| Assigned: | Wednesday, October 16 |
| Due: | Monday, October 28, 11:59pm |
| Points: | 600 |
This assignment gives you experience using structures and arrays and writing a program with two modules.
It is important to follow the algorithm and design described here. Do not make up a different algorithm or design. You will need to define two structure types and eight functions, including main.
You will need to submit three files, mst.cpp, equiv.cpp and equiv.h. Submit all three files, even if I already have equiv.cpp and equiv.h from Assignment 4.
Read the entire assignment before you start working on it. Be sure to follow the instructions.
An undirected graph (or simply a graph) is a collection of numbered vertices connected by a collection of edges. An edge connects exactly two different vertices to one another. You can draw a diagram of a graph by showing the vertices as circles and the edges as lines connecting the vertices. Here is an example.
I will use variables such as u, v, v1, v2 to stand for vertex numbers. Each variable can refer to any vertex. For example, variable v1 might refer to vertex 4.
A path from vertex v1 to vertex vk is a sequence of vertices (v1, v2, …, vk) such that there is an edge between vi and vi+1, for i = 1, …, k − 1.
We will only be concerned with connected graphs. A graph is connected if there is a path from every vertex to every other vertex. For example, in the above graph, there is a path from 3 to 4 that goes (3, 1, 2, 4). There are other paths from 3 to 4 as well. For any two vertices u and v, there is a path from u to v in the example graph, so it is connected.
A weighted graph is a graph in which each edge has a number attached to it, called the weight of the edge. Here is a diagram of a weighted graph.
If e is an edge, we write W(e) to indicate the weight of e.
A spanning tree of a connected graph is obtained by deleting as many edges as possible without making the graph so that it is no longer connected. That is, we still need to have a path from each vertex to each other vertex. For example, the following is a spanning tree of the above weighted graph.
If graph G has n vertices then every spanning tree of G has n − 1 edges.
The weight of a spanning tree is the sum of the weights of its edges. For example, the weight of the above spanning tree is 55. Here is another spanning tree for the same graph. Its weight is 50.
Obviously, some spanning trees have smaller total weight than others. A minimal spanning tree is a spanning tree with the smallest possible weight.
There is a well-known algorithm, called Kruskal's algorithm, for computing a minimal spanning tree of a weighted graph G. It goes as follows, expressed in conceptual, not physical, terms.
Make a list (or array) of the edges in G. Suppose there are m edges. Sort the edges from small weight to larger weight, giving list (e0, e1, …, em − 1) of edges. Because they have been sorted by weight, we know that (W(e0) ≤ W(e1) ≤ … ≤ W(em − 1)
Start with an empty graph T. It has all of the vertices of G, but no edges yet. Edges will be added to T until, at the end, it is the minimal spanning tree.
For i = 0, …, m − 1 do
Look at edge ei. Suppose that it connects vertices u and v.
If there is not already a path in T between vertices u and v, then add edge ei to T. Otherwise, do not add it to T.
When the algorithm is done, T is a minimal spanning tree of G.
Kruskal's algorithm is an example of a greedy algorithm, which is an algorithm that tries to optimize on a large scale by optimizing on a small scale. When Kruskal's algorithm chooses an edge to add to T, it chooses the edge with the smallest weight that does the job (because it looks at the edges in increasing order of weight). That is the cheapest thing to do for this step. There is no thought about whether that decision will be a good one in the long run.
It turns out that Kruskal's algorithm works, and always finds a minimal weight spanning tree. Watch out, though. Many greedy algorithms for other problems at first appear to be reasonable, but fail to produce a correct result on some inputs.
Write a program that reads, from the standard input, a description of a weighted graph with integer weights, in the input format shown below. Then the program should write, on the standard output:
the original graph G;
the computed minimal spanning tree T of G;
the total weight of T.
To show a graph (or tree), show the number of vertices, the number of edges and a list of all of the edges, with their weights.
Make it clear just what each part of the output is. Don't make a reader guess what he or she is looking at. The output of your program might look like this.
Input graph: There are 5 vertices and 6 edges vertices weight 1 2 9 1 3 12 2 4 18 2 3 6 2 5 20 3 5 15 Minimal spanning tree: There are 5 vertices and 4 edges vertices weight 2 3 6 1 2 9 3 5 15 2 4 18 The total weight of the spanning tree is 48.
Your program must follow the design in the section "A refinement plan." It can have additional functions, but it must have at least the functions indicated in the design. Do not add extra responsibilities to the required functions.
Kruskal's algorithm needs to be able to determine whether two vertices are connected by the edges that have already been added to the tree.
