CS 314 - Specification 11 - Graphs

Programming Assignment 11 - Individual Assignment. You must complete this assignment on your own. You may not acquire from any source (e.g.  another student,  an internet site, Large Language Model or Generative AI such as chatGPT or Copilot) a partial or complete solution to a problem or project that has been assigned. You may not show another student your solution to an assignment. You may not have another person (current student, former student, tutor, friend, anyone) “walk you through” how to solve an assignment. You may get help from the instructional staff. You may discuss general ideas and approaches with other students but you may not develop code together. Review the class policy on collaboration from the syllabus.

• Placed online: Tuesday, November 11
• Due: no later than  11 pm, Thursday, December 5 (You CAN use slip days on assignment 11 if you have some left. Some semester do not allow the use of slip days when the assignment is due the last day of class, but that is not the case this term.)
• General Assignment Requirements

The purposes of this assignment are:

1. implement two graph algorithms
2. use graphs to rank college football teams based on the results from a single season

Description: In this assignment you will implement two instance methods for a graph class and one method in a client of the graph class that evaluates properties of the graph.

When finished turn in your Graph.java, FootballRanker.java, and GraphAndRankTester.java files.

Provided Files:

Background: Prior to the 2014 season, division 1 college football was one of the few college sports where the champion was not determined by a tournament run by the NCAA. Instead the BCS, the Bowl Championship Series, used various polls and surveys are used to determine the "best" two teams who then played each other in the final game of the season. The ranking algorithm the BCS used consisted of two human surveys PLUS six algorithms carried out by computers. (Note, the algorithms used by the BCS were not allowed to take into account the score of games, only whether a team won or lost a game. The people running the BCS thought it unsporting to encourage stronger teams to run up the score on  weaker teams.) As of 2017 division 1 college football uses a 4 team playoff at the end of the season. The four teams are picked by a committee of humans. The new selection committee claims to not use any algorithms in determining its rankings. There are still a large number of individuals that publish ranking based on algorithms.

In this assignment you will complete a Graph class and a FootballRanker class that determines the "best" team based on the graph formed by the division 1 college football teams and the games they played against each other. Centrality of vertices in the graph with some adjustments are used to determine the "best" team. Your program shall rank teams three ways and compare the results to the Associated Press end of season poll, another poll completed by humans with no algorithmic input.

The three ranking algorithms  for the assignment are:

1. Rank each team by calculating the number of other teams they are connected to and the average shortest UNWEIGHTED paths to those connected teams. Determine the average UNWEIGHTED path length by dividing the total sum of all shortest, unweighted path lengths (number of edges in path) by the number of vertices (teams) connected.
2. Rank each team by calculating the number of other teams they are connected to and the average shortest WEIGHTED path to those connected teams. Determine the average WEIGHTED path length by dividing the total sum of all shortest, weighted path lengths by the number of vertices (teams) connected.
3. Rank each team using the result from method 2 and then dividing the average weighted path length by the team's win / loss percentage.

Most of the program is already done. There are three methods you must complete are:

1. Complete the instance method dijkstra in the Graph class. This method finds the shortest weighted path from the given start vertex to all other vertices in the graph using Dijktsra's Shortest Weighted Path Algorithm. Read the method documentation thoroughly to understand all the requirements of the method. Recall the class slides contain the pseudocode for this algorithm.
    
2. Complete the instance method findAllPaths in the Graph class. This method updates each vertex in the Graph so that each vertex stores the number of other vertices it is connected to. (In the assignment this is also referred to as the number of paths from the vertex.) The method also finds the sum of all the shortest weighted and unweighted paths from the vertex to every other vertex it connects to. If the boolean parameter named weighted is false these values will equal each other. Read the comments for the numVertexConnected, totalUnweightedPathLength, and totalWeightedPathLength instance variables in the Vertex class to help understand the purpose of findAllPaths.
   
   Each Vertex object has instance variables to store this data. The method must call the findUnweightedShortestPath or dijkstra methods as appropriate. The Vertex class, the nested Path class, and  the getPath(String) method are useful for completing and testing this method. A lot of the support code is already done for you. You must determine how to use the existing code. You are of course, free to write you own support code if you prefer, but your output must match the expected output.
   
