Computer Science 3300
Section 001
Fall 2013
Programming Assignment 5

Assigned: Monday, November 11
First version due: Monday, November 25, 11:59pm
Second version due: Friday, December 6, 11:59pm

Read the entire assignment before starting on it.

Table of contents

  1. Background: graphs and spanning trees
  2. Program Requirements
  3. Algorithmic issues
  4. Design issues
  5. Implementation issues
  6. Handling connections: an abstract data type
  7. An implementation plan
  8. Notes on grading
  9. Submitting your work
  10. Extra credit (Read only after you have the program working)
  11. Why should I believe that Kruskal's algorithm works? (This is not required for writing the program.)
  12. Motivation for computing minimal spanning trees (you can skip this)

Background: graphs and spanning trees

You used graphs in assignment 3. This problem uses them again, so here is a review.

graph is a collection of vertices connected by a collection of edges. An edge connects exactly two vertices to one another. You can draw a picture of a graph by showing the vertices and the edges connecting them. Here is an example. The vertices are shown as circles with numbers in them and the edges are lines connecting the vertices.

We will only be concerned with connected graphs. A graph is connected if there is a path from every vertex to every other vertex, following edges. 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.

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 picture of a weighted graph.

A spanning tree of a 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.

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 59. Here is another spanning tree for the same graph. Its weight is 48.

Obviously, some spanning trees have smaller weight than others. A minimal spanning tree is a spanning tree with the smallest possible weight.

Program Requirements

Write a program that reads a description of a weighted graph and prints the edges that are part of a minimal spanning tree of that graph. It should also print the total weight of the minimal spanning tree. The input should be read from the standard input (cin), and the output should be written to the standard output (cout).

Input format

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
indicates that there is an edge between vertices 2 and 4, and that its weight is 50. The end of the input is signaled by a line that contains just a 0. An input describing graph

might look like this.

1 2  9
1 3 12
2 4 18
2 3  6
2 5 20
3 5 15

You can assume that there are no more than 100 edges, but that number must be easy to change. To increase this to 200, for example, should only require changing one line of your program.

Output format

The output of your program for this input might look like this.

The input graph has 5 vertices, and its edges are as follows.

  vertices    weight
  1   2            9
  1   3           12
  2   4           18
  2   3            6
  2   5           20
  3   5           15
A minimal spanning tree uses the following edges.

  vertices    weight
  2   3            6
  1   2            9
  3   5           15
  2   4           18

The total weight of the spanning tree is 48.

Remarks on input format

Your program is not required to insist that the three numbers that describe an edge are on one line, but it can if it wants to. For example, if your program allows an input like the following, that is acceptable. But your program is not required to allow such inputs. (You will find it easier to allow these additional input forms.)

1 2  
3 12
2 4 18  2 3  6
2 5 20
3 5 15
The only difference between this and the previous input is the placement of line breaks.

Important note. The first number in the input is the number of vertices, not the number of edges. The only way you can determine the number of edges is by reading them and counting them.

Important note. The last line has only one number on it, not three.

Algorithmic Issues

For this problem, the algorithm is sufficiently difficult that it is a good idea to think about the algorithm before proceeding with the program design.

Kruskal's algorithm

There is a well-known algorithm, called Kruskal's algorithm, for computing a minimal spanning tree of a weighted graph. It goes as follows.

  1. Sort the edges from small weight to larger weight. Call the sorted list of edges e0, ..., em-1. So, for example, e0 is the edge with the smallest weight. (See below for an easy way to sort an array of edges.)

  2. Start with an empty graph K. It has all of the vertices, but no edges yet. Edges will be added to it until, at the end, it is the minimal spanning tree.

  3. For i = 0, ..., m-1 do

    1. Look at edge ei. Suppose that it connects vertices u and v.
    2. If there is not already a path in K between vertices u and v, then add edge ei to K. Otherwise, do not add it to K. (How to determine whether there is already a path is discussed below.)

