We start from the previous diagram and add weight costs on every single edge. The graph is a tuple consisting of a vertex list and an edge list. Vertices are listed from left to right. Edges are stored as triples with starting node, ending node, and weight.

We build the edge list by scanning nodes left to right so each undirected edge appears only once. Then we convert the representation to use indexes instead of node values. This lets you jump to any node in constant time when using a vector or array for the vertex list.

This makes operations on edges and nodes much faster than linear scans for both. Next up: directed graphs, both unweighted and weighted.

0:00 Introduction to Weighted Graphs
1:00 Adding Edge Weights
3:16 Creating the Vertex List
5:19 Building the Edge List
10:34 Completing the Value-Based List
10:44 Why Value-Based Is Slow
13:04 Converting to Index-Based
15:29 Benefits of Index Lookups
17:49 Wrap-Up and Next Videos
18:22 Thank You and Outro

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Hey there! Hello there. Hey there. In the last video we talked about representing

an unweighted and undirected graph using the edge list representation inside your

machine. Now let’s talk about how to do that exact same thing with undirected

weighted graphs.

seeing on the screen right now is basically the last diagram that I drew on my previous video

where we talked about how to represent unweighted undirected graphs in the machine. You don’t

necessarily need to watch that video to understand this video but you probably want to watch a few

of my other previous graph videos to know the basics of graphs and edge lists and edge-based

paths and node-based paths and all that stuff. So for now I’m just going to continue from this

document. So let’s see, I’m going to duplicate

I’m going to duplicate this real fast I am going to just kind of modify the captions here because

we’re not going to do unweighted anywhere we’re going to say a weighted undirected graph did I

get that backwards I think sometimes people say undirected and then the weighted part after so

I’m just going to I’m just going to change it so it’s still undirected which means the edges don’t

have arrows so that’s fine but it’s going to be weighted now which means I have to assign a cost

of traveling along every single edge.

So I am going to just

so i am going to just sort of like erase a bunch of this stuff real fast erase these notations

uh you know what i’m going to erase all of this because i want you to you know see it

from start to finish okay so now i’m going to add weight costs on every single edge

um in the last video what i mentioned is that the the edge list representation of a graph in

the machine uh is basically your graph is a tuple that’s the orange you know marking

a tuple that’s the orange you know markings up here your graph is a tuple where the tuple consists

of a vertex list and also an edge list so we’ll do that from scratch together again but um let

me get the weights on there first what color do you want i don’t know i’m going to put arbitrary

numbers on all the edges just to say that every edge has a cost remember in a graph when you

travel along an edge if the edge has a weight then you consider that weight to be part of the cost

usually of traveling there so like if this was a network diagram

traveling there so like if this was a network diagram maybe these weights represent pings or

lags or bandwidth cost or something like that if this is like an airport diagram maybe the weights

represent gasoline or time or something like that so i’m going to do like four there we can repeat

weights if we really want to i’m going to line this up a little bit better that 12 looks gross

okay i think we’re okay now i have 14 all right did i forget anything raise your hand if i forgot

something

raise your hand if i forgot something oh i forgot okay did you raise your hand okay

i’m going to do the 13 and 6 uh let’s just put a 2 weight there you could put ones everywhere

if you wanted to i mean any any number is fine um also in an unweighted graph you kind of can

consider the number of hops to be a path cost which means you can kind of consider every uh

edge to just have a weight of one but for now we’re just gonna say this is the weighted graph

Okay, so the first…

okay so the first thing we need to do uh after that is make a vertex list so our vertex list

is just going to be a list of all of our vertices all of our nodes in the graph

how do we list them so for starters like we usually say vertex list but for me personally

this this particular data structure just that we’re going to type up on the bottom

i usually like to consider that a vector or an array at least because you can jump into a

Thank you.

