In this program, you will implement some useful algorithms for graphs that represent friendships (e.g. Facebook). A
friendship graph is an undirected graph without any weights on the edges. It is a simple graph because there are no self
loops (a self loop is an edge from a vertex to itself), or multiple edges (a multiple edge means more than edge between a
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pair of vertices).
The vertices in the graphs for this assignment represent two kinds of people: students and non-students. Each vertex will
store the name of the person. If the person is a student, the name of the school will also be stored.
Here’s a sample friendship graph:
| | |
| | |
(ming,penn state)—-(nick,penn state)—-(ricardo,penn state)
Note that the graph may not be connected, as seen in this example in which there are two “islands” or cliques that are not
connected to each other by any edge. Also see that all the vertices represent students with names of schools, except for
rachel and samir, who are not students.
1. Shortest path: Intro chain
sam wants an intro to aparna through friends and friends of friends. There are two possible chains of intros:
The second chain is preferable since it is shorter.
If sam wants to be introduced to michele through a chain of friends, he is out of luck since there is no chain that
leads from sam to michele in the graph.
Note that this algorithm does NOT have any restriction on the composition of the vertices: a vertex along the shortest
chain need NOT be a student at a particular school, or even a student. In other words, this algorithm is not about
students, let alone students at a particular school. So, for instance, you may need to find the shortest path (intro
chain) from nick to samir, which will be:
which consists of two penn state students, one rutgers student, and one non-student.
2. Cliques: Student cliques at a school
Students tend to form cliques with their friends, which creates islands that do not connect with each other. If these
cliques could be identified, particularly in the student population at a particular school, introductions could be made
between people in different cliques to build larger networks of friendships at that school.
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In the sample graph, there are two cliques of students at rutgers:
Note that in the full graph these are not islands since samir connects them. However, since samir is not a student
at rutgers, it results in two cliques of rutgers students that don’t know each other through another rutgers student.
At penn state, there is a single clique of students:
(ming,penn state)—-(nick,penn state)—-(ricardo,penn state)
Also, a single clique of students at ucla:
And a single clique of students at cornell:
3. Connectors: Friends who keep friends together
If jane were to leave rutgers, sam would no longer be able to connect with anyone else–jane was the “connector”
who could pull sam in to hang out with her other friends. Similarly, aparna is a connector, since without her, sergei
would not be able to “reach” anyone else. (And there are more connectors in the graph…)
On the other hand, samir is not a connector. Even if he were to leave, everyone else could still “reach” whoever they
could when samir was there, even though they may have to go through a longer chain.
Definition: In an undirected graph, vertex v is a connector if there are at least two other vertices x and w for
which every path between x and w goes through v.
For example, v=jane, x=sam, and w=bob.
Finding all connectors in an undirected graph can be done using DFS (depth-first search), by keeping track of two
additional quantities for every vertex v. These are:
dfsnum(v): This is the dfs number, assigned when a vertex is visited, dealt out in increasing order.
back(v): This is a number that is initially assigned when a vertex is visited, and is equal to dfsnum, but can
be changed later as follows:
When the DFS backs up from a neighbor, w, to v, if dfsnum(v) > back(w), then back(v) is set to
If a neighbor, w, is already visited then back(v) is set to min(back(v),dfsnum(w))
When the DFS backs up from a neighbor, w, to v, if dfsnum(v) ≤ back(w), then v is
identified as a connector, IF v is NOT the starting point for the DFS.
If v is a starting point for DFS, it can be a connector, but another check must be made – see
the examples below. The examples don’t tell you how to identify such cases–you have to
figure it out.
Here are some examples that show how this works.