When the algorithm is done, K is a minimal spanning tree for your graph. See below for an argument that Kruskal's algorithm works.

Determining connections

A tricky issue with Kruskal's algorithm is how to determine whether two vertices are already connected by a path. Suppose we write u ~ v is there is a path between u and v. You might have seen the notion of an equivalence relation (in CSCI 2427). Notice that ~ is an equivalence relation.

  1. (~ is reflexive) u ~ u. That is, you can always get from u to itself (by taking no edges).

  2. (~ is symmetric) If u ~ v then v ~ u. That is, if there is a path from u to v, then there is also a path from v to u.

  3. (~ is transitive) If u ~ v and v ~ w then u ~ w. If there is a path from u to v and also a path from v to w, then there is a path from u to w.

An equivalence relation is always characterized by its equivalence classes. That is, at any point in time, you can characterize which vertices are connected to which others by writing a collection of sets of connected vertices. For example, suppose that there are 6 vertices. Sets

{1,3,4}, {2,5}, {6}
indicate that vertices 1, 3 and 4 are connected to one another, vertices 2 and 5 are connected to one another, and vertex 6 is not connected to any other vertex. Only vertices that are in the same set are connected to one another.

Look at what Kruskal's algorithm needs to do with this collection of sets.

  1. It needs to be able to ask whether two vertices are in the same set. That is, are they already connected by a path?

  2. It can add an edge, which causes two of the sets to be combined. For example, if it starts with

    {1,3,4}, {2,5}, {6}
    and adds an edge from 3 to 5, then sets {1,3,4} and {2,5} get combined, yielding a new collection of sets
    {1,2,3,4,5}, {6}

If we can figure out a way to handle a collection of disjoint sets, with an operation to test whether two numbers are in the same set, and another operation that combines two of the sets in to one, then we will be able to implement Kruskal's algorithm. An algorithm that manages sets and that handles the operations required by Kruskal's algorithm is discussed below.

Design Issues

Now, with the basics of the algorithms outlined, we can design the program. This program should have three modules.

  1. Module 1 implements the equivalence handling. It should consist of two files, equiv.h and equiv.cpp. In equiv.h, define a structure type Equiv, holding a pointer to an array of integers and the size of that array. See the description of the equivalence manager algorithm for the intent of the array. Write contracts and prototypes for two functions, together(e, x, y) and combine(e, x, y). The idea is that together(e, x, y) returns true if x and y are in the same set, as described by equivalence relation e; and combine(e, x, y) modifies e so that the sets containing x and y are combined into a single set. The headings are as follows.

       bool together(Equiv& e, int x, int y);
       void combine(Equiv& e, int x, int y);
    File equiv.cpp contains implementations of those functions, plus helper functions. Every function, including helper functions, must have a clear contract that tells what it accomplishes and how each of its arguments affects what it accomplishes.

  2. Module 2 implements Kruskal's algorithm. Keep it simple. It has two files, graph.h and graph.cpp. In file graph.h, create a structure type Graph, where a graph stores

    You will also want a structure type that represents one edge, holding two vertex numbers and a weight. Create the following functions, and add prototypes and contracts for them to graph.h. Put the implementations of these functions into graph.cpp.

    1. A function that reads information about a graph, in the format shown in the requirements section, and that returns a graph holding the information that it read.

    2. A function to print a graph, by telling how many vertices it has, how many edges it has, and then showing the edges. It should not print any specific heading. For example, it should not say that it is printing the input graph. Just provide information about the graph.

    3. A function to compute the total weight of a graph, by adding up the weights of the edges. It should return the weight, and should not write anything.

    4. A function that takes a graph G as a parameter and returns another graph that is a minimal spanning tree of G. This function contains an implementation of Kruskal's algorithm.

    Note that graph.cpp will need to include equiv.h, since it needs the equivalence manager as a tool.

  3. Module 3 contains only a main program. Call it main.cpp. It should include graph.h, but not equiv.h. The main program should be short. It just does the following steps.

