So in this case, there's a cycle, zero to five to four to six back to zero, and this

other cycle's two, zero, one, three, two there are two, four, six those are all

cycles. So how hard is it to find a cycle in a

graph in these categorizations well that one, It's very simple.

This one, maybe any programmer could do. Maybe.

You have to have the graph representation, But you have to use DFS.

Well, you could figure out a way to do it without, probably,

But anyway it's really simple with DFS. You, you don't even need to hire an expert

for finding a cycle. Alright here's a classic graph processing

problem that dates back to the eighteenth century.

So it's this town in Konigsberg in Prussia at the time where there's an island,

And the river kinda comes in and branches around the island and then goes out in two

branches. And there's a bunch of bridges, five

bridges onto the island, two from the banks and one across, to the this third

peninsula and then there's two bridges crossing that way,

So a total of seven bridges one, two, three, four, five, six, seven.

And Euler who's a famous mathematician would go out on Sunday stroll in this

place and came up with the idea. You know could anyone find a way to go on

a Sunday stroll and cross each one of these bridges exactly once.

So that's, often talked of as, the, original graph processing problem.

So, in terms of graphs, it's is there a cycle that uses every edge exactly once.

Given a graph, is there a cycle that uses every edge exactly once?

A and actually, a Euler, proved, a let's say first theorem in graph theory, a if

its connected, and all the vertices have even degree, you can always do it.

In this case you can't because there's a vertex with odd degree.

So that's, that's the answer to the existence, is there a cycle.

But suppose you wanted to find the cycle. So you can go ahead and check the degree

of every vertex. We looked at easy code for that to know

that there exists a cycle but how about finding one that uses every edge exactly

once. So in this case, here's the cycle that

uses every edge exactly once. And this graph every vertex has even

degree, and if you go zero, one, two, three, four, two, zero, six, four, five,

zero you get to every edge exactly once. That's an Eulerian Cycle. so how about

that one? Is that any programer a or do you have to

hire an expert, or is it impossible? Well this one we have it listed as a

typical, diligent algorithm student can do it.

But it's a, it's a, a bit of a challenge. It's a, an interesting program and again

once you get through the bipartite graph on you can think about this one.

A it makes some sense what the algorithm does, but it might take you a few tries to

get the code debugged, Let's say then you'll find code for it, it

looks like. Alright.

So that's a Eulerian Cycle What about if you want to visit every vertex exactly

once? So, you don't care about going over all

the edges, you just want to get to all the places.

That's called the, in this case there is a way.

For this graph zero, five, three, four, six, two, one, zero.

So that's a way to get to every vertex exactly once.

This is sometimes called the traveling salesperson problem on graphs.

If the sales, traveling salesperson has to get to every city and wants to just go

there once without retracing steps. So how about that one?

The, every edge is more to visit, it might seem more challenging.

And, actually, maybe if you have any experience with this, you realize that

this one is intractable. That's called the Hamiltonian Cycle

problem, and it's a classical NP-Complete problem.

We'll be talking about NP-Complete problems at the end of the course, but

basically the idea is that nobody knows an efficient solution to this problem for

large graphs. And it's a frustrating situation that

we'll talk about. But, you definitely not gonna, solve it by

just being a diligent algorithm student a and not even hiring an expert will get it

solved, no matter how much the expert charges.

So the intuition on finding a cycle that visits every edge once.

Yeah, you could do it. Find a cycle that visits every vertex

once, probably not. That's the kind of challenge that we face

when addressing applications of graph processing.

Here's another example. Problem is, given two graphs you want to

know are they identical except for the way that we need the vertex.

So here's an example of, a these two graphs don't look all that identical, at

all. But if you, take zero here and rename it

four and one and rename it three and like that,

Then sorry zero here and name it four and one and rename it three and like that.

Then you'll see that they are the same graph.

They represent the same connections. And in so many applications where, maybe

the vertex names are, are a bit arbitrary or you just wanna know.

Really the interest is in the structure of the connections, you might wanna know if

just the way that I name the vertex makes the graph different?

Or if I have two classes that have two different kinds of interactions, is it the

same interactions that's independent of the people or scientific experiment

studying a property of the universe or whatever, you might wanna know, is that

connection structure the same or not? That's called the Graph Isomorphism

Problem. How difficult you think that one is?

So, you know, you could, there you can try all possible ways of renaming the

vertices, But there's really a lot of ways, in

factorial ways. Way too many to try for a huge graph.

Is there efficient way to do it, Or is it intractable like the Hamiltonian

Path Problem. Where it's in this category that nobody

knows an efficient algorithm for but there could be one.

Actually for Graph Isomorphism, that's one that's stumped mathematicians and computer

scientists for many years. Nobody knows even how to classify this

problem. We don't know if it's easy or if it's in a

class of problems that are, we don't know how to solve but there could be a

solution. We can't show that it's impossible or

guaranteed to be difficult. Nobody knows how to classify this problem.

Again, pointing out that, even for a relatively simple problem to state.

The state of our knowledge and understanding the properties of algorithms

to solve such problems, is, it's incomplete for sure.

So one last one, here's a graph processing challenge.

So this graph, when it's laid out it's got two edges that cross between, between

three and four and zero and five. And in general if you have a graph that

you've got, so say the social networking graph of a small class, you want to study

that graph and look at it, you want to draw it on the plane and maybe you don't

want, you want to do it without having edges crossed.

So in this case there is a way to place the vertices in the plain so that when you

draw the edges no two of them cross. So, how difficult is that problem, even is

it possible to do or not. So, that's a classic problem in

graph-processing that, came up, from, the first time that people were ever sending

graphs in computers. And the answer to this one is also,

interesting to contemplate. There's a linear time algorithm known for

this based on DFS. So that means the running times, you could

run it on huge graphs. You could know, whether or not you could

lay it out in the plane. It was discovered, by Tarjan in the

1970's. And you heard that name come up again, in

graph processing. Based on DFS,

But, if you really wanna get this done, you need to hire an expert, 'cause that is

quite a complex algorithm, And probably, beyond, what a diligent

algorithm student, or a professor, might accomplish.