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In this lesson, we walk through the steps of implementing a version of the graph data structure in Python using two classes: Vertex and Graph.

Graphs: Python

Our pathfinding method is almost complete. Let's take a step back and think how a passenger in Harwick station could find their way to Callan. 

First, they'd look for all the stations connected to Harwick. If one of those stations was Callan, they're in luck! 

Otherwise, they would look for the connections from each of those stations **excluding Harwick because they've already crossed it off their list**.

We'll take the same strategy with our pathfinding method.

```py
while len(start) > 0:
  current_vertex = start.pop(0)
  # current_vertex is end_vertex
    # a path exists!
  # current_vertex is not end_vertex
    # add vertices connected to 
    # current_vertex onto the list
    # to keep searching for a path
```



Finding a Path II

Our pathfinding method works when a path exists, but there is a serious bug! **We're not accounting for cycles in our graph.**

Consider the following undirected Graph:

```py
railway = Graph()

callan = Vertex('callan')
peel = Vertex('peel')
ulfstead = Vertex('ulfstead')
harwick_station = Vertex('harwick')

railway.add_vertex(callan)
railway.add_vertex(peel)
railway.add_vertex(harwick)
railway.add_vertex(ulfstead)

railway.add_edge(peel, harwick)
railway.add_edge(harwick, callan)
railway.add_edge(callan, peel)

railway.find_path(peel, ulfstead)
```

Peel connects to Harwick and Callan. We look for a path starting at `peel`, adding `harwick` and `callan` to the `start` list. Next, we look at `harwick`, adding `peel` and `callan`. We'll keep adding the same vertices, and our loop will never terminate. 

When a path exists, `return True` will exit the loop. When a path **does not** exist, it's a major problem! 

We'll use a dictionary to track which stations we've visited. This will stop our passengers from riding around in circles.

Refactoring Path-Finding

We're keeping things organized by storing our classes in separate files, so let's do a brief review on how to use code from another file.

Python gives us the ability to _import_ code from another file. Here's how we can use our `Vertex` class from within **graph.py**.

```py
# from <file_name> import <class>

# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# inside of ./vertex.py file
class Vertex
  # code for the class...

# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# inside of ./graph.py file
from vertex import Vertex

my_vertex = Vertex("Import Accomplished!")
```



Import Interlude

We'd like our `Graph` class to be able to set edges between the stored vertices. An instance of `Graph` is either directed or undirected, which affects how edges work between two vertices.  As a reminder, the `Graph` class defaults to being undirected with edges being set in both directions.

The good news is our `Vertex` class already has an `.add_edge()` method, so we can delegate most of the work. 

The `Graph` version of `.add_edge()` will take a "from" and a "to" vertex, and set the appropriate edges by calling the `Vertex` instance's own `.add_edge()`. 

```py
undirected_railway = Graph()
directed_railway = Graph(True)

callan_station = Vertex('callan')
peel_station = Vertex('peel')

undirected_railway.add_vertex(callan_station)
undirected_railway.add_vertex(peel_station)

directed_railway.add_vertex(callan_station)
directed_railway.add_vertex(peel)

directed_railway.add_edge(callan_station, peel_station)
# directed graph will set the edge one-way
print(callan_station.get_edges())
# ['peel']
print(peel_station.get_edges())
# []

undirected_railway.add_edge(callan_station, peel_station)
# directed graph will set the edge both ways
print(callan_station.get_edges())
# ['peel']
print(peel_station.get_edges())
# ['callan']
```


Building the Graph II

So far our `Vertex` class has stored edges inside of a dictionary with keys of the connected vertex's name and the value simply set to `True`. 

We can make our implementation support edge weights with a few small changes. To keep this class as flexible as possible, we'll introduce a default `weight` argument to `.add_edge()` in the `Graph` and `Vertex` classes. With no explicit `weight` argument, it will default to `0`. We'll then set the appropriate value in the dictionary to that `weight`. 