    1. Read a graph. This should be one line of code, since there is already a function that does this job.
    2. Print the graph (two lines of code, one for a heading saying that this is the original graph, and another line to call the graph printer).
    3. Compute the minimal spanning tree of the graph (one line).
    4. Print the minimal spanning tree of the graph (two lines, since you need to say what is being printed).
    5. Print the weight of the minimal spanning tree (one line).

Implementation Issues

Edges and vertices

Do not confuse edges with vertices. Keep track of the number of vertices in a graph separately from 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.

Sorting an array of edges

You will need to sort the array of edges according to the weights of the edges. There is a function in the C standard library called qsort that will sort an array. It implements a variant of the Quicksort algorithm. You should include header file <cstdlib> to use qsort.

Qsort is a general sorting function, designed to be able to sort any type of array into any desired order. In order to achieve that degree of generality, 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. In general, you write

   qsort(base, numElements, elementSize, compare);

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*);
Now statement
   qsort((void*) Arr, n, sizeof(Edge), (QSORT_COMPARE_TYPE) compareEdges);
will sort array of edges Arr according to weights, with smaller weights toward the beginning of the array.


Your function, compareEdges, does not have type QSORT_COMPARE_TYPE, since compareEdges takes two parameters of type const Edge*, and qsort says that it should take two parameters of type void*. But you can tell the compiler to ignore this, and to let you pass your compareEdges function to qsort, by doing a cast, which tells the compiler to treat a value of one type as if it is a value of a different type. In most cases, this is a very dangerous thing to do, but you have to do it to use qsort.

Handling Connections: An Abstract Data Type

Programs often need to use tools that are provided as abstract data types. An abstract data type is a type of data along with some operations. An object that belongs to the data type remembers some information, and the operations either get or modify that information.

You need an abstract data type that manages equivalence classes of positive integers. This section describes the abstract data type in a little more detail, and tells you a simple way to implement it.

The conceptual view

The conceptual view is how somebody who uses the abstract data type thinks about it. You create an Equiv object by telling it a positive integer n. You write

  Equiv e(n);
to create an object e that manages an equivalence relation (a collection of disjoint sets) over 1, ..., n. Initially, the sets are {1}, {2}, ..., {n}.

Two functions are available.

together(e, x, y)

This returns true if x and y are currently in the same set in object e, and false if they are in different sets.

combine(e, x, y)

This makes x and y equivalent, combining the sets that contain them.

The following shows an example that does a sequence of combine and together operations. After each combine operation, the new collection of sets is shown. The result of a together operation is either true or false. The example is for the case where n = 7.

Operation Result
  {1} {2} {3} {4} {5} {6} {7}
combine(e,3,6) {1} {2} {3,6} {4} {5} {7}
combine(e,4,5) {1} {2} {3,6} {4,5} {7}
together(e,3,6) true
together(e,4,6) false
combine(e,3,5) {1} {2} {3,4,5,6} {7}
combine(e,3,1) {2} {1,3,4,5,6} {7}
together(e,3,4) true
together(e,1,6) true
together(e,2,4) false
combine(e,3,5) {2} {1,3,4,5,6} {7}

Notice that you are allowed to combine two numbers that are already in the same set. Nothing happens then.

Important note. Your graph.cpp module uses the equivalence manager. It must only use the operations described here, under the conceptual view of the equivalence manager. It must not make use of any of the implementation details discussed under the algorithm. For example, graph.cpp should make no mention of the concept of a boss or a leader.


There is a simple way to implement the Equiv abstract data type. The idea is to have a leader of each set. For example, if numbers 3, 5 and 8 are in the same set, then they might decide that 5 is their leader. You write a function leader(e, k) that returns the leader of the set that contains k in collection of sets e. Then, to find out whether two numbers are in the same set, just check whether they have the same leader.

