Previously I talked about guidelines for good unit tests in an educational setting. In this post I mainly talked about cautionary tales, but this time I want to try to give you a positive example how these guidelines can be applied to increase the usefulness of a unit test both for students and for teachers.

## The exercise

In my current algorithms and data structures course I wanted to change the exercise about merge sort, since it was very hard for many students and very easy to plagiarize. I wanted to tackle both of these problems: Make the problem more manageable for a struggling student and make it harder to simply copy an online solution. In the real world one of the main application areas of merge sort are sequential data structures that can only be accessed from one end, such as linked lists. Here, merge sort usually outperforms quick sort and other options. Additionally, while a merge sort for arrays is very hard to implement without copying the array, this becomes almost trivial for linked lists, since only the links between elements have to be altered.

So the exercise I came up with was to implement merge sort, but for a custom doubly linked list structure and as an in-place algorithm that only allocates O(1) additional space (not counting the O(log n) space required for the recursion stack).

Solutions for this exercise can also be found online, but you have to use a larger google query (java merge sort doubly linked list instead of java merge sort) and as of this writing the first three search results all have major problems disqualifying them for a copy-and-paste-plagiarism.

• GeeksForGeeks and Techie Delight use a recursive implementation of the merge step, which is very elegant but fails with a StackOverflowError for large lists, because it requires O(n) stack space.
• The answers to this question on StackOverflow either do not operate in-place or do not show the code for the custom list classes.

## The unit test

Let’s start by remembering the guidelines we established:

1. Use test-driven design and look at your own mistakes
2. Give more feedback than usual
3. Provide a deep inspection of data structures in error messages
4. Anticipate structural mistakes
5. Provide guiding restrictions
6. Only test what is essential for a correct solution
7. Fix the order of your unit tests
8. Avoid dependencies between tests if possible

For the discussion of the unit tests I will follow this structure and explain how I tried to implement each of these lessons. If you want to have a look at the result, you can download the unit tests as JAR archive.

### 1. Use test-driven design and look at your own mistakes

OK, I admit: I did not use a fully test-driven approach as I should have. The reason for this is laziness and impatience. I quickly wanted to see if the exercise could work, but as we will see later the full suite of unit tests was quite complex and comprised about 500 lines of code - compared to 50 lines of code that were required for a working solution. What I did do however, was to implement a few rudimentary tests and take note of my mistakes:

• My first approach did the split correctly, but failed to combine the sorted sublists and therefore just returned the minimum of the list.
• Somewhere in the middle of fixing the first error, I produced a list that could not be printed because the link structure contained an infinite loop.
• In the first working solution, I forgot to update the link to the last element in the list, leading to a wrong backward traversal.
• I accidentally broke the in-place property by creating a dummy object for each of my merge calls resulting in O(n) memory allocated in the heap.

This already gives us a few important hints for the unit tests:

• Do not assume that the links form a correct list structure. Instead prepare for loops and other problems.
• Always traverse the list from both sides.
• You can break the in-place property in more subtle ways than copying the list.

### 2. Give more feedback than usual

The minimal feedback for this exercise would be to just compare a forward traversal of the expected list and the actual resulting list. However, as we have seen before, an additional backward traversal can reveal errors that would not be obvious during a forward traversal. Even more importantly, the student should not just get a timeout error for loops in his link structure, but there should also be a error message that specifically targets this case.

We can even go a step further. A student may “break” the list in any possible way with wrong or missing links. This is where a class invariant comes in handy. Students should get a notice every time their resulting list hurts any of the following conditions:

• Link symmetry: If a node a links to a node a.next == b, then it must hold that b.prev == a. Equivalently if a.prev == b then it must hold that b.next == a.
• First is first: If first != null then it must hold that first.prev == null.
• Last is last: If last != null then it must hold that last.next == null.
• First to last: The node last must be reachable by following next links from first (and vice versa).

### 3. Provide a deep inspection of data structures in error messages

The previous lesson directly leads us to an idea how the deep inspection of our doubly linked list should look. In my head or on paper I visualize correct and broken list structures as nodes with arrows that represent the next and prev references. However, as the following example of an incorrect link shows, it is not easy do this visualization in a plain text message in an exception:

_______
↓       |
a ⇄ b ⇄ c → d ⇄ e

