deven367/babylm-100M-qed · Datasets at Hugging Face

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Kruskal's gonna just throw out the desire to have a connected subgraph at each step of the iteration.
Kruskal's algorithm will be totally content to grow a tree in parallel with lots of simultaneous little pieces, only having them coalesce at the very end of the algorithm.
So in Prim's algorithm, while we were only allowed to pick the cheapest edge subject to this constraint of spanning some new vertex.
In
Kruskal's algorithm we're just going to pick the cheapest edge that we haven't
looked at yet.
Now, there is an issue, of course, we wanna construct a spanning tree at the end.
So, we certainly don't wanna create any cycles, so we'll skip over edges that will create cycles.
But other than that constraint, we'll just look at the cheapest edge next in Kruskal's algorithm and pick it if there is no cycles.
So let's look at this five vertex example.
Again, there is no starting point.
We're just gonna look at the cheapest edge overall.
So that's obviously this unit cost edge and we're gonna include that in our tree.
Right?
Why not?
Why not pick the cheapest edge?
It's a greedy algorithm.
So what do we do next?
Well, now we have this edge of cost two, that looks good, so let's go ahead and pick that one.
Cool.
Notice these two edges are totally disjoint.
Kay.' So we are not maintaining a connectivity of our subgraph at each iteration of Kruskal's algorithm.
Now, it just so happens that when we look at the next edge, the edge of cost three, we will fuse together the two disjoint pieces that we had previously.
Now, we happen to have one connected piece.
Now, here's where it gets interesting.
When we look at the next edge, the edge of cost four, we notice that we're not allowed to pick the edge of cost four.
Why?
Well, that would create a triangle with the edges of costs two and three, and that of course is a no-no.
We want to span the tree at the end of the day, so we can't have any cycles.
So we skip over the four because we have no choice, we can't pick it, we move on to the five and the five is fine.
So when we pick the edge of cost five, there's no cycles, we go ahead and include it.
And now we have a spanning tree and we stop or if you prefer, you could think of it that we do, we do consider the edge of cost six.
That would create a triangle with the edges of costs three and five, so we skip the six.
And then, for completeness, we think about considering the seven, but that would form a triangle with the edges of costs one and five, so we skip that.
So after this single scan through the edges in assorted order, we find ourselves with these four pink edges.
In this case, it's a spanning tree and as we'll see, not just in this graph but in every graph it's actually the minimum cost spanning tree.
So, with the intuition hopefully solidly in place, I don't think the following pseudocode will surprise you.
We want to get away with a single scan through the edges in short order.
So, obviously in the preprocessing step, we want to take the unsorted array of edges and sort them by edge cost.
To keep the notation and the pseudocode simple, let me just, for the purposes of the algorithm, description only, rename the edges one, two, three, four, all the way up to m conforming to this sorted order, right?
So, the algorithm's just gonna scan through the edges in this newly found sorted order.
So we're gonna call the tree in progress capital T, like we did in Prim's algorithm.
And now, we're just gonna zip through the edges once in sorted order.
And we take an edge, unless it's obviously a bad idea.
And here a bad idea means it creates a cycle, that's a no-no, but as long as there's no cycle, we go ahead and include the edge.
And that's it, after you finish up the for loop you just return the tree capital T. It's easy to imagine various optimizations that you could do.
So for example, once you've added enough edges to have a spanning tree.
So n - one edges, where n is the number of vertices, you can get ahead, go ahead and abort this for loop and terminate the algorithm.
But let's just keep things simple and analyze this three line ver sion of Kruskal's algorithm in this lecture.
So just like in our discussion of Primm's algorithm, what I wanna do is first just focus on why is Kruskal's algorithm correct?
Why does it output a spanning tree at all?
And then, the spanning tree that it outputs?
Why on earth should it have the minimum cost amongst all spanning trees?
That's when we're, once we're convinced of the correctness, we'll move on to a naive running time implementation and then finally, a fast implementation using suitable data structures.
[sound].
Stanford University. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; We have one of the most unique charter school contexts in California.
In that, we serve one of the highest indices of
language learners which is upwards of 80% of our student population coupled with students in poverty who are 95% of our students qualify for our free or reduced lunch. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; Two months ago an e-mail came to me from the
Assistant to the Secretary of Education Arne Duncan asking, do any of us knows any teachers who are true heroes, who would during teacher appreciation week deserve a call directly from the Secretary commending them on the amazing work they do on behalf of America's children. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; In my eleven years working in high-need urban areas I have never encountered an educator who exhibits the type of dedication and passion and commitment that Misla Barco brings to our school on a daily basis.
She is a true hero in every sense of the word. &amp;gt;&amp;gt; I love teaching.
I think I was born to be a teacher.
[foreign]
[laugh]
I'm happy for the school because it's a great recognition for the program.
It's a great recognition for the work that stands for the students supporting the high school so I'm very happy.
[foreign]
For our students to graduate from high school is a huge step because their parents, they didn't even go to elementary school.
We are changing people's lives. &amp;gt;&amp;gt; She is one of a kind.
She is a really great teacher.
She is like what we call our big mama. &amp;gt;&amp;gt; I think this class is very helpful because I just came from Mexico few years ago. &amp;gt;&amp;gt; Her teaching methods are
like unique.
She could teach us anything in a fun way and we'll get it faster than other classes.
[clapping] &amp;gt;&amp;gt; Teaching in a intense charter school setting can almost be like dog years for a teacher.
So spending four years here can be like spending ten years at uh, you know school that exhibits less challenges and it's not quite as intense. &amp;gt;&amp;gt; [foreign]. &amp;gt;&amp;gt; So, Misla being the longest of any veteran in our school having spent nine years here, I think that alone kind of encapsulates her dedication to our school's mission as well as our young people. &gt;&gt; [foreign] &gt;&gt; For more please visit us at stanford.edu.
The President:
Hello, Buckeyes!
(applause)
Yes.
It is good to be back at The Ohio State University.
(applause)
I want to thank --
Audience Member:
I love you!
The President:
I love you back.
(applause)
I am thrilled to be here.
I want to thank a couple of people.
First of all, the outstanding Mayor of Columbus,
Michael Coleman, is here.
(applause)
I want to thank OSU Provost Joe Alutto.
(applause)
And I just got this extraordinary tour from
Giorgio Rizzoni, who's the director of the Center for
Automotive Research.
So give him a big round of applause.
(applause)
Now, let's face it, a presidential visit isn't even close to being the biggest thing this weekend on campus.
(laughter)