Weighted edges allow us to make graphs that represent rail systems with a travel-time between stations.

```py
railway = Graph()

callan = Vertex('callan')
peel = Vertex('peel')
harwick = Vertex('harwick')

railway.add_vertex(callan)
railway.add_vertex(peel)
railway.add_vertex(harwick)

# Travel-time between callan and peel: 12
railway.add_edge(callan, peel, 12)
# Travel-time between harwick and callan: 7
railway.add_edge(harwick, callan, 7)

print(callan.edges)
# { 'peel': 12 }
print(harwick.edges)
# { 'callan': 7 }
```



Adding Weight

In this lesson, we'll build a robust implementation of the graph data structure. With just two classes, `Vertex` and `Graph`, we can implement a variety of graphs that satisfy the requirements of many algorithms. Let's detail the functionality we require from these classes:* `Vertex` stores some data.* `Vertex` stores the edges to connected vertices and their weight.* `Vertex` can add a new edge to its collection.* `Graph` stores all the vertices.* `Graph` knows if it is directed or undirected.* `Graph` can add a new vertex to its collection.* `Graph` can add a new edge between stored vertices.* `Graph` can tell whether a path exists between stored vertices.Since we're dealing with multiple classes, we'll use multiple files for this implementation. To keep the concepts grounded in a real-world application, we'll build up a transportation network of railroads and train stations as we go.

Introduction to Graphs

We'll continue building out the `Vertex` class. Remember, it's responsible for knowing which other vertices are connected. These connections are the edges of our graph implementation. 

A key in the `Vertex` instance's `edges` dictionary represents a connection to that other vertex. For now, we can just set the value to be `True`.

```py
grand_central = Vertex('Grand Central Station')
forty_second_street = Vertex('42nd Street Station')

print(grand_central.get_edges())
# []

grand_central.add_edge(forty_second_street)
print(grand_central.edges)
# { "42nd Street Station": True }
print(grand_central.get_edges())
# ["42nd Street Station"]
```

Let's add this functionality to our `Vertex` class!


Building the Vertex II

Our railway has grown to four stations with two connecting tracks. How can we tell a passenger which stations are reachable from Harwick?

```py
undirected_railway = Graph()

callan_station = Vertex('callan')
peel_station = Vertex('peel')
ulfstead_station = Vertex('ulfstead')
harwick_station = Vertex('harwick')

undirected_railway.add_vertex(callan_station)
undirected_railway.add_vertex(peel_station)
undirected_railway.add_vertex(harwick_station)
undirected_railway.add_vertex(ulfstead_station)

undirected_railway.add_edge(peel_station, harwick_station)
undirected_railway.add_edge(peel_station, callan_station)
```


Our `Graph` class needs to determine whether a path exists between two vertices. A path means two vertices which are connected by a sequence of one or more intermediary edges and graphs.




Finding a Path I


Let's start with our `Vertex` class. This class is responsible for storing information about individual vertices in our graph. In our railway network, instances of `Vertex` will represent train stations.

An instance of `Vertex` will store data and a Python dictionary which tracks other `Vertex` instances connected to `self`. 



Building the Vertex

Fantastic work! We've implemented a robust graph data structure in Python. Our two classes, `Vertex` and `Graph` are capable of representing the typical variations in graphs that occur in many different algorithms.`Vertex`:* Uses a dictionary as an adjacency list to store connected vertices.* Connected vertex names are keys and the edge weights are values.* Has methods to add edges and return a list of connected vertices. `Graph`:* Can be initialized as a directed graph, where edges are set in one direction.* Stores every vertex inside a dictionary	* Vertex data is the key and the vertex instance is the value.* Has methods to add vertices, edges between vertices, and determine if a path exists between two vertices.

Graph Review


We've built a class to store information and connections between individual vertices, but we need another class that keeps track of the big picture. 

Our `Graph` class will track **all** the vertices and handle higher level concerns like whether the graph is directed, requiring edges to have a set direction, or undirected, allowing bi-directional movement across edges.

We'll start by giving `Graph` the functionality to add vertices. We'll use an internal `.graph_dict` property to store every vertex by its value pointing to the vertex instance itself. 

We want to do the following:

```py
grand_central = Vertex("Grand Central Station")
railway = Graph()

print(railway.graph_dict)
# {}
railway.add_vertex(grand_central)
print(railway.graph_dict)
# {"Grand Central Station": grand_central}
```