For two vertices u and v, say that u ~ v just when there is a path between u and v. Then ~ is an equivalence relation. (~ is reflexive: the path from u to u has length 0 and is just (u). It should be easy to see that ~ is also symmetric and transitive.)
Use your equivalence relation manager from the previous assignment to keep track of the equivalence classes of ~. Initially, there are no edges, so each vertex is in an equivalence class by itself. When Kruskal's algorithm considers adding an edge between vertices u and v, it consults the equivalence relation manager. If u ~ v, then the edge between u and v is omitted.
If u and v are not equivalent, then you add this edge to the tree and tell the equivalence relation manager to combine the equivalence classes of u and v.
Do not copy your equivalence manager into your file that implements Kruskal's algorithm. Keep it in separate files equiv.cpp and equiv.h. Include "equiv.h" in mst.cpp. Do not include equiv.cpp in mst.cpp!
Start by creating a directory to hold assignment 5. Copy equiv.h and equiv.cpp from Assignment 4 into that directory. Download files
into that directory.| Development plan |
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1. Create a file called mst.cpp
Copy and paste the template into it. Edit the file. Add your name and the assignment number. If you will use tabs, say how far apart the tab stops are. Add line #include "equiv.h" |
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2. Document the program
Write a comment in mst.cpp telling what this program will do when it is finished. Include information to help a reader understand what this program does and how to use it, including:
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3. Vertex numbers and edge numbers
Do not confuse edges with vertices. Keep track of the number of vertices in a graph separately from the number of edges. Remember that the first number in the input is the number of vertices, not the number of edges. The vertices are numbered 1, …, n, where n is the first number in the input. You will need to have an array of edges. There are usually (but not always) more edges than vertices. It is a good idea to start numbering the edges from 0, not from 1. So put the first edge that you read at index 0 in the array of edges. |
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4. Representing a weighted graph
Start by defining two structure types, Edge and Graph, in mst.cpp. Be sure to document each.
Both Edge and Graph must be documented by saying
Don't make the reader guess. |
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5. Adding an Edge
In mst.cpp, define a function insertEdge(u, v, w, g) that adds an edge between u and v of weight w to graph g. If the edge array is full, insertEdge must refuse to add the edge. Here is a suitable contract for insertEdge. Note its similarity to the preceding paragraph. // insertEdge(u, v, w, g) inserts an edge of weight w // between vertices u and v into graph g. // // If there is not enough room in g to add the edge, // then insertEdge does nothing. Pass g to insertEdge using pointer. If you pass g by value, insertEdge will modify a (shallow) copy of your graph, which is not what you want. So the type of parameter g should be Graph*. Here is a suitable heading for insertEdge. void insertEdge(int u, int v, int w, Graph* g) Make sure that insertEdge does the whole job of inserting an edge. Don't force its caller to do part of the job. Increasing the number of edges in g is part of the job. |
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6. Input and Input Format
In mst.cpp, define a function readGraph(e) that reads a description of a graph from the standard input and returns a poiner to that graph. It should create the graph with enough room to hold e edges. The graph will need to be allocated in the heap, since it will be used after readGraph returns. First, write a heading and a contract for readGraph. Then write the function definition. The input starts with a line that tells how many vertices the graph has. If there are five vertices, then those vertices have numbers 1, 2, 3, 4 and 5. In general, if there are n vertices, then they are numbered 1, …, n. Following the first line are the edges, one per line. Each edge line has three integers on it. Line 2 4 50indicates that there is an edge of weight 50 between vertices 2 and 4. The weights are integers. The end of the input is signaled by a line that contains just a 0. Input 5 1 2 9 1 3 12 2 4 18 2 3 8 2 5 20 3 5 15 0describes graph
ReadGraph must use insertEdge to add each edge to the graph. It is tempting to read three numbers per line, no matter what. On the last line, only one number will be read successfully. But programs tend to be modified, and, wherever it is not too much work, it is a good idea to be prepared for modifications. What if the program is changed so that there is more input after the graph? You don't want readGraph to read into the what comes after the graph. Write readGraph so that it does not depend on the line containing just 0 to be the last thing in the input. That is easy to do. Read the first number and check it before you read the next two. |
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7. Output and Output Format
In mst.cpp, define and document a function writeGraph(G) that writes a description of graph G on the standard output. See the sample output above for a reasonable output format. WriteGraph should not say that the graph is "the input graph". It should just show the graph. Naming the graph is the responsibility of whoever calls writeGraph. |
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8. Main
In mst.cpp, define a main function that just reads a graph and writes it. Test the program. Do not move on until this works. Do not skip testing this! |
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9. Sorting an Array of Edges