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Example 1: (B is a connector)
Neighbors for a vertex are stored in adjacency linked lists like this:
The DFS starts at A.
dfs @ A 1/1 (dfsnum/back)
dfs @ B 2/2
dfs @ C 3/3
neighbor B is already visited => C 3/2
dfsnum(B) <= back(C) [B is a CONNECTOR] nbr A is already visited => B 2/1
dfsnum(A) <= back(B) [A is starting point of DFS, NOT connector in this case] Example 2: (B is a connector) A--B--C The same example as the first, except DFS starts at B. This shows how even thought B is the starting point, it is still identified (correctly) as a connector. The trace below is not complete because it does not show HOW B is determined to be a connector in the last line - that's for you to figure out. Neighbors are stored in adjacency linked lists as in Example 1. dfs @ B 1/1 dfs @ C 2/2 nbr B is already visited => C 2/1
dfsnum(B) <= back(C) [B is starting point, NOT connector] dfs @ A 3/3 nbr B is already visited => A 3/1
dfsnum(B) <= back(A) [B is starting point, but IS a CONNECTOR in this case] Example 3: (B and D are connectors) A---B---C | | E---D---F Neighbors stored in adjacency linked lists like this: A: B B: E,C,A C: D,B D: F,E,C E: D,B F: D DFS starts at A. dfs @ A 1/1 dfs @ B 2/2 dfs @ E 3/3 dfs @ D 4/4 dfs @ F 5/5 nbr D is already visited => F 5/4
dfsnum(D) <= back(F) [D is a CONNECTOR] nbr E already visited => D 4/3
dfs @ C 6/6
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nbr D already visited => C 6/4
nbr B already visited => C 6/2
dfsnum(D) > back(C) => D 4/2
dfsnum(E) > back(D) => E 3/2
nbr B is already visited => E 3/2
dfsnum(B) <= back(E) [B is a CONNECTOR] nbr C is already visited => B 2/2
nbr A is already visited => B 2/1
dfsnum(A) <= back(B) [A is starting point, NOT a connector in this case] Example 4: (B and D are connectors) A---B---C | | E---D---F Same graph as in Example 3, but neighbors are stored in adjacency linked lists in a different sequence: A: B B: A,C,E C: B,D D: C,E,F E: B,D F: D DFS starts at D, Connectors are still correctly identified as B and D. dfs @ D 1/1 dfs @ C 2/2 dfs @ B 3/3 dfs @ A 4/4 nbr B is already visited => A 4/3
dfsnum(B) <= back(A) [B is a CONNECTOR] nbr C is already visited => B 3/2
dfs @ E 5/5
nbr B is already visited => E 5/3
nbr D is already visited => E 5/1
dfsnum(B) > back(E) => B 3/1
dfsnum(C) > back(B) => C 2/1
nbr D is already visited => C 2/1
dfsnum(D) <= back(C) [D is starting point, NOT connector] dfs @ F 6/6 nbr D is already visited => F 6/1
dfsnum(D) <= back(F) [D is starting point, is a CONNECTOR] Implementation Download the attached friends_project.zip file to your computer. DO NOT unzip it. Instead, follow the instructions on the Eclipse page under the section "Importing a Zipped Project into Eclipse" to get the entire project, called Friends, into your Eclipse workspace. Here are the contents of the project: A class, friends.Friends. This is where you will fill in your code, details follow. A class, Graph, that holds the graph on which the the friends algorithms operate. The file Graph.java defines supporting classes Friend and Person that are used to store a graph in adjacency linked lists format. The file Graph.java also defines a class called Edge that you are free to use in your implementation in the Friends class. You will NOT change ANY of the contents of Graph.java. 4/20/20, 2:30 AM Page 5 of 8 Classes structures.Queue and structures.Stack that you may use in your implementation in the Friends class. You will NOT change ANY of the contents of Stack.java and Queue.java. Every graph that on which you might want to run your algorithms will have the following input format - the sample graph input here is for the friendship graph shown in the Background section above. (The Graph class constructor should be passed a Scanner with the input file as its target.) 