To implement the leader function, use another idea that is related to leaders. Each number that is not a leader has a boss. The boss of a number might be that number's leader, or it might not. But the boss is closer to the leader. You find your leader by going to your boss, then to your boss's boss, etc, until you hit a number that does not have a boss. To indicate that a leader does not have a boss, we will say that its boss is 0. So, if e.boss[k] is k's boss (in e), then you can find e's leader using the following loop.

  r = k;
  while(e.boss[r] != 0) r = e.boss[r];
Now r is k's leader.

When you combine two sets, you start by getting the leaders of those sets. Be careful; if the leaders are the same, do not make any changes. But if the leaders are different, then make one of those leaders be the boss of the other. For example, if x and y are two different leaders, then

  e.boss[x] = y;
makes y be x's new boss.

It is critical that you only change the boss of a leader. If x is not a leader, do not change x's boss.

Remember that, if you are asked to combine two values that already have the same leader, do not make any change.

An Implementation Plan

Here is a suggestion for how to write this program. I have included a time estimate for how long you might realistically expect each step to take. Your times will vary!

  1. [3 hours] Write the type definitions, contracts and headings in equiv.h. Implement the functions in equiv.cpp, including functions leader, together and combine. They should be very short and simple. Add one more function that prints the boss array, for testing. Add debug prints to the combine function so that it can show the boss array before and after it runs.

    Write a separate main program that tests your implementation of this module. Print the result of each together call. Be sure to try cases where you are combining things that are already together. Test thoroughly. Do not throw away your tester, and do not move on until this module works.

    Read: Handling connections: an abstract data type.

  2. [3 hours] Create types Edge and Graph in graph.h. An Edge just remembers the two vertices that it connects and its weight. A graph needs to remember how many vertices it has, how many edges it has, and an array of Edges. Include a constructor that builds an array of a fixed size. You will want to tell the constructor how many vertices there are, but make it set the graph to have no edges initially. The size of the array of edges should be a named constant, such as maxEdges. For example,

       const int maxEdges = 100;
    defines maxEdges to be 100. Do not forget to write const. Make the definition of maxEdges global so that it can be shared by different functions. Global constants are not a problem.

    Add prototypes (headings) to graph.h for the readGraph and printGraph functions.

    In graph.cpp, define a function that adds an edge to a graph. Then implement the functions that read and print a graph. You should be able to adapt your readGraph function from assignment 3. But do not use an adjacency list for this program. Just put the edges into the array of edges. You should find readGraph and printGraph to be simple functions.

    Now write part of the main function in main.cpp. Just have it read the graph and print it. Do not move on until it works.

    Read: Input format, Remarks on input format, Output format and Synopsis of module 2.

  3. [0.5 hours] Implement the function that computes the total weight of a graph. Add a line to your main program that computes and prints the total weight of the input graph. Test it.

    Read: Function to compute the total weight of the graph.

  4. [3 hours] Implement the function that computes the minimal spanning tree, using Kruskal's algorithm. It needs to take a graph as a parameter and yield another graph, which will hold the minimal spanning tree.

    Add debug prints that allow you to print the list of edges. Show the list before and after you sort it, to ensure that you are using qsort correctly.

    Since you already have the equivalence manager, Kruskal's algorithm is very short and simple. Make sure not to make it too complicated. The following is a sketch to show how simple it is. It presumes that things have certain names, but you should be able to see how to adapt it to your names.

      Sort the edges in the graph G.
      Make a new graph K with no edges (but with room to add edges).
      Make a new equivalence manager E to manage G.numVertices numbers.
      For i = 0,...,G.numEdges-1
        u = G.edges[i].vertex1
        v = G.edges[i].vertex2
        If E says that u and v are not together then
          Tell E to combine u and v.
          Add edge G.edges[i] to K.
        End if
      End for

    Modify your main program to use it, and to do the entire job, according to the requirements. Test it on several different graphs. Do not try just one graph. Be warned that most students under-test their programs, and turn in programs that contain errors.

    Read: Kruskal's algorithm, Sorting the array of edges

  5. Now that you have a working version, save it and keep it. For example, create a subdirectory to hold the basic, working version. In Linux, command

      cp -R olddir newdir
    will make a copy of directory olddir, calling the copy newdir. For example,
      cp -R mst mst1
    makes a copy of directory mst and all of its contents, and calls the copy mst1.