Kruskal's gonna just throw out the desire to have a connected subgraph at each step of the iteration.

Kruskal's algorithm will be totally content to grow a tree in parallel with lots of simultaneous little pieces, only having them coalesce at the very end of the algorithm.

So in Prim's algorithm, while we were only allowed to pick the cheapest edge subject to this constraint of spanning some new vertex.

In

Kruskal's algorithm we're just going to pick the cheapest edge that we haven't

looked at yet.

Now, there is an issue, of course, we wanna construct a spanning tree at the end.

So, we certainly don't wanna create any cycles, so we'll skip over edges that will create cycles.

But other than that constraint, we'll just look at the cheapest edge next in Kruskal's algorithm and pick it if there is no cycles.

So let's look at this five vertex example.

Again, there is no starting point.

We're just gonna look at the cheapest edge overall.

So that's obviously this unit cost edge and we're gonna include that in our tree.

Right?

Why not?

Why not pick the cheapest edge?

It's a greedy algorithm.

So what do we do next?

Well, now we have this edge of cost two, that looks good, so let's go ahead and pick that one.

Cool.

Notice these two edges are totally disjoint.

Kay.' So we are not maintaining a connectivity of our subgraph at each iteration of Kruskal's algorithm.

Now, it just so happens that when we look at the next edge, the edge of cost three, we will fuse together the two disjoint pieces that we had previously.

Now, we happen to have one connected piece.

Now, here's where it gets interesting.

When we look at the next edge, the edge of cost four, we notice that we're not allowed to pick the edge of cost four.

Why?

Well, that would create a triangle with the edges of costs two and three, and that of course is a no-no.

We want to span the tree at the end of the day, so we can't have any cycles.

So we skip over the four because we have no choice, we can't pick it, we move on to the five and the five is fine.

So when we pick the edge of cost five, there's no cycles, we go ahead and include it.

And now we have a spanning tree and we stop or if you prefer, you could think of it that we do, we do consider the edge of cost six.

That would create a triangle with the edges of costs three and five, so we skip the six.

And then, for completeness, we think about considering the seven, but that would form a triangle with the edges of costs one and five, so we skip that.

So after this single scan through the edges in assorted order, we find ourselves with these four pink edges.

