Copy equiv.h and equiv.cpp from Assignment 4 into that directory. Put all of the files for this assignment in that directory.

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.

Write a comment in mst.cpp telling what this program will do when it is finished. Include the input format (below), an example input and an example output. when it is finished.

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. You will need to have an array of edges. 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. Any bugs that result from ignoring this advise are on you.

To help to avoid confusing vertex numbers with edge numbers, create types vertexNum and edgeNum as follows.

  typedef int vertexNum;
  typedef int edgeNum;

Do not confuse a vertex number with a count of the number of vertices in a graph. If you talk about vertex number 1, then you use type vertexNumber. If a variable represents an edge number, it should have type edgeNum.

Edge weights are not vertex numbers or edge numbers. Their type should be int.

Start by defining two structure types, Edge and Graph, in mst.cpp. Be sure to document each.

1. An object of type Edge represents one edge in a graph. It has three fields: two vertex numbers and a weight. Include a parameterless constructor that sets all three fields to 0.

   An Edge stores two vertex numbers. What should their types be?

2. An object of type Graph represents a weighted graph. It has four fields:

   • the number of vertices in the graph,
   • the number of edges in the graph,
   • an array of Edge structures that holds the edges,
   • the physical size of that array of edges.

   Include a constructor Graph(nv) that yields a graph with nv vertices and no edges. The constructor must create the array of edges.

   You can assume that there are no more than 100 edges, but that number must be easy to change. To increase the maximum number of edges to 200, for example, should only require changing only one line of your program. To achieve that, create a named constant that is the maximum number of edges. For example,

      const int maxEdges = 100;
   

   defines constant maxEdges with a value of 100. Any time you need to refer to the maximum number of edges, use maxEdges, not 100.

   Note. The physical-size field should be set to maxEdges by the constructor. When you want to know the size of the array of edges in g, use g.physicalSize (or whatever you have called that field), not maxEdges. The reason is that it is is possible that you add another Graph constructor later that creates an array whose size is not maxEdges. You would prefer not to be forced to edit the rest of the code when you do that.

   Do not confuse the physical size of the array with the current number of edges.

All structure types must be documented by saying (1) what an object of this structure type represents and (2) what property each field represents.

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 should 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.

It is critical that g be passed by reference to insertEdge. Otherwise, insertEdge will modify a copy of your graph, which is not what you want.

Here is a suitable heading for insertEdge.

  void insertEdge(vertexNum u, vertexNum 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.

In mst.cpp, define a function readGraph(G) that takes a graph G and reads a description of a graph from the standard input, storing it into G.

First, write a contract. 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 50

looks like this.

5
1 2  9
1 3 12
2 4 18
2 3  8
2 5 20
3 5 15
0

Function readGraph must use insertEdge to add each edge.

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.

WriteGraph will need a variable that holds an index in the array of edges. That variable is an edge number. So it should have type edgeNum.

In mst.cpp, define a main function that just reads a graph and writes it. Test it. Do not move on until this works. Do not skip testing this!

In mst.cpp, define and document a function sortEdges(G) that sorts the array of edges in graph G. Use library function qsort to do this. (Should the contract say that sortEdges uses qsort?)

Function qsort is part of the C standard library. It is a general-purpose sorting function that sorts an array using a variant of the Quicksort algorithm. You should include header file <cstdlib> to use qsort.

Qsort needs information about the array and how to sort it. It also needs for you to perform some conversions that, in general, would be questionable, but that work correctly here.

Qsort takes four parameters. Function call

   qsort(base, numElements, elementSize, compare);

• Parameter base is a pointer to the array that you want to sort. You should convert this pointer to type (void*).

• Parameter numElements is the number of elements in the base array. It is the logical size of the edge array.

• Parameter elementSize is the number of bytes occupied by each element. Since each thing in the array has type Edge, elementSize should be sizeof(Edge). (In general, sizeof(T) is the number of bytes that something of type T occupies.)

• Parameter compare is a function. (C++ allows you to pass a function as a parameter to another function. You do not run the function yourself; qsort will run it for you, many times. You are just lending the tool to qsort, not using the tool yourself.)

  Function compare takes two parameters, which will be pointers to particular members of the array when qsort calls it. compare(A, B) should return

  A negative number

  if *A should come before *B in the sorted array
  

  A positive number

  if *A should come after *B in the sorted array
  

  0

  if you do not care what order they are in
  

  Your comparison function can look as follows, since you want to sort according to the weights of the edges.

    int compareEdges(const Edge* A, const Edge* B)
    {
      return A->weight - B->weight;
    }
  

To run qsort, define a new type, QSORT_COMPARE_TYPE, as follows. It is the type of the compare parameter that qsort expects to receive.

   typedef int (*QSORT_COMPARE_TYPE)(const void*, const void*);

   qsort((void*) Arr, n, sizeof(Edge), (QSORT_COMPARE_TYPE) compareEdges);

In mst.cpp, define and document a function minimalSpanningTree(G) that will take a Graph G as a parameter and return a minimal spanning tree of G (also of type Graph).

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. Are the edges sorted correctly? Do not skip testing this!

Now modify your minimalSpanningTree function definition so that it uses Kruskal's algorithm to compute a minimal spanning tree of G. Create a new graph K without any edges.

Look at each edge in G and decide whether to add it to K. If so, then add it to K. You already have a function that adds an edge to a graph. Use it.

MinimalSpanningTree will need a variable that holds an index in the edge array. What should its type be? Think about it.

See the next step, "determining connections," to see 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). Don't be naive about copies of graphs. You do not need to copy G. Read about shallow copying.

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, each vertex is in an equivalence class by itself. When Kruskal's algorithm calls for adding an edge between u and v, tell the equivalence relation manager to merge u and v. To ask if there is a path between u and v, just ask if u and v are in the same equivalence class.

Do not copy your equivalence manager into your file that implements Kruskal's algorithm. Keep it in separate files equiv.cpp and equiv.h. Just include "equiv.h" in mst.cpp. Do not include equiv.cpp in mst.cpp!

Now test this. Is the minimal spanning tree correct? Try more than one graph! Many students have submitted programs that they thought were correct but that failed some tests.

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.