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CS1332-Homework 10 Graph Algorithms Solved
For this assignment, you will be coding 4 different graph algorithms. This homework has quite a few files in it, so you should make sure to read ALL of the documentation given to you before asking a question.
Graph Data Structure
You are provided a Graph class. The important methods to note from this class are:
• getVertices provides a Set of Vertex objects (another class provided to you) associated with a graph.
• getEdges provides a Set of Edge objects (another class provided to you) associated with a graph.
• getAdjList provides a Map that maps Vertex objects to Lists of VertexDistance objects. This Map is especially important for traversing the graph, as it will efficiently provide you the edges associated with any vertex. For example, consider an adjacency list map where vertex A is associated with a list that includes a VertexDistance object with vertex B and distance 2 and another VertexDistance object with vertex C and distance 3. This implies that in this graph, there is an edge from vertex A to vertex B of weight 2 and another edge from vertex A to vertex C of weight
3.
Vertex Distance Data Structure
In the Graph class and Dijkstra’s algorithm, you will be using the VertexDistance class implementation that we have provided. In the Graph class, this data structure is used by the adjacency list to represent which vertices a vertex is connected to. In Dijkstra’s algorithm, you should use this data structure along with a PriorityQueue. At any stage throughout the algorithm, the PriorityQueue of VertexDistance objects will tell you which vertex currently has the minimum cumulative distance from the source vertex.
Search Algorithms
Breadth-First Search is a search algorithm that visits vertices in order of “level”, visiting all vertices one edge away from start, then two edges away from start, etc. Similar to levelorder traversal in BSTs, it depends on a Queue data structure to work.
Depth-First Search is a search algorithm that visits vertices in a depth based order. Similar to pre/post/inorder traversal in BSTs, it depends on a Stack data structure to work. However, in your implementation, the Stack will be the recursive stack. It searches along one path of vertices from the start vertex and backtracks once it hits a dead end or a visited vertex until it finds another path to continue along. Your implementation of DFS must be recursive to receive credit.
Single-Source Shortest Path (Dijkstra’s Algorithm)
The next algorithm is Dijkstra’s Algorithm. This algorithm finds the shortest path from one vertex to all of the other vertices in the graph. This algorithm only works for non-negative edge weights, so you may assume all edge weights for this algorithm will be non-negative.
There are two main variants of Dijkstra’s Algorithm related to the termination condition of the algorithm. The first variant is where you depend purely on the PriorityQueue to determine when to terminate the algorithm. You only terminate once the PriorityQueue is empty. The other variant, the classic variant, is the version where you maintain both a PriorityQueue and a visited set. To terminate, still check if the PriorityQueue is empty, but you can also terminate early once all the vertices are in the visited set. You should implement the classic variant for this assignment. The classic variant, while using more memory, is usually more time efficient since there is an extra condition that could allow it to terminate early.
Minimum Spanning Trees (MST - Prim’s Algorithm)
An MST has two components. By definition, it is a tree, which means that it is a graph that is acyclic and connected. A spanning tree is a tree that connects the entire graph. It must also be minimum, meaning the sum of edge weights of the tree must be the smallest possible while still being a spanning tree.
By the properties of a spanning tree, any valid MST must have |V | − 1 edges in it. However, since all undirected edges are specified as two directional edges, a valid MST for your implementation will have 2(|V | − 1) edges in it.
Prim’s algorithm builds the MST outward from a single component, starting with a starting vertex. At each step, the algorithm adds the cheapest edge connected to the incomplete MST that does not cause a cycle. Cycle detection can be handled with a visited set like in Dijkstra’s.
Self-Loops and Parallel Edges
In this framework, self-loops and parallel edges work as you would expect. If you recall, self-loops are edges from a vertex to itself. Parallel edges are multiple edges with the same orientation between two vertices. These cases are valid test cases, and you should expect them to be tested. However, most implementations of these algorithms handle these cases automatically, so you shouldn’t have to worry too much about them when implementing the algorithms.