uh because you can jump into a specific index in constant time which is really really fast if you

don’t know uh big o and time complexities check out my other videos but um it’s just really really

fast to jump to an index when you’re using an array or a vector but it’s kind of slow when

you’re using an actual linked list or some other data structure that makes you scan through it so

i’m going to say that my vertex list is actually a vector but i’ll keep calling it a list anyway

so we got the vertices uh we have to start naming them okay so the first thing is i’m going to do

we got the three there and we got the 13 and we got the 6 and the 12 and the 15 i’m just naming

all of the nodes in my graph i like to go from left to right because that makes it easier for

me to debug what i’m doing um you probably in your code i said this in the last video you probably

don’t want to store the actual value in your vertex list you probably actually want to create

an object of type node or of type vertex and give it the template data

and give it the template data type of in this case integer and then set the 13 inside of the class

it’s not as fast and memory efficient as just putting a number there but uh you know for me

i like more flexibility and i like to be able to do more powerful things with my stuff so i’m going

to say that 13 and all of the other nodes really that represents to me i made a brand new instance

of a class called node or vertex and then i set it up however i wanted and one of the things i did

set it up, we’ll stick to number 30.

things i did to set it up we’ll stick the number 13 inside of the class but for the for the diagram

it’s fine like this okay so we have our vertex list that’s the v if you’re looking at the uh

you know the v comma e the v part now we need to do the edge list so i’m going to say here’s my

edge list and we’re going to use the edge list representation like i talked about in previous

videos, which is basically a tuple where one tuple describes one edge. So in our

So, in our graph that we have right now, we definitely need a starting point and an ending point.

So, let me just start by looking at the 3 node, and I want to describe one of the edges for the 3.

So, the 3 goes to the 18 and the 13 and the 12.

I’m going to scan my eyes from left to right to make it a little more simple.

So, the 3 node, it connects to the 13 node, so I’m going to put a 3, 13.

the nodes doesn’t matter in an undirected graph so i could have easily put 13 and 3 it’s fine

you don’t want to duplicate it by putting 13 comma 3 sorry you don’t want to put 3 comma 13 and then

later put 13 comma 3. you don’t want to double it up because that would be like representing two

separate edges i’m just going to sort it in my mind before so that i don’t have to sort it again

later when i’m reading the graph which makes it faster for read operations but so 3 goes to 13

And then we have to represent the weight, so the tuple gets a little more complicated.

so the tuple gets a little more complicated than the last video just by a little bit i’m going to

say 3 comma 13 comma the weight which is just a 1 here and i think i mentioned this in my other

videos like a while back but uh the weight doesn’t have to be integer it could be a float it could be

any data type you wanted really as long as your algorithms support it and you have you know your

operator overloads all done and all that could be a full class just for the weight if you wanted to

but for this video it’s just going to be the weight is an integer so that’s the cost of

traveling

is an integer so that’s the cost of traveling along that one edge so notice how all of our

tuples have three items each it’s a triple tuple so i’m going to continue trying to represent uh

what the three touches so the three uh it touches the 13 and it also touches the 12 so i’m going to

say 3 goes to 12. what does that edge cost it costs us six because that’s the little pink six

there we’re done with that tuple so what else does three touch uh touches the 13 and then it touches

the

touches the 13 and then it touches the uh 12 and then it touches the 18. so i’m going to put an 18

there what does that edge cost it costs a four because there’s a pink little four at the very

top of the graph surprise you didn’t know that okay so then uh i’m sorry we’re going to move on

to the next node so we did the three already so i’m done with that i’m going to look at the 13.

where does the 13 connect it connects to the three but that’s to the left

because i’m this is why i like to scan my eyes from left to right i already know stuff on the

left is handled i don’t have to do it again so forget about the three i already described it

up here in the very first tuple three comma thirteen comma one i don’t want to describe it

again so forget about the three now the thirteen connects to the six so thirteen comma six and what

does that cost it costed two for our arbitrary labeling thirteen doesn’t touch any other node

so we’re done with the thirteen row and by the way i’m doing a line new lines here you don’t have to

to do that, I just

here you don’t have to do that i just like to i like to do one you know row per per start node

to make it look cooler and nicer okay so the uh 13 is done now let’s do the six

six touches the 13 but remember in the previous row we just handled that so i’m not going to do

it again the six to the right touches three other nodes you can see there are three lines

protruding on the right side of the six so um i’m just going to take those from left to right

It’s.

I’m just going to take those from left to right.

It’s touching the 12 at a cost of 14.

The 6 is also touching the 15 at a cost of 12, I think.

And the 6, you know, if I make mistakes, just leave it in a comment.

I will leave the video up so you can always remember my humiliation.

But I’m going to try my best.

I did not practice this.

6 touches the 15, and then the 6 touches the 18.

So 6 touches 18, and then it costs 9.