  6. If you have time, implement the extra credit parts. Test them carefully. Do not make the mistake of just writing something and then doing shoddy testing. Actually look at the results of your tests. Notice that you are modifying equiv.cpp. Instead of just running your application, try running your tester that was designed for testing equiv.cpp. You will get better testing that way. Be warned that most students implement the improvements incorrectly because they do not take time to think carefully about what they are doing.

  7. Submit your work.

Notes on grading

Here are some things to do and mistakes to avoid, with a range of points that you might lose (out of 100) for each one.

  1. Write your name, the course and the assignment number in a comment at the top of the program, along with the tab stop distance. (up to 2 points)

  2. Failure to compile (100 points). Your program is expected to compile. A program that has fatal compile errors will receive a grade of 0.

  3. Compiler warnings (up to 5 points). Your program should not have any compiler warnings.

  4. Failure to follow the design (up to 100 points). The design requires you two write certain functions and to organize the program into modules in a particular way. It is not acceptable to substitute a different design.

    The design does not call for global variables, with the possible exception of variables for debugging. Do not use global variables. Using even one global variable means that you are not following the design, and can lead to a very high loss of points. You can use global constants.

  5. Extra credit. (15 points) If your program works, producing the correct results, and has either no problems or only minor problems, and you have correctly implemented both of the improvements to the implementation of the Equiv abstract data type, then you will receive 15 bonus points.

    If your program does not work or has serious problems, then you cannot receive points for the extra credit part. So concentrate on making the program right before you work on the improvements. You will receive more points for a program that works without the extra credit than you receive for one that does not work and does the extra credit modifications.

  6. Partial program. If only part of the program is written, you will receive some credit, depending on how much is written, and how well that part is written. Do not count on a great deal of credit for a partial program that cannot be run, or that only reads and echoes the input.

  7. Incorrect results (up to 40 points). The program should produce correct results.

  8. Memory faults (up to 40 points). If your program encounters a memory fault then it does not do what it is supposed to do. Ensure that you do not use an uninitialized or dangling pointer. Do not use an index that is outside the bounds of an array.

  9. Violating the abstraction of the Equiv abstract data type (up to 20 points) If graph.cpp or main.cpp contain anything that is supposed to be private to the equiv.cpp module, then you have violated the Equiv abstraction. You will lose substantial points for that.

  10. Private functions advertised in header file (up to 3 points). File equiv.h should say as little as possible about things that should only be used in equiv.cpp. It should contain prototypes for together and combine, but not for leader, since the leader function is only for internal use in equiv.cpp, and is not part of the abstract data type.

  11. Poor or missing contracts (up to 15 points). Every function must have a clear and precise contract. A contract tells what a function does, not how it works. A contract tells precisely how every one of its parameters affects what the function does, and refers to parameters by name. If the function returns anything, then the contract explains what it returns and how that is affected by the parameters.

  12. Functions do not do what they are supposed to do, or return incorrect results according to their contracts (up to 7 points per function).

  13. Crippled functions (up to 8 points). A crippled function is one that does not really do its whole job, but requires its caller to do part of the job.

  14. Poor indentation (up to 10 points). You are expected to indent your function definitions correctly. Minor problems will not lose much, but seriously bad indentation can lose 10 points, and a complete lack of indentation can lose much more than that.

  15. Failure to explain tabs (2 points). Be sure that you either do not use tabs in your program or that you tell where your tab stops are at the top of the program. Different editors are set up with different tab stops. For example, write

      // tab stops: every 4 characters
    If you do not know where the tab stops are, do not use tabs.

  16. Excessively complicated algorithms (up to 5 points per algorithm).

  17. Excessive inefficiency (up to 5 points).

  18. Long lines, margin comments (up to 5 points). Avoid long lines. As a rule of thumb, keep lines at most 80 characters long. Avoid margin comments (comments written in the right margin telling what the code to the left of them does). Those comments rarely say anything useful, and they make lines long and clutter the program. If you need to write a comment inside a function, write a paragraph comment telling what the next paragraph of code does.

