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Из курса от партнера University of California San Diego

Algorithms on Graphs

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University of California San Diego

Algorithms on Graphs

951 оценки

Курс 3 из 6 — Specialization Data Structures and Algorithms

If you have ever used a navigation service to find optimal route and estimate time to destination, you've used algorithms on graphs. Graphs arise in various real-world situations as there are road networks, computer networks and, most recently, social networks! If you're looking for the fastest time to get to work, cheapest way to connect set of computers into a network or efficient algorithm to automatically find communities and opinion leaders in Facebook, you're going to work with graphs and algorithms on graphs. In this course, you will first learn what a graph is and what are some of the most important properties. Then you'll learn several ways to traverse graphs and how you can do useful things while traversing the graph in some order. We will then talk about shortest paths algorithms — from the basic ones to those which open door for 1000000 times faster algorithms used in Google Maps and other navigational services. You will use these algorithms if you choose to work on our Fast Shortest Routes industrial capstone project. We will finish with minimum spanning trees which are used to plan road, telephone and computer networks and also find applications in clustering and approximate algorithms.

Из урока

Paths in Graphs 2

This week we continue to study Shortest Paths in Graphs. You will learn Dijkstra's Algorithm which can be applied to find the shortest route home from work. You will also learn Bellman-Ford's algorithm which can unexpectedly be applied to choose the optimal way of exchanging currencies. By the end you will be able to find shortest paths efficiently in any Graph.

Fastest Route6:41

Naive Algorithm10:46

Dijkstra's Algorithm: Intuition and Example7:52

Dijkstra's Algorithm: Implementation3:39

Dijkstra's Algorithm: Proof of Correctness4:51

Dijkstra's Algorithm: Running Time7:15

Познакомьтесь с преподавателями

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering

Hi, in this lecture, you will learn the algorithm for finding fastest routes,

which is used, for example, when you open your navigation app to get home from

work faster or by logistics companies when they want to deliver

the goods to their customers in time, but use less cars and less people.

So let's first state the problem.

It's really easy to do.

For example, what is the fastest route to get home from work, or

more generally, what is the fastest route from point A to point B right now?

And below, you see a screenshot from a popular navigation app,

Yonix Navigator, which is given a task of getting the fastest

route from the point you are currently in, which is marked by a circle

with the letter y in it, to the finish, which is your destination.

And it suggests to you a few variants.

So the default suggestion is the fastest route, but there are two others.

And there may be, for example, a route which uses less turns or

the route which is easiest for a novice driver.

And sometimes, there can be a shorter route,

which is not the fastest one, but shortest one in terms of distance.

So in this problem, we can represent the city or the country as a graph,

where nodes are some positions in the country.

For example, the crossroads and

the edges are the roads that connect those crossroads.

And there are some weights on those edges and

the weight on an edge is, in this case, the time in seconds or

in minutes you need to get from start of the edge to the end of the edge.

And what you need to find is the shortest path in this graph,

but not shortest path in the graph in terms of the number of edges,

shortest path in terms of the sum of the weights of those edges.

So if you want to get from A to B in the smallest amount of time possible,

then you get those by sum paths with sum edges and

sum weights on those edges corresponding to the time.

And you just sum up those weights of the edges, and

you get the total time it will take you to get from A to B.

You want the fastest route?

So you need the shortest path in terms of the sum of the weights of

edges on this path.

Let's see now,

does the breadth-first search algorithm from the previous lesson maybe help us?

So let's look at this graph, here,

we see that node A has a direct edge from A to point B, where you need to get.

And so, what our breadth-first search would say is that you already know

the optimal path from A to B because there is a direct edge, and so there is

no point in getting to some other nodes, just go directly from A to B, right?

Well, this doesn't work, for example, in this case,

let's suppose that it takes 5 hours to get from A to B, and

the corresponding weight of the edge from A to B is 5.

And let's suppose there is another node, C,

such that there is an edge from A to C, which takes 2 hours to go through it.

And another edge from C to B, which also takes 2 hours to get through it.

And of course, this looks a little bit strange, but

if there is a traffic jam on the road from A to B, and roads from A to C and

from C to B are free roads, where there is no traffic jam, then this can happen.

And so, then we see that going from A to B through C will take just 4 hours,

while going directly from A to B takes 5 hours, and so

our breadth-first search algorithm doesn't work.

So maybe it's vice-versa, maybe it's always better to work around,

to go around some node, not to go directly.

But that's also not the case, because if the edge from A to B is 3 hours,

and edges from A to C and from C to B didn't change, and

we just removed the traffic jam from A to B and it became 3 hours instead of 5,

it is now better to go directly from A to B.

So there is now universal rule to go directly or not to go directly.

We need something more clever than that to solve our shortest or

fastest through the problem.

Now let's gain some intuition about this problem.

So let's look at this graph below.

And assume that we stay at origin node S.

And we only know that the edge from S to B is going to take 3 hours and

from S to C is going to take 5 hours.

So can we be sure that the distance from S to C is equal to 5,

that it will take 5 hours for the optimal route from S to C, okay?

So no, this is not the case, because for example,

the edge from B to C can have weight 1, and

so, the shortest route from S to C will be from S to B and

from B to C, which is only 4 hours instead of 5 hours if we go directly.

So we cannot be sure that the fastest route from S to C is 5 hours.

Now, another question is can we be sure that the distance from S to

B is equal to 3?

And in this case, the answer is yes,

because if we don't go directly from S to B, what other options do we have?

We can go only from S to C and then go by some other edges to come to B,

but if we go from S to C, we spent already 5 hours.

And after that, we also spend some time to get to B.

And the time is now negative.

So we cannot spend less than 3 hours time in the end.

We actually cannot spend less than 5 hours in the end.

So going from S to B directly by 3 hours is beneficial for us.

In terms of the graph and the weights of the edges in it,

it means that there are no negative weights on the edges.

All the weights are non-negative numbers.

And so, we cannot decrease the length of the path by going through them.

So if we have already a path of length 3 and

all other paths start from spending more time,

we cannot improve this direct buff.

And in the next video, we'll use this idea to create a negative

algorithm that solves our fastest route problem.

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