the heck oh there we go i need

i need accessibility controls you know it’s true actually i really need glasses i’ve needed glasses

for a couple of years maybe like five years and for some reason i just can’t get myself to go so

what’s happening now is when people show me memes on their phones my face goes it goes backwards

like this it’s like a it’s like a repellent for my face if you show me a meme i’m like

me a meme i’m like not not fun to live that way so i gotta get glasses should i get a chain i want

a gold chain that’s like studded and like blinged out and then like i have the glasses here only

when i need to read and then it hangs the rest of the time and it maybe it plays music or something

i don’t know i don’t think i’m gonna go that far so uh when we did undirected unweighted i’m just

trying to make sure that i didn’t forget anything there’s no cheesy broccoli in this video directed

and unweighted. Yeah, I guess that was it.

weighted yeah i guess that was it okay so we’re at the end of this particular video i’m going to do

one more before we shift topics the next one is going to be how to represent a directed and

weighted graph inside the machine um using an adjacency matrix okay thank you for watching

this video i hope you learned a little bit of stuff and had a little bit of fun i’ll see you

soon i’m gonna go have a burrito no i’m not i’m just gonna keep recording hey everybody

hey everybody thanks for watching this video again from the bottom of my heart i really

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