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    • Graph Algorithms : 60

    • Graphs : 59

    • Streams : 58

    • Generics : 57

    • Implementing a Map : 56

    • Hashing : 55

    • Binary Search : 54

    • Quicksort : 53

    • Merge Sort : 52

    • Sorting Algorithms : 51

    • Practice with Recursion : 50

    • Trees and Recursion : 49

    • Trees : 48

    • Recursion : 47

    • Lists Review and Performance : 46

    • Linked Lists : 45

    • Algorithms and Lists : 44

    • Lambda Expressions : 43

    • Anonymous Classes : 42

    • Practice with Interfaces : 41

    • Implementing Interfaces : 40

    • Using Interfaces : 39

    • Working with Exceptions : 38

    • Throwing Exceptions : 37

    • Catching Exceptions : 36

    • References and Polymorphism : 35

    • References : 34

    • Data Modeling 2 : 33

    • Equality and Object Copying : 32

    • Polymorphism : 31

    • Inheritance : 30

    • Data Modeling 1 : 29

    • Static : 28

    • Encapsulation : 27

    • Constructors : 26

    • Objects, Continued : 25

    • Introduction to Objects : 24

    • Compilation and Type Inference : 23

    • Practice with Collections : 22

    • Maps and Sets : 21

    • Lists and Type Parameters : 20

    • Imports and Libraries : 19

    • Multidimensional Arrays : 18

    • Practice with Strings : 17

    • null : 16

    • Algorithms and Strings : 15

    • Strings : 14

    • Functions and Algorithms : 13

    • Practice with Functions : 12

    • More About Functions : 11

    • Errors and Debugging : 10

    • Functions : 9

    • Practice with Loops and Algorithms : 8

    • Algorithms I : 7

    • Loops : 6

    • Arrays : 5

    • Compound Conditionals : 4

    • Conditional Expressions and Statements : 3

    • Operations on Variables : 2

    • Variables and Types : 1

    • Hello, world! : 0


    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()) {

    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?

    Wikipedia defines a graph as:

    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:

    Let’s look at these example a bit more:

    Types of Graph
    Types 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. Graphs
    Trees 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 Recursion
    Graph 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 Strings, graphs can be hard to work with using iterative solutions! Recursion is a much better approach…

    Recursive Graph Size
    Recursive 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
    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 Strings that you extract from the CSV.

    Your class should parse this String and provide a single instance method isConnected. isConnected accepts two Strings 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.