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# Graphs

import cs1.graphs.UnweightedGraph;
import cs1.graphs.GraphNode;
import java.util.Arrays;
UnweightedGraph<Integer> graph = UnweightedGraph.circleUndirectedGraph(
Arrays.asList(1, 2, 4)
);
for (GraphNode<Integer> node : graph.getNodes()) {
System.out.println(node);
}

This lesson introduces a new data structure: graphs. Graphs are extremely useful, both for modeling certain types of data, and for enabling certain algorithms. In fact, trees, which we’ve been studying previously, are themselves a type of graph. Graphs are also great for practice with recursion. So let’s get started!

## What is a Graph?What is a Graph?

A graph data structure consists of a finite (and possibly mutable) set of vertices (also called nodes or points), together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph. These pairs are known as edges (also called links or lines), and for a directed graph are also known as edges but also sometimes arrows or arcs. The vertices may be part of the graph structure, or may be external entities represented by integer indices or references.

Let’s look at some diagrams that will help visualize this new data structure, and introduce some important terminology.

### What Are Graphs For?What Are Graphs For?

Graphs can be used to model a variety of real-world entities, which is part of what makes them so useful to computer scientists. Some examples include:

• Social graphs, where each node is a person, edges represent friendships or acquaintances, and understanding the graph helps better understand the structure of human relationships.
• Internet routing, where each node is a machine connected to the internet, edges represent connections between machines, and understanding the graph helps data travel around the world.
• Transit graphs, where each node is a place on Earth, edges represent transit connections between different locations, and understanding the graph help you travel around the world.

Let’s look at these example a bit more:

### Types of GraphTypes of Graph

Depending on the properties of their edges, graphs can be directed or undirected, weighted or unweighted. We’ll focus our attention on undirected unweighted graphs. But let’s discuss the differences briefly.

### Trees v. GraphsTrees v. Graphs

We’ve discussed trees previously, and it turns out that trees are one specific type of graph. Now that we have been introduced to graphs, we can define a tree more precisely.

## Graph RecursionGraph Recursion

Previous we introduced recursion, a problem solving technique that works by breaking down large problems into smaller pieces, solving them, and then combining the results. Graphs are another data structure that we frequently will examine recursively. Let’s see why:

In addition, like trees, compared to lists and arrays and `String`s, graphs can be hard to work with using iterative solutions! Recursion is a much better approach…

### Recursive Graph SizeRecursive Graph Size

To help you prepare for our next homework problem, let’s discuss the recursive approach to counting the number of nodes in a graph. Specifically, we’ll review the idea of graph traversal, and how not to get stuck along the way.

## Solve: Undirected Graph Size (Practice)

Created By: Geoffrey Challen
/ Version: 2021.10.0

Create a public class named `GraphSize` that provides a single static method `size`. `size` receives an unweighted graph containing `Integer` values using `cs1.graphs.UnweightedGraph`, so an `UnweightedGraph<Integer>`.

To complete this problem you'll need to implement graph traversal. From a given node, you want to visit all of its neighbors except any nodes that you've already visited. If you use a `Set` to track the nodes that you've visited, then you can simply return the size of that `Set` when you are finished. We've provided some starter code to get you off on the right track.

For reference, `cs1.graphs.UnweightedGraph` has the following public properties:

And `cs1.graphs.GraphNode` has the following public properties:

import cs1.graphs.GraphNode;
import cs1.graphs.UnweightedGraph;
import java.util.HashSet;
import java.util.Set;
public class GraphSize {
public static int size(UnweightedGraph<Integer> graph) {
// Handle null
Set<GraphNode<Integer>> nodes = new HashSet<>();
traverse(graph.getNode(), nodes);
return 0;
}
private static void traverse(GraphNode<Integer> node, Set<GraphNode<Integer>> nodes) {
// Add this node to the set
// Iterate through all the neighbors
// If the neighbor hasn't been visited, call traverse on it and pass it the visited set
}
}

## Solve: Cities Are Connected

Created By: Geoffrey Challen
/ Version: 2021.10.0

Create a public class `Connections` with a single public constructor that accepts a `String`. The `String` contains, in CSV format, a list of cities and other cities that they are connected to. So, for example, the input:

``````Champaign,Chicago,St. Louis
Chicago,Detroit,Milwaukee
St. Louis,Champaign,Cincinnati
``````

Means that Champaign is connected to Chicago and St. Louis, and that Chicago is connected to Detroit and Milwaukee, and so on. Essentially the CSV serializes a directed graph, where the first item on each line is a node and the other items represent other nodes that it is connected to. This is one way of serializing a directed, unweighted graph. If the `String` passed to the constructor is `null`, throw an `IllegalArgumentException`. Make sure to `trim` all the `String`s that you extract from the CSV.

Your class should parse this `String` and provide a single instance method `isConnected`. `isConnected` accepts two `String`s and returns `true` if the first city is connected to the second based on the graph passed to the constructor. So, given the input above, `isConnected("Champaign", "Chicago")` would return `true`, but `isConnected("Chicago", "Champaign")` would return `false`. (Note that the graph is not necessarily symmetric.) If either `String` passed to `isConnected` is `null`, or if you don't have connection information for the source, you should throw an `IllegalArgumentException`.

Note that only the cities that appear first on each line in the CSV should be treated as cities you have connection information for. So, for example, even though "Detroit" appears as a destination from "Chicago" in the data set above, we do not have a line starting with "Detroit", and therefore a call to `isConnected` with "Detroit" as the first parameter should throw an `IllegalArgumentException`.

Our suggestion is to use a private field to store a data structure that you populate in your constructor. This will keep your `isConnected` method simpler. Good luck, and have fun!

## More Practice

Need more practice? Head over to the practice page.