In this case, it's a spanning tree and as we'll see, not just in this graph but in every graph it's actually the minimum cost spanning tree.

So, with the intuition hopefully solidly in place, I don't think the following pseudocode will surprise you.

We want to get away with a single scan through the edges in short order.

So, obviously in the preprocessing step, we want to take the unsorted array of edges and sort them by edge cost.

To keep the notation and the pseudocode simple, let me just, for the purposes of the algorithm, description only, rename the edges one, two, three, four, all the way up to m conforming to this sorted order, right?

So, the algorithm's just gonna scan through the edges in this newly found sorted order.

So we're gonna call the tree in progress capital T, like we did in Prim's algorithm.

And now, we're just gonna zip through the edges once in sorted order.

And we take an edge, unless it's obviously a bad idea.

And here a bad idea means it creates a cycle, that's a no-no, but as long as there's no cycle, we go ahead and include the edge.

And that's it, after you finish up the for loop you just return the tree capital T. It's easy to imagine various optimizations that you could do.

So for example, once you've added enough edges to have a spanning tree.

So n - one edges, where n is the number of vertices, you can get ahead, go ahead and abort this for loop and terminate the algorithm.

But let's just keep things simple and analyze this three line ver sion of Kruskal's algorithm in this lecture.

So just like in our discussion of Primm's algorithm, what I wanna do is first just focus on why is Kruskal's algorithm correct?

Why does it output a spanning tree at all?

And then, the spanning tree that it outputs?

Why on earth should it have the minimum cost amongst all spanning trees?

That's when we're, once we're convinced of the correctness, we'll move on to a naive running time implementation and then finally, a fast implementation using suitable data structures.

[sound].

Stanford University. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; We have one of the most unique charter school contexts in California.

In that, we serve one of the highest indices of

language learners which is upwards of 80% of our student population coupled with students in poverty who are 95% of our students qualify for our free or reduced lunch. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; Two months ago an e-mail came to me from the

Assistant to the Secretary of Education Arne Duncan asking, do any of us knows any teachers who are true heroes, who would during teacher appreciation week deserve a call directly from the Secretary commending them on the amazing work they do on behalf of America's children. &amp;gt;&amp;gt; [foreign] &amp;gt;&amp;gt; In my eleven years working in high-need urban areas I have never encountered an educator who exhibits the type of dedication and passion and commitment that Misla Barco brings to our school on a daily basis.

She is a true hero in every sense of the word. &amp;gt;&amp;gt; I love teaching.

I think I was born to be a teacher.

[foreign]

[laugh]

I'm happy for the school because it's a great recognition for the program.

It's a great recognition for the work that stands for the students supporting the high school so I'm very happy.

[foreign]

For our students to graduate from high school is a huge step because their parents, they didn't even go to elementary school.

We are changing people's lives. &amp;gt;&amp;gt; She is one of a kind.

She is a really great teacher.

She is like what we call our big mama. &amp;gt;&amp;gt; I think this class is very helpful because I just came from Mexico few years ago. &amp;gt;&amp;gt; Her teaching methods are

like unique.

She could teach us anything in a fun way and we'll get it faster than other classes.

[clapping] &amp;gt;&amp;gt; Teaching in a intense charter school setting can almost be like dog years for a teacher.

So spending four years here can be like spending ten years at uh, you know school that exhibits less challenges and it's not quite as intense. &amp;gt;&amp;gt; [foreign]. &amp;gt;&amp;gt; So, Misla being the longest of any veteran in our school having spent nine years here, I think that alone kind of encapsulates her dedication to our school's mission as well as our young people. &gt;&gt; [foreign] &gt;&gt; For more please visit us at stanford.edu.

The President:

Hello, Buckeyes!

(applause)

Yes.

It is good to be back at The Ohio State University.

(applause)

I want to thank --

Audience Member:

I love you!

The President:

I love you back.

(applause)

I am thrilled to be here.

I want to thank a couple of people.

First of all, the outstanding Mayor of Columbus,

Michael Coleman, is here.

(applause)

I want to thank OSU Provost Joe Alutto.

(applause)

And I just got this extraordinary tour from

Giorgio Rizzoni, who's the director of the Center for

Automotive Research.

So give him a big round of applause.

(applause)

Now, let's face it, a presidential visit isn't even close to being the biggest thing this weekend on campus.

(laughter)