Visualizations of Graphs
Graph Data Structure
You are provided a Graph class. The important methods to note from this class are:
• getVertices provides a Set of Vertex objects (another class provided to you) associated with a graph.
• getEdges provides a Set of Edge objects (another class provided to you) associated with a graph.
• getAdjList provides a Map that maps Vertex objects to Lists of VertexDistance objects. This Map is especially important for traversing the graph, as it will efficiently provide you the edges associated with any vertex. For example, consider an adjacency list map where vertex A is associated with a list that includes a VertexDistance object with vertex B and distance 2 and another VertexDistance object with vertex C and distance 3. This implies that in this graph, there is an edge from vertex A to vertex B of weight 2 and another edge from vertex A to vertex C of weight
3.
Vertex Distance Data Structure
In the Graph class and Dijkstra’s algorithm, you will be using the VertexDistance class implementation that we have provided. In the Graph class, this data structure is used by the adjacency list to represent which vertices a vertex is connected to. In Dijkstra’s algorithm, you should use this data structure along with a PriorityQueue. At any stage throughout the algorithm, the PriorityQueue of VertexDistance objects will tell you which vertex currently has the minimum cumulative distance from the source vertex.
Search Algorithms
Breadth-First Search is a search algorithm that visits vertices in order of “level”, visiting all vertices one edge away from start, then two edges away from start, etc. Similar to levelorder traversal in BSTs, it depends on a Queue data structure to work.
Depth-First Search is a search algorithm that visits vertices in a depth based order. Similar to pre/post/inorder traversal in BSTs, it depends on a Stack data structure to work. However, in your implementation, the Stack will be the recursive stack. It searches along one path of vertices from the start vertex and backtracks once it hits a dead end or a visited vertex until it finds another path to continue along. Your implementation of DFS must be recursive to receive credit.
Single-Source Shortest Path (Dijkstra’s Algorithm)
The next algorithm is Dijkstra’s Algorithm. This algorithm finds the shortest path from one vertex to all of the other vertices in the graph. This algorithm only works for non-negative edge weights, so you may assume all edge weights for this algorithm will be non-negative.
There are two main variants of Dijkstra’s Algorithm related to the termination condition of the algorithm. The first variant is where you depend purely on the PriorityQueue to determine when to terminate the algorithm. You only terminate once the PriorityQueue is empty. The other variant, the classic variant, is the version where you maintain both a PriorityQueue and a visited set. To terminate, still check if the PriorityQueue is empty, but you can also terminate early once all the vertices are in the visited set. You should implement the classic variant for this assignment. The classic variant, while using more memory, is usually more time efficient since there is an extra condition that could allow it to terminate early.
Minimum Spanning Trees (MST - Prim’s Algorithm)
An MST has two components. By definition, it is a tree, which means that it is a graph that is acyclic and connected. A spanning tree is a tree that connects the entire graph. It must also be minimum, meaning the sum of edge weights of the tree must be the smallest possible while still being a spanning tree.
By the properties of a spanning tree, any valid MST must have |V | − 1 edges in it. However, since all undirected edges are specified as two directional edges, a valid MST for your implementation will have 2(|V | − 1) edges in it.
Prim’s algorithm builds the MST outward from a single component, starting with a starting vertex. At each step, the algorithm adds the cheapest edge connected to the incomplete MST that does not cause a cycle. Cycle detection can be handled with a visited set like in Dijkstra’s.
Self-Loops and Parallel Edges
In this framework, self-loops and parallel edges work as you would expect. If you recall, self-loops are edges from a vertex to itself. Parallel edges are multiple edges with the same orientation between two vertices. These cases are valid test cases, and you should expect them to be tested. However, most implementations of these algorithms handle these cases automatically, so you shouldn’t have to worry too much about them when implementing the algorithms.
Visualizations of Graphs
1 file (142.4KB)