So now we’re probably done with the 6.

we’re probably done with the six um moving on uh one node to the right now we’re going to look at

the 12 okay so the 12 it touches 3 and 13 and 6 but those are already handled so forget it

the 12 touches the 15 we did not handle that right yeah because we’re going from left to right

uh and and then it costs us from going from 12 what am i doing it doesn’t touch the 15 oh my god

so the 12 touches the

uh so the 12 touches the uh the three which is already handled and it touches the

six which is already handled and then it only goes to the right to the 18. okay

anyway you know when i make mistakes on camera and i say do over i always go i don’t know why

so the 12 and the 18 uh it’s going to cost us three so then we’re done with the 12th so i’m

I’m going to go to the next one. We’ll look at the 15. The 15. 6 is already handled. 8.

6 is already handled, 18 is not handled yet, so I’m going to put 18 there, and it cost us $88, and then the 18 is the last one.

There’s nothing to do, because every node is on the left of the 18.

If you carefully, if you pause the video and look at every single node, you’ll see it’s already represented in a previous node.

So we’re officially done with the basic edge list representation of this graph, this undirected but weighted graph.

We need to do something else, though.

we need to do something else though i mentioned in the last video i’m going to repeat it in case

you didn’t watch the last video but we need to make sure that uh oh i guess i forgot to separate

those two things that means i have to duplicate the vertex list over here on the right side too

okay um so on the left this is okay but it’s kind of slow because what if we’re trying to look at uh

you know one particular edge let’s say that we we want to see what’s going on with this uh this

edge right here. Let’s say the 3 and the 13 edge, for whatever reason.

13 edge for whatever reason um i don’t know green 3 and the 13 edge okay so we’re looking at that

edge maybe for some reason we had just finished scanning uh maybe that’s not a fun edge well

what’s going on hello uh oh i forgot to set up a thing where you can make your cursor really

visible okay um actually so so this is not the most fun edge let me let me see about the 15 and

and 18-inch.

18 edge excuse me i had to take a drink of water so we’re going to look at this edge connecting

the 15 and the 18 node that’s a little more interesting so you can imagine maybe you were

doing a little scan uh if you know big o complexity by now we had to check out every single edge in

this edge list until we found the one that connected the 15 and the 18 so this was a linear

time scan based on the number of edges so this is just it’s scalable but it’s not very fast right so

We ended up kind of looking at this one.

so we ended up kind of looking at this one right here not fast um but what if we want to examine

uh some properties of the nodes or the vertices uh involved like the 15 and the 18.

here in this representation on the left we have the values we have the 15 value and the 18 value

we don’t really have a pointer to a full node object if that’s what you’re doing in your code

so um how do we get those those node objects we’re gonna have to do another linear scan we’re gonna

I’m gonna have to look at this one right here.

to look at this one right here oh that doesn’t match the value of 15 look at this one look at

this one look at this one we’re gonna have to scan every single node until we realize that it’s those

two at the end this is true even if it’s a vector uh holding your vertices because if we don’t know

the index then we have to actually just sweep through the whole vector so that’s really slow

it’s still kind of scalable but uh it’s not the best instead let’s update the the representation

on the right to have index

on the right to have indexes we did this in the last video but basically if you just kind of look

at this three node here the three node well let me let me copy this one more time the three node

corresponds to index zero which i forgot to write down my bad let me write that down for you real

fast get rid of that get rid of that get rid of that okay so i’m just going to update this vertex

list to just show you that if we have a vector or in a

that if we have a vector or an array then every single item has an index here’s index one index

two index three and four and five and then i guess i’ll just copy paste that to the other one since

i was too dumb to make an extra separated graph there um separated item okay so the uh the 15

sorry no the 13 the 3 and the 13 i guess i got to get rid of that that’s distracting me

I guess I gotta get rid of that, that’s distracting me.

The three and the 13,

those correspond to indexes zero and one

in the vertex list, right?

So zero and one, that’s the three and the 13.

So I’m just gonna change the values,

the T-type values, to the raw indexes.

It’s gonna seem a little weird at first,

and I’m gonna leave the weight alone,

because that’s totally fine.

We’re not gonna jump to a weight,

unless maybe you had a weight list, a weight vector,

if you wanted to make instances of weights.