  19. More than one line needs to be changed to alter the maximum number of edges allowed (up to 3 points). Make sure that you use a named constant for the maximum number of edges. For example, say

      const int maxNumberOfEdges = 100;
    and always use maxNumberOfEdges as the maximum number of edges, not 100.

  20. Packing functions together (up to 1 point). Put a blank line between function definitions. There is no need to pack them together. That makes the program more readable.

    But also do not put huge blank sections in your program consisting of several blank lines in a row. A long space is two blank lines. Longer than that is excessive.

  21. system("pause") (up to 5 points). Some programmers write system("pause") at the end of main to compensate for a flaw in the design of a Windows development environment. Since that is a nonstandard thing, I must edit your program to get rid of it before I can test your program. I should not have to fix your program for any reason.

Submitting Your Work

To turn in this program, log into one of the Linux computers in the lab. (You can log in remotely.) Ensure that your files are there. Change to the directory that contains those files. Then issue the follow commands.

For part (a) use

~abrahamsonk/3300/bin/submit 5a equiv.cpp equiv.h graph.cpp graph.h main.cpp
and for part (b) use
~abrahamsonk/3300/bin/submit 5b equiv.cpp equiv.h graph.cpp graph.h main.cpp

Extra Credit

For extra credit, implement two improvements on the algorithm for managing equivalence classes. But do so carefully.

The improvements make the equivalence manager more efficient when it is working on very large collections of sets.

For simplicity, instead of e.boss, I will just write boss in this section. Assume we are talking about a particular e.

The first improvement: improving leader

The first improvement involves a change to the leader function. After the leader function scans through a chain of boss links to find the leader of a number, it should go back through the chain and put the genuine leader in the boss array for every number that was looked at in the chain. That way, subsequent leader computations will go much more quickly. For example, if the boss array contains

  boss[1] = 2
  boss[2] = 4
  boss[3] = 0
  boss[4] = 6
  boss[5] = 3
  boss[6] = 0
  boss[7] = 4
  boss[8] = 5
then computing the leader of 1 requires chaining through 1, 2, 4, 6, stopping at leader 6. The improvement changes the contents of the boss array to the following, by installing the correct leader (6) of each of the numbers that was looked at.
  boss[1] = 6
  boss[2] = 6
  boss[3] = 0
  boss[4] = 6
  boss[5] = 3
  boss[6] = 0
  boss[7] = 4
  boss[8] = 5
You can do this with another loop that rescans through the chain, the same way the chain was scanned the first time, but now putting the leader into the array as you go. Alternatively, modify the leader function to be recursive, then just change the boss after each recursive call. (In the past, most students who did this improvement using a loop got it wrong, so be careful. Most students who did it with recursion got it right.)

Notice that we have not scanned the entire boss array from beginning to end! Boss[8] is still 5, even though the leader of 8 is 3. Only the numbers that were looked at in the original scan have their boss values changed. If you try to change everything in the array, you make the implementation slower, not faster.

Also notice that it was not just the boss of 1 that was changed. All of the numbers that were examined in the chain have their bosses set to their leaders. For example boss[2] was changed too.

Be sure to test your improved leader function. It is easy to write it incorrectly. Try it by hand to see whether it seems to do the right thing.

The second improvement: improving combine

Each number that is a leader has a collection of constituents who have that number as their leader. For example, in the above array, number 3 is a leader, and the constituents of 3 are 3, 5 and 8. So number 3 has three constituents, counting itself. The constituent count of a leader tells how many numbers are in the set that contains that leader.

When doing a combine operation, you find two values s and t that are leaders. You can then either change the boss of s to t (so s is no longer a leader) or change the boss of t to s (so t is no longer a leader). Either one will accomplish the goal of combining the two sets, but the choice of which to do influences the efficiency of the implementation. The best choice is to change the boss of the number that has the fewest constituents. That tends to keep the lengths of the boss chains up to the leader short.

