On the shortest path.

The shortest path from I to K followed by the shortest path from K to J or the shortest path from I to J whichever those two things are shortest, is the shortest path from I to J possibly using K and things less than K.

Any shortest path is going to be a simple path. Right. Why?

That is also on the shortest path, I see.

At this point, even though we have a shortest hop path to B, we don't have a shortest link path to B.

But the pairwise shortest path problem is of independent interest.

So if the shortest path actually had on it a repeat, it was not simple path, then it can't be a shortest path, there must be a shorter path still, so that's a contradiction.

As you can see, the shortest path will lead along here.

And for breadth first search, optimal would mean finding the shortest path.

If you think it's guaranteed to find the shortest path, check here.

Male narrator So we move on, we look for another shortest path.

That the shortest path between them is no more than K steps.

The sum of results, so all the hubs' shortest path distance is stored?

We had talked about shortest paths in a graph, but now let's extend this notion to the shortest weighted path in our graph.

And you can see that it doesn't necessarily find the shortest path of all.

To Shortest

One is that up to this point, our shortest path has been the shortest number of hops, and there's a one hop path from A to B, which also happens that it's actually fairly expensive.

So all pairs we would like to know the distance the length of the shortest path.

So this is really the key step. It is finding the distance of the shortest path.

Well, if you're doing routing in a communication network, that piggybacks on classical shortest path algorithms.

A value function associates to each grid cell the length of the shortest path to the goal.

Now, given that we can execute a shortest path from any given node, then m logn time.

It's called dynamic programming, and just like A star, it's going to find you the shortest path.

Finding the shortest path in a weighted, undirected connected graph took time m times the logarithm of n.

Resize Widgets to Shortest

The big question now is can we write a program that finds the shortest path from start to goal?

So now we pull it off, do the goal test, and now we found our path to the goal, and it is, in fact, the shortest possible path.

The length of the shortest path from I to K using only those less than K, we have that already.

All we know from what we've done so far is that the shortest path from A to D is four.

So, in this particular case, the code here is only going to tell us the length of the shortest path.

Dijkstra is the name of the individual who first described and analyzed this algorithm for a single source shortest path.

So next, we're going to extend this idea of finding a path from V₁ to V₂ to the idea of finding a shortest path from V₁ to all other nodes.

Great geniuses have the shortest biographies.

It's the shortest route to Boston.

It's the shortest route to Paris.

Shortest route to the prison... March!

Andrew Goldberg described a data structure that can be used to very rapidly compute shortest path distances between nodes in a network.

It's one of the shortest paths from a to g, but it's by no means the only path from a to g.

This could also be infinity which indicates that there is no bound on the length of the shortest path it's infinitely large.

We're going to write now focus on a particular one which is the average shortest path length to all the other nodes.

So to find this quantity, we're going to need to calculate the shortest path length from V₁ to all the other nodes.

I wonder which way is the shortest.

Actually, the shortest one is one month.

We can get to B faster than 15. We can get to B in 13. Now, is that the shortest possible path for B?

The shortest path from I to K only using nodes that are less than K then we follow another path now from K to J only using nodes whose numbers are less than K.