You could do that, I’m not gonna do it.

do that i’m not going to do it so then again the 3 is a 0 the 12 is index 3 the 3 is another 0

the 18 is index 5. i’ll just do this for every single row so the 13 here is index 1 and then

the 6 index 2 notice how i’m just looking up here at the indexes of the vertex list and i’m just

changing the node values to the indexes where the nodes actually are in the vertex list so that’s a

So two.

vertex list so that’s a 2 and then the 12 is going to be a 3 and then the 6 is also going to be a 2

and the 15 is going to be a 4 and then the 6 is going to be a 2 and 18 is going to be 5

on to the next row the 12 node is index 3 the 18 node is index 5 uh the 15 node is index 4 and the

18 node is index 5 so whoa whoa correct me if i made a typo in the comments please i’ll release

another video as a thank you.

i’ll release another video as a thank you but uh now i’ve converted the whole entire representation

on the right side to an index based uh uh you know edge list so i’m gonna do this one over

here is by value and the one on the right is by index why is the one on the right better now

remember at some point we might have been scanning to find an edge well what did i just do wrong oh i

clicked over there okay we might have been scanning to find you know some edge so we’re like

Like, subscribe, and hit the bell icon to be notified when I post a new video. Thanks for watching, and I’ll see you in the next one.

Bye-bye.

edge so we’re like scanning scanning scanning scanning and then maybe we decide we’re trying

to figure out what’s going on with the two and the not the two and the four the uh the six and

the fifteen node right so that’s uh the six and the fifteen node that’s basically you know this

right here uh so now what if we want to do something with the nodes we want to do something

with the six node do something with the 15 node we don’t want to scan through the vertex list again

we don’t want to go scan scan scan scan scan that’s linear time that’s not as fast as it could

Could be.

That’s linear time, that’s not as fast as it could be.

So instead, we’ve already converted here so that we have indexes.

So if I look at the 2 and the 4, the 2 and the 4 represent indexes, not node values.

So this 2 tells me that I can go to index 2 to get the 6 node,

and the 4 tells me I can go to index 4 to get the 15 node.

Notice how I did not have to scan the vertex list to figure out where those nodes are.

And if you’re using a vector for your vertex list,

list then you can jump to every index in constant time so now if we were going to you know search

and do a bunch of stuff on edges we probably only had to spend linear time on the edge list but then

we only spent constant time uh to to manipulate each node whereas it would have been linear plus

linear uh in the past like on the left side that would have been uh you know if we’re going to do

stuff with with both the edges and the nodes it would have been o of linear time based on the

number of edges if we’re gonna do every

on the number of edges if we’re going to do every you know something to every single edge

plus linear time based on number of vertices but over here it’s going to be linear time of the

edges plus just constant time which basically means it’s going to be you know big o of just

the number of edges and i’m talking about a you know a fantasy scenario where we’re just going

to scan a bunch of stuff you can imagine there’s a whole bunch of different operations you could

do on a graph, and it would be very advantageous and fast.

very advantageous and fast if you could just jump to a node if you wanted to be able to manipulate

it directly um let’s see so i did i think 20 minutes already so i think this is going to be

it for for today’s video um and i’m going to do another video after this where we start working

with directed graphs so i’m going to do directed unweighted and then directed weighted so by the

time you watch all four of these videos

by the time you watch all four of these videos you’ll know how to represent all four types of

graphs in the machine okay anyway so 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 next time tell your friends eat a chocolate

okay i’m outie hey everybody thanks for watching this video again from the bottom of my heart i

really appreciate it i do hope you did learn something and have some fun uh if you could do

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please, a small little favor. Could you please subscribe and follow this channel or these videos

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just wake up because i know somebody subscribed or followed it just wakes me up and i get filled

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watching this video and enjoy the cool music as as i fade into the darkness which is coming for us all

Teksting av Nicolai Winther

Thanks for watching!

Thank you.

The post How to Represent Undirected Weighted Graphs Using Edge Lists appeared first on NeuralLantern.com.

Hey there, let’s talk about paths inside of a graph. We draw a directed weighted graph and compare node-based paths to edge-based paths. Node lists alone cannot tell which edge you took when parallel edges have different costs. Edge-based paths use tuples of start, destination and weight so every route is clear.

We also cover simple paths with no repeated edges or vertices, cyclic paths that start and end on the same node, and how a cycle proves the graph is cyclic. Then we explain DAGs, directed acyclic graphs, with examples that stay acyclic and ones that gain cycles.