Modify the data structure so that each number has not only a boss, but also a count of its constituents. A number that is not a leader has no constituents. So now the information is an array of structures, where each structure contains a boss and a constituent count.

A picture of the initial array, before any combines have been done, might look like this. Notice that each number has one constituent, itself.

  index    boss  numConstituents
    1        0           1
    2        0           1
    3        0           1
    4        0           1
    5        0           1
    6        0           1
    7        0           1

When doing a combine, compare the constituent counts of the two leaders. Change the boss of the leader with the fewer constituents. (If they have the same number of constituents, then the choice is arbitrary.) If you change things so that the boss of s becomes t, then remember that all of the constituents of s become new constituents of t.

If you do combine(e,3,5) in the above array, you might arbitrarily decide to make the boss of 3 be 5. Then the array looks like this.

  index   boss  numConstituents
    1        0           1
    2        0           1
    3        5           0
    4        0           1
    5        0           2
    6        0           1
    7        0           1
If you now do combine(e,5,1), you must change the boss of 1, since it has fewer constituents than 5. The array ends up looking like this.
  index   boss  numConstituents
    1        5           0
    2        0           1
    3        5           0
    4        0           1
    5        0           3
    6        0           1
    7        0           1
As before, only change the boss of a number that is currently a leader. If you now do combine(e,2,1), you must realize that you are really being asked to combine 2 and 5, since 5 is the leader of 1. Since 5 has more constituents, you change 2's boss, yielding
  index   boss  numConstituents
    1        5           0
    2        5           0
    3        5           0
    4        0           1
    5        0           4
    6        0           1
    7        0           1
As you can see, this improvement tends to lead to shorter chains of bosses before the leader is found.

Suppose you continue by combining 6 and 7. You might get the following.

  index   boss  numConstituents
    1        5           0
    2        5           0
    3        5           0
    4        0           1
    5        0           4
    6        7           0
    7        0           2
Now combine 1 and 6. Their leaders are 5 and 7. Since 5 has more constituents, change the boss of 7 to be 5. The new information is as follows.
  index   boss  numConstituents
    1        5           0
    2        5           0
    3        5           0
    4        0           1
    5        0           6
    6        7           0
    7        5           0
Notice that 5 now has six constituents. Also notice that, although 6 is one of 5's constituents, its boss is 7. The boss chains are shorter, but you still need to do the leader calculation using a loop (or recursion).

Efficiency before and after improvements

If you do neither improvement, then the equivalence manager can take time proportional to n2 to process n combine and together requests. With the improvements, the time is no worse than proportional to n(f (n)) where f (n) is a very slowly growing function of n, so slow that, for all practical values of n, f(n is no more than 6.

Why Should I Believe that Kruskal's Algorithm Works?

Algorithm design requires creativity, and often requires simply feeling around until you hit on something that looks like it might work. But be careful not to stop after you have something that feels reasonable. A lot of algorithms that initially look like they will work are incorrect. You don't want to jump into writing a program that implements the algorithm when the algorithm is wrong. That is a waste of time. It is important to offer some careful reasoning about why the algorithm works before you try to implement it.

Kruskal's algorithm adds the edges starting with the cheapest ones, and, since it is supposed to find a cheapest possible spanning tree, it seems to be doing something right. But that is not justification that the algorithm works. It is only justification for investigating it to see whether it works.

This section offers an argument that Kruskal's algorithm is correct. It should be clear that Kruskal's algorithm finds a spanning tree, if one exists, since it only skips over edges between vertices that are already connected by a path. The issue is whether that spanning tree is minimal. The argument is necessarily of a mathematical nature, as are most such arguments. It will probably take more than one reading for a person who is inexperienced with this kind of argument to make sense of it. But it is worth getting used to the idea of checking your algorithms carefully.