In mst.cpp, define and document a function sortEdges(G) that sorts the array of edges in graph G into nondescending order by weight. Use library function qsort to do this. (Should the contract say that sortEdges uses qsort?) See Using qsort. |
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10. Building a Minimal Spanning Tree
In mst.cpp, define and document a function minimalSpanningTree(G) that takes a graph G (passed by pointer) as a parameter and returns a pointer to a minimal spanning tree of G. Document minimalSpanningTree as it will be when finished. It will return a minimal spanning tree of G. For now, only make it sort the edges of G and return G. You will write more later. Modify main so that it computes the minimal spanning tree (by calling minimalSpanningTree) and prints that tree. Test this, keeping in mind that minimalSpanningTree currently just sorts the edges. You can use the automated tester. Are the edges sorted correctly? Do not skip testing this! |
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11. Kruskal's Algorithm
Modify your minimalSpanningTree function definition so that it uses Kruskal's algorithm to compute a minimal spanning tree of G. Create a new graph T (in the heap) without any edges. Look at each edge in G and decide whether to add it to T. If so, then add it to T. You already have a function that adds an edge to a graph. Use it. See the determining connections for how to decide whether to add an edge to the minimal spanning tree. The only modification of G that minimalSpanningTree(G) is allowed to do is to sort the array of edges. You are not allowed to destroy the array of edges in G in order to build the array of edges of the minimal spanning tree! Graph G must be intact after running minimalSpanningTree(G). Making a shallow copy of G won't help. Do not copy G. |
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12. Getting the Total Weight
In mst.cpp, define and document a function to compute the total weight of a graph by adding up the weights of the edges. It should return the weight, and must not write anything. Add a call to this function in main. Test it. |
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13. Proofread your program.
Proofread your contracts. Make sure that there are no spelling or grammatical errors. Be sure that each contracts explains how each parameter affects the function, and refer to each parameter by its name. |
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14. Submit your work.
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You can use the following commands to compile, run or test your program.
Consider adding your own tests. You can put test inputs into files, and I strongly recommend that you do that. Just redirect the standard input to a file. Command
./mst <graph1.txtruns executable file mst with its standard input coming from file graph1.txt.
As always, your program must follow the coding standards.
Check the following.
Be sure that your program reads from the standard input and writes to the standard output.
Do not ignore compiler warnings. If you do not understand what a warning means, ask about it.
It is crucial that equiv.cpp works correctly. If you still need to fix it, then fix it. You cannot get mst.cpp to work with a faulty equivalence relation manager.
But do not modify equiv.h or equiv.cpp unless you are only fixing errors. No minimal-spanning-tree code should find its way into equiv.h or equiv.cpp.
Pay attention to contracts. In the past, many students have become lax about contracts in this assignment, and it has cost them a lot of points.
Be sure to document your structure type definitions. Say what an object of the structure type represents and what information each field holds.
Ensure that you are implementing Kruskal's algorithm correctly. Test your program on more than one graph. Test your program on a graph where all of the edges are used in the minimal spanning tree. Check the output. Is it right? Don't just assume that the output is correct.
Make sure that arrays are large enough. If array A has size n then A[n] is out of bounds.
Your mst.cpp file must not make use of the fact that type ER is a pointer or an array. It should restrict itself to the interface of the equivalence relation manager module. You can expect to lose a lot of points if you ignore this.
Each function can have at most one loop in its body. Do not use one loop to simulate two nested loops by manipulating the loop parameters.
Each function body must have no more than 16 noncomment lines.
A function body must not change the value of a call-by-value parameter. It can change a call-by-reference parameter or the value pointed to by a call-by-pointer parameter.
Make sure that each function does its whole job., not just part of it job. Make sure that a function does not have undocumented requirements.
Avoid code duplication and duplicate calls.
To turn in your work, log into xlogin, change your directory for the one for assignment 5, and use the following command.
~abrahamsonk/2530/bin/submit 5 mst.cpp equiv.cpp equiv.hAfter submitting, you should receive confirmation that the submission was successful. If you do not receive confirmation, assume that the submission did not work.
Be sure to submit all three files.
Command
~abrahamsonk/2530/bin/submit 5will show you what you have submitted for assignment 5.
You can do repeated submissions. New submissions will replace old ones.
Late submissions will be accepted for 24 hours after the due date. If you miss a late submission deadline by a microsecond, your work will not be accepted.
To ask a question about your program, first submit it, but use assignment name q5. For example, use command
~abrahamsonk/2530/bin/submit q5 mst.cpp equiv.cpp equiv.hInclude a data file if appropriate. Don't expect me to create your data file.
Then send me an email with your question. Do not expect me to read your mind. Tell me what your questions are. I will look at the files that you have submitted as q5. If you have another question later, resubmit your new file as assignment q5.