This sets up algorithms for later videos. Leave a comment with topics you want next. Thanks for watching and please subscribe.

00:00 Introduction to Graph Paths
00:28 Drawing a Sample Graph
02:25 What is a Path
02:42 Node-Based Paths
03:27 Adding Weights and Parallel Edges
04:34 Example Paths with Different Costs
07:19 Edge-Based Paths
07:35 Representing Paths with Tuples
11:35 Order Matters in Directed Graphs
12:52 Simple Paths
15:34 Cyclic Paths
18:22 Directed Acyclic Graphs DAGs
19:08 Building a DAG Example
21:41 Adding a Cycle to Break the DAG
23:15 Conclusion and Future Videos

=-=-=-=-=-=-=-=-=

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Hey there, let’s talk about paths inside of a graph.

Okay, so first thing that I’ll say is you probably should have watched my other videos already so that you understand at least the basics of graphs or maybe you already know and you just came here for a little extra.

So let’s just draw a sample graph here.

I’m going to do like random nodes.

Let’s see what am I supposed to do.

I’m supposed to like full screen that.

I’m going to draw some random nodes with some random values inside of them.

So I’m just going to do like a three here.

Maybe like a three like that.

And then I’m going to duplicate this a few times.

So I’m going to go.

This is a valid graph.

It’s just not a connected graph as we talked about in my last video.

So I’m just going to do this.

I guess that’s okay.

I’m going to change all these numbers.

Even though your graph should be able to support duplicate values in the nodes,

it’s going to be really hard to explain.

So I’m just going to change all this stuff.

9 here and then like 22.

I think that’s probably good.

So I’m going to do this.

And this.

Okay.

So still a valid graph, but there’s no chance that we could have a path anywhere in this

entire graph.

anywhere in this entire graph so that sucks we could by the way do a path

with a undirected graph but I kind of feel like I want to do direction on all

the edges so let’s see let’s do direction and also weight I’m gonna like draw an

arrow here and arrow here there I’m gonna draw edges first in a way that it

like I could find a path almost anywhere I want so I’m just going to like draw a bunch of edges

maybe like another edge over here nope that’s wrong how about this maybe another edge going up

here and an edge going up here and an edge going down there I’m just going to draw a whole bunch

of edges in the hopes that we’ll have a decent graph with lots of path possibilities okay I guess

Oh, no, no, this is not a complete graph.

I’ve just drawn a graph where you can kind of get everywhere.

Okay, so, well, I mean, what is a path?

It’s just sort of like, you know, some sort of a representation of how do you get from

one start node to some destination node, like a map.

So the first thing that we could do is we could say, let’s make a path based on nodes.

I’m going to say node-based paths.

Actually, let me get rid of this and put it on the next slide.

We’ll do this, duplicate it, that way I have the original.

Let’s pretend that our path is going to be just a list of nodes that we travel through.

So I’m going to say node-based and we’ll have one path, we’ll say path1 and then path2,

maybe we’ll do like a second path.

well at first we might think this is totally fine but uh later on it’ll get confusing in terms of

which edge did we actually take so let me show you let me show you what i mean i’m first going

to add weights to all the edges so i’m just going to do like you know four and a three over here

and then like a one and a two little one and then like a six and then like a four i’m just adding

random weights or random costs to all of the edges and um then eventually let’s see when i

let’s see when I finish doing all of that eventually we can probably realize that we

could add more than one edge going from the same start to the same destination node but give it a

different weight so that we have a little bit more of a choice so suppose for the sake of argument

that we have this six and the four here the six can go to the four at a cost of four but maybe the

a cost of one. I don’t know why it’s there. I don’t know why the four is there. Also,

we’ll just say this is our graph. So you can have edges like this that go to the same start

and destination nodes, but have different weights. So a path that is based on nodes

doesn’t really have enough information to tell us which edge we took. Let me show you what I mean.

So suppose we want to get from node, I don’t know, node three, I’ll put an S here for start.

for start. And then I guess I’ll put a D for destination or donut. I’ll put a D there on

node number 22. So we’ll say we’re going to go from node three to node 22. So start and destination

a path based, sorry, a node based path, path one, we could say maybe let’s go from the three

to the six to the four to the 12 to the nine to the 22 and that tells you all the notes that

you would hop on another path could be let’s see that first path when we went from three one six

four twelve nine uh 22 when we go from the six to the four we could have a second path here

that goes from three to one to six to four but takes a different edge this is kind of what i’m

this is kind of what i’m trying to show you notice that we have two different paths i’m going to do

one in uh maybe red so it’s going to be three one six four so from three to one and then six

and then we’ll travel along that four edge once we get to the four we’ll go 12 and then once we

Then the other path, path number two, let’s say that we chose the other edge.