   The method also finds and stores the "longest shortest path" in the Graph. Because the method must find the shortest path between all pairs of vertices relying on the given findUnweightedShortestPaths method or the dijkstra you complete based on the value of the parameter weighted, we can keep track of the longest of the shortest paths we find. The length of the paths shall be based on their weighted cost. For the findUnweightedShortestPaths this is the same as the number of edges in the path. Use the getPath method to create a Path object for the longest (highest cost) path you find as you search the shortest paths. Use the instance variable named longest to refer to this Path object.
   
   Consider the following example:
   
   
   
   The graph is weighted, but undirected. If we ran findAllPaths method and sent false for weighted the cumulative sum statistics in the vertex objects would be:
   

   Vertex	numVertexConnected	totalUnweightedPathLength	totalWeightedPathLength
   A	6	10.0	10.0
   B	6	9.0	9.0
   C	6	10.0	10.0
   D	6	8.0	8.0
   E	6	13.0	13.0
   F	6	10.0	10.0
   G	6	10.0	10.0
   J	1	1.0	1.0
   K	1	1.0	1.0

   
   If we ran findAllPaths method and sent true for weighted the cumulative sum statistics in the vertex objects would be:
   

   Vertex	numVertexConnected	totalUnweightedPathLength	totalWeightedPathLength
   A	6	20.0	56.0
   B	6	15.0	51.0
   C	6	12.0	42.0
   D	6	12.0	43.0
   E	6	17.0	73.0
   F	6	11.0	39.0
   G	6	17.0	68.0
   J	1	1.0	4.0
   K	1	1.0	4.0

   
    
    
3. Complete the printRootMeanSquareError method in the FootballRanker class. The FootballRanker class computes rankings for teams based on the three approaches described above. This method relies on the findAllPaths method which in turn relies on the dijkstra method, so you must complete those two methods before this one will work properly.  The  printRootMeanSquareError  method in the FootballRanker compares the computed rankings from findAllPaths with the end of season poll from the Associated Press. The AP's ranks are stored in list parameter sent to this method (position in list is equivalent to rank when using zero based indexing) and the graph ranks are in the TreeSet. Iterate through the AP ranks and determine the root mean square error between the AP ranks and the calculated ranks. The root mean square error is determine by taking the difference between the AP rank and the predicted rank, squaring the difference, adding all of these squared differences together, dividing by the number of teams, and taking the square root. The mathematical formula:
   
   
   
   where, n is the number of teams, x1 is the AP rank for the ith ranked team, x2 is the predicted rank for that team based on our graph calculations for the ith team, and n is the number of teams. If a team is ranked in the AP poll, but not in our predictions assign it a rank equal to one more than the total number of teams in the graph predictions. (The FootballRanker cuts some teams from the predictions if they do not have enough direct or transitive wins. (wins based on connections in the graph)). The method prints out the data as shown in the sample output and returns the root mean square error rounded to the nearest tenth.

Your class must pass the tests in GraphAndRankTest class.

Add your own tests to the GraphAndRankTest  class and delete the provided tests so the TAs only see your tests.

Answer the question at the top of the GraphAndRankTest  class. (Put some thought into this.) If you are not familiar with the structure of the college football system in the US you may have to do some research. It is helpful to understand the divisions within college football.

Checklist: Did you remember to:

• review and follow the general assignment requirements?
• work on the assignment individually and not use any LLM and / or chatbot such as chatGPT or Copilot?
• fill in the headers in your Graph.java, FootballRanker.java, and GraphAndRankTester.java
• ensure program passes the provided tests and matches the sample output exactly?
• add you own tests to GraphAndRankTester and delete the provided tests?
• answer the question at the top of the GraphAndRankTester class?
• ensure your program does not suffer a compile error or runtime error?
• turn in the three required files (Graph.java, FoorballRanker.javam and GraphAndRankTester.java) to the proper GradeScope assignment (#11) no later than  11 pm, Thursday, December 5?