15 sam|y|rutgers jane|y|rutgers michele|y|cornell sergei|y|rutgers ricardo|y|penn state kaitlin|y|rutgers samir|n aparna|y|rutgers ming|y|penn state nick|y|penn state bob|y|rutgers heather|y|penn state rachel|n rich|y|ucla tom|y|ucla sam|jane jane|bob jane|kaitlin kaitlin|nick bob|samir sergei|aparna samir|aparna aparna|ricardo nick|ricardo ming|nick heather|nick michele|rachel michele|tom tom|rich The first line has the number of people in the graph (15 in this case). The next set of lines has information about the people in the graph, one line per person (15 lines in this example), with the '|' used to separate the fields. In each line, the first field is the name of the person. Names of people can have any character except '|', and are case insensitive. The second field is 'y' if the person is a student, and 'n' if not. The third field is only present for students, and is the name of the school the student attends. The name of a school can have any character except '|', and is case insensitive. Names of people and schools are unique. The last set of lines, following the people information, lists the friendships between people, one friendship per line. Since friendship works both ways, any friendship is only listed once, and the order in which the names of the friends is listed does not matter. You will complete the following static methods in the Friends class, to implement the three algorithms in the previous section. (All of these methods take a Graph instance as a parameter, aside from other possible inputs detailed below.) Methods 1. (35 pts) shortestChain Input: Name of person who wants the intro, and the name of the other person. For instance, inputs could be 4/20/20, 2:30 AM Page 6 of 8 "sam" and "aparna" for the graph in the Background section. (Neither of these, nor any of the intermediate people in the chain, are required to be students, in the same school or otherwise.) Result: The shortest chain (list) of people in the graph starting at the first and ending at the second, returned in an array list. For example, if the inputs are sam and aparna (sam wants an intro to aparna), then the shortest chain from sam to aparna is [sam,jane,bob,samir,aparna] (This represents the path sam--jane--bob--samir--aparna) If there is more than one shortest path, ANY of them is acceptable. If there is no way to get from the first person to the second person, then the returned list is empty (null or zerosized array list). 2. (25 pts) cliques Input: Name of school for which cliques are to be found, e.g. "rutgers" Result: The names of people in each of the cliques, in any order, returned in an array list of array lists. Moreover, the cliques themselves could be in any order in the top level array list. For the example cited in the Cliques part of the Algorithms section above, with input rutgers, the result is: [[sam,jane,bob,kaitlin],[sergei,aparna]] In other words, an array list that has two cliques, each of which is an array list. The names in the clique array list can be in any order. So, the same cliques could have been returned as: [[jane,sam,kaitlin,bob],[aparna,sergei]] and it would be correct. The cliques themselves can be in any order within the top level array lists, so the following variation (for example) is also acceptable: [[sergei,aparna],[sam,jane,bob,kaitlin]] However, names must not be repeated in a clique. If there are no students in the input school, the result is empty (null or zero-sized array list). 3. (40 pts) connectors Input: None Result: Names of all connectors, in any order, returned in an array list. If there are no connectors, the result is empty (null or zero-sized array list). In the sample friendship graph of the Background section, the connectors list is [jane,aparna,nick,tom,michele]. Any other perumtation of the names in the list is fine, since the order does not matter. Names in the list must not be repeated. Implementation Rules 4/20/20, 2:30 AM Page 7 of 8 Do NOT change ANY of the contents of Graph.java, Queue.java, and Stack.java. In Friends.java, you may NOT MAKE ANY CHANGES EXCEPT to (a) fill in the body of the required methods, or (b) add private helper methods. Note: You may use the Stack or Queue classes that are imported in Friends.java, as needed. But you are not required to use either. Testing In the previous assignments, we provided a driver application (e.g. Evaluator, TrieApp) and test files to help you with the testing process. Now that you are familiar with this testing process, you can use your experience to write your own driver, and construct your own test files to test your Friends methods. Be careful to place the test files directly under the project folder (NOT inside src). You will not be submitting your driver code, or any of the test files. So feel free to experiment with them as needed. Submission Submit your Friends.java file. 4/20/20, 2:30 AM Page 8 of 8