We chose the six to the four edge that only cost us one.

So it would kind of look the same for the most part, except when we finally get to that

other node, we’re actually going to be going this way, right?

And then we’re going to go up and then to the right and then down.

So if you look at this carefully for a little while, maybe pause the video.

These are actually two totally different paths with two totally different costs.

different costs. So, um, you know, the cost of the orange, let me write it in orange.

It’s going to be five plus one. So six, and then a seven there, and then a eight, nine there. And

then, uh, then I guess like a 15 there and then a 16 there. So the cost of the orange pass is going

to be 16. If we switch to the red path, the only difference is the choice of edge we took between

we took between the six and the four node.

So we would increase our cost by three.

So the red path would be a cost of 19.

But how can you tell the difference in a path

that is based on a node?

It doesn’t have enough information, right?

So what we’re gonna do instead

is we’re gonna use edge-based paths.

Edge-based paths.

In some places, you know, you can use node-based paths.

That’s totally fine.

fine but for me I’m going to say let’s do edge based paths edge based paths and now we just have

to figure out how to add that extra information into the path so going from the three to the one

we’re actually traveling along an edge right we’re traveling along that five edge so let’s just stick

a tuple in there a tuple meaning just a little collection of values for those of you in c++ if

items we’re going to use three item tuples tuples not always limited to three items it could be two

three four even probably one um or much more so i’m going to say let’s use a tuple that has three

items per item or tuple and we’ll just say that we specify the start node and then the destination

node so we start at three we go to one and then the weight of the edge that we choose to travel

but we’re putting enough information into our hop

that this can be considered an edge-based path.

So when we go to the one node,

we ended on the one node, sorry, we’ve gone to the one node.

We just ended there.

So we could start on the one node for the next tuple

because then we’re going to go to the,

what are we doing first, red?

I think we’re doing red first, yeah.

We go from the one node to the six nodes.

So I’m going to go comma six.

and then we choose which edge to travel along that’s just the one edge and so 161 means start

at the one node and at the six node and take the one weighted edge then do the same thing

again for the next tuple so we’re at the six now where are we going to go we’re going to go to the

four and then we choose the edge that costs four and then for the next tuple we’re on the four

I’m going to say 12 is the destination and which edge did we choose the two even though there’s

only one choice at least we have enough information for all edges in our entire path so then again

we’re at the 12 we’re going to go to the 9 so I’m going to put a common 9 there and then we choose

to take the edge that uh what was that cost six I can’t read anymore um let me actually just look

real fast six I guess I overrode it I’m just going to assume that it’s a six um I’ve got

I guess I could erase it real fast

or move it or something

okay let’s see what is that

yeah it’s a 6 okay

so we’re on the 12 we’re going to go to the 9

and we choose to take the edge that costs 6

then we’re on the 9

and we choose to go to the 22

using the edge that costs 2

and then this last thing doesn’t need to be done

because we have the entirety of the path

in our edge based list

So I’m actually just going to copy paste this entire path and stick it into the second path

because remember the only difference between the first path and the second path that we

chose is when we were going from 6 to 4 we chose the 1 edge instead of the 4 edge.

So now look at this, I can just change that 4 to a 1 and now we definitely have two different

paths and because we’re using edges to describe our path we can tell the difference.

You know in a computer program we would know exactly which edge to travel along or even

which edge to travel along or even just a user moving their finger along a graph or whatever.

So in my opinion, edge-based paths are a little bit better.

They’re also better because if you practice writing down edge lists like this,

then you get a lot closer to a representation of a graph in your machine called an edge list.

So in an edge list, oops, I put ed light.

I don’t know why I did that.

And now the computer’s not responding.

So my mistake is immortalized.

my mistake is immortalized. Edge list representation of a graph is basically a graph is equal to

a tuple where one item in the tuple is a list of vertices and another item in the tuple

is a list of edges. So if you learn how to do edge-based paths, then you’re already a step

closer to being able to represent a graph inside of your computer program in the machine.