We will simplify the argument by assuming that no two edges of a graph can have the same weight. That assumption is fairly easy to get rid of, and the argument is essentially the same, but the assumption reduces the complexity of the argument a bit.

The argument

Before trying to show that an algorithm works, you typically try looking for some counterexamples. Try to find an input that the algorithm gets wrong. Suppose that you try that for a while with Kruskal's algorithm, and are not able to find any counterexamples. Then imagine somebody else searching for counterexamples. What if that other person finds one? Think about that (unknown, or generic) counterexample, and try to convince yourself that it cannot really be a counterexample; that is, no counterexample exists.

CLAIM. Kruskal's algorithm finds a minimal spanning tree of a graph.

The argument is a kind of thought experiment. Suppose that somebody else claims to have found a counterexample demonstrating that Kruskal's algorithm does not work. We ask that person to give us the weighted graph G that the algorithm gets wrong, and the actual minimal spanning tree T for that graph. Obviously, if Kruskal's algorithm fails to work, then somebody should be able to come up with both G and T, where T is the correct minimal spanning tree, and where Kruskal's algorithm gets the wrong answer.

Compute the spanning tree K produced by Kruskal's algorithm. It must be different from T or this is certainly not a counterexample.

If G has n vertices, then every spanning tree of G has exactly n-1 edges. (Try arguing that.) Since K and T do not have exactly the same edges, there must be some edge that is in T but not in K.

Look at the edges of G in the same order e0, e2, ..., em-1 that they are looked at by Kruskal's algorithm (in ascending order of weight), and select the first one ei that is in T but not in K. Suppose that edge ei connects vertices u and v. Why wasn't ei added to tree K? Because vertices u and v were already connected by a path through edges that Kruskal's algorithm had looked at earlier. Look at that path, and look at how the same vertices are connected in T.

At least one of the edges ej in this path between u and v must not be in tree T. (Otherwise, T would have a cycle, and a tree cannot have any cycles.) Also, j < i, since the edges shown for K were added before edge ei was considered. Since the edges are looked at in ascending order of weight, ej must have a smaller weight than edge ei.

Make a modified tree T' by removing edge ei from T and adding edge ej. All of the vertices are still connected. (You can still get from any vertex shown in the picture to any other, so you have not cut anything off from anything else.) So T' is another spanning tree. But T' has smaller total weight than T, since we replaced a heavier edge by a lighter one. Obviously, T is not a minimal spanning tree, because T' is a spanning tree with smaller total weight.

That completes the argument. No matter what purported counterexample somebody gives us, we have shown that it cannot be a genuine counterexample. So no counterexample exists, and Kruskal's algorithm is correct for every graph.

Motivation for Computing Minimal Spanning Trees
(you can skip this)

Suppose that we have a collection of towns, and we must build railroad tracks so that it is possible for a train to get from any town to any other town. Our budget for building tracks is small, so we choose to build as little track as possible. One approach to deciding which tracks to build is to construct a weighted graph, where the vertices are the towns, and the weight of the edge from town A to town B is the length of the railroad track that would need to be built to connect towns A and B directly. Then a minimal spanning tree of that graph would be a good choice for connecting the towns, since it is the cheapest way to connect all of the towns using railroad tracks between towns.

A similar problem occurs when a house is being wired for electricity. All of the outlets on a given circuit need to be connected to the circuit panel, and to each other. To connect them with a minimum amount of wire, you might build a graph having a vertex for each outlet and for the circuit panel, with the weight between two vertices being the wiring distance between those vertices. Then get a minimal spanning tree for the graph.

(A minimal spanning tree is not always the best solution to either of those problems. You can often do better by introducing new railroad track junctions or wiring junctions, called Steiner points. For example, if you have three towns in a triangle, you can connect them together by sending tracks to a point in the middle of the triangle. The middle point is a Steiner point. But good placements of Steiner points can be difficult to find, so a minimal spanning tree is a reasonable compromise.)

There are other uses of minimal spanning trees. For example, they are used as part of an algorithm that finds good (but not optimal) traveling salesman tours of graphs.