So another thing that I want to point out is that since this graph has direction,

the order of the start node and the destination node absolutely matters for example if we have

this three comma one here indicating we start at three and we end at one well that means there must

be an edge that points from three to one otherwise this is an invalid path if for some reason

if for some reason let me get rid of this here there’s too much stuff on the screen

three this is invalid because it’s saying that we have to start at the one node and end at the three

node but the one node has no has no edge going to the three node so this is totally invalid

does that make sense i hope it makes sense to you if this was an undirected graph where every single

edge did not have direction then the order actually wouldn’t even matter you would just

to see if an edge existed.

So let’s see, we talked about different paths.

You know, typically you would want to choose

the lowest cost path,

but there are algorithms that we can use

in future videos to compute paths.

Let’s see, something called a simple path

is where every single edge in the path is distinct.

So for example, I don’t know,

let’s say we had like a really gross path here.

Let me get rid of, let me get rid of some of this.

Let me get rid of some of this.

Get rid of the oranges maybe.

So that this is a little slightly easier to read.

Let’s see.

Did we choose?

Oh, I have to fix this.

Let’s fix the three and then the one.

And then we chose the four.

So this is the red path indeed.

So a simple path is, nope, I accidentally erased like an extra edge or something.

So I got to do one from the four to the 12.

Okay.

to the 12 okay let me just do that real fast from the 4 to the 12. at a cost of 2. okay so a simple

path is something where you don’t repeat edges so i can also write down simple path here simple path

if this was not a simple path then we would just basically have to repeat an edge somewhere

so um if any of the edges or vertices were repeated this would not be a simple path so let me just i

would not be a simple path so let me just I don’t know let me just update this real fast let’s say

we do oh what should we do let’s do I don’t know when we go from three to one to six to four

let’s go back from four to six and then we’ll take the other edge back to four and then we’ll

go up to 12 so this will be like um not a simple path if we say gosh I should have written this

this down before i drew it three to one and then one to six and then six to four and then four back

to six okay so i’m going to add another tuple so starting at four going back to six and taking

that three edge i think is what we’re looking at and then another tuple to get back when we go from

six to four and then take the um one cost edge and then once we’re at four then we can go four

cost of two and then 12 to nine at a cost of six and then nine to 22 at a cost of two okay so

i mean i’m just saying you can make a path where you just repeat a whole bunch of the edges and

nodes and you’re just like going in a circle forever like whatever you really want it’s not

necessarily going to be an efficient path or something that an algorithm should produce

but you can have a path like that maybe your character is just going crazy in the game and

because we repeat stuff on the previous slide.

Since we did not repeat anything, this is a simple path.

Okay, let’s see.

Something else that I wanna talk about is a cyclic path.

A cyclic path is basically a path

where you start and end on the same node.

Being able to find a cyclic path in a graph

is a way that you can prove that the graph is itself cyclic.

is itself cyclic. Let me try to get rid of this here. Yeah, there we go. Okay, so just

pretend for the sake of argument that I’m just kind of looking around and I’m trying to find,

you know, some path that proves I can start at the same node and end on that node without

repeating edges. So let’s see. Right, so I’m going to do take the three and go to the six.

to the six and then so i want to try if i start on the three then i want to try to find a way to get

back to the three again so i go from three to six and then i can go from six to four

and then if i go from four to six over here i’m not repeating that edge going from six to four i’m

using a new edge from four to six and then i can go from six to three again not actually repeating

means I have found a cycle in the graph. Let me just describe this path real fast in terms of

edges. So we started on three. Oops. We started on three and then we went to six and it cost us three.

And then once we were on six, we went to four. So I’m going to go from six to four. And we chose

the edge that cost us four. And then once we were on the four, we just went backwards to the six.

four going to six and we took the edge that cost three once we were on six we took the six to three

edge so six to three whoops six to three edge and that cost us three yeah did I do that right

when we go from the three to the six in the beginning I wrote down a three but I should

have written down a two hopefully you cringed on my behalf right because I was I think that red

on the left side so my brain was looking at the three but we were really taking the arrow that

points down from the three to the six so there’s a two there let me just double check one more time

six to four at cost of four four to six at cost of three six two three at cost of three

so we have started and ended uh on the same node and therefore we have found a cycle in this graph

which means this graph is cyclic what is cyclic just means it’s a graph that has a cycle

It’s a graph that has a cycle anywhere inside of it.

So this is a cyclic graph.

Make sense?

Okay, why is this important?

Well, there’s kind of like a special type of graph called a DAG,

which means a directed…

Here, let me write this down real fast.

On the next slide, so that we don’t get confused.

On the next slide, I’m going to say we have a DAG

is basically a directed…

directed, acyclic graph. What does A mean when it’s in front of something? It just means it’s

not that thing. So acyclic means not cyclic and then graph. So in some algorithms you are required

to, you know, like have a DAG before you can even use the algorithm or if you’re being tested

somewhere. So let’s just start by, you know, drawing like a little bit of a graph and we’ll

have a graph that qualifies. So I have a little graph here. If it was empty,

I don’t know if you could call it a DAG or not. I think I would because it’s empty. It could be

directed. So we have like a graph here. I would call this a DAG because it’s a graph. We don’t

have any edges, but like I could assume that this would be a directed graph. But just to make this

a better example, let’s just say we have like two nodes here. And you could also probably assume

DAG because you could assume that there are no edges in this graph but if we did add an edge

a second from now then it would be directed not the best example but so let’s just add an edge here

oh actually let’s add an edge that’s undirected real fast from from two to six and then maybe

I’ll just kind of duplicate some of these we’ll do a couple more so I can just make this more

so ask yourself is this a DAG directed acyclic well there are no cycles that I can see in the

graph there’s no paths that you can find to kind of like end where you started but this is a it is

a graph and I don’t see any cycles but it’s not directed because these edges don’t have direction

so that means right away we have to upgrade every single edge to have direction so I guess I was

going to draw a bunch more edges and now I’m just going to only draw this one edge

So this is definitely a DAG if we just sort of do it like this, maybe another edge like this,

because I don’t know about you, but I cannot find any path that allows me to start and end

on the same node. Let’s just double check it. We’ll do the two. Can we find our way back to

the two? No, because there are no incoming edges to the two. All right, that’s good. Let me just

do a check here. What about the six? If we go out to the six, from the six to the four to the

6 from the 6 to the 4 to the 11 it would have been dangerous if we could get to the 2 but we

can’t even get to the 2 so that’s fine that means the 6 is okay we cannot find our way back there

the 4 we can go to the 11 but we can’t go any further we’re at a dead end we can’t even go to

the 6 that means the 4 can’t find its way back to itself where you can’t find a path that goes back

the 11 has only incoming edges so that’s automatically you know a dead end so all of

nodes qualify as we could not start and also end at that node with a single path so that’s good

that means this is a DAG a directed acyclic graph if you want to make this into not a DAG just figure

out how to add a path so that you end up with a cycle so what was the first thing we did the 2

goes to the 6 goes to the 4 goes to 11 then it’s a dead end we could either add an edge from 11 to 2

to make a cycle but that’s a little bit too easy right because if you see any pair of nodes

right because if you see any pair of nodes with arrows kind of pointing back and forth between the

two nodes then right away that alone is a cycle so I’m just going to maybe do an arrow or an edge

pointing from four to two then it doesn’t look so much like a cycle although it’s not too hard

to realize that there is a cycle in this graph in this graph so this is not a DAG let me show you

and then from four to two.

And therefore we would have a cycle.

So this whole graph is not considered acyclic.

This graph has a cycle.

And if we want to do an edge-based path,

we don’t have weights on this graph right now.

So we could describe the path as such.

We could say, start at two, go to six.

And if your edge list is of a graph

that doesn’t even have weight at all,

then you don’t really need that third item in the tuple.

item in the tuple you can just do a start node and a destination node this does kind of preclude

you from having parallel edges you know two different edges that point from the same start

node to the same end node so as i go from two to six and then from six to uh four and then from

four to two and then we have a cycle and that’s it okay i think this is all i really wanted to

talk about for this particular video in future videos we’re going to do algorithms and then

some others up already that have graph properties and connectedness and definitions and things like

that. So leave a comment asking about some algorithm or some property of a graph that

you might be interested in. Otherwise, thank you so much for watching this video. I hope you learned

a little bit of stuff. And I hope you had a little bit of fun. I’ll see you in the next video.

Yeah, I’m gonna go eat some cookies. No, I shouldn’t do that. But I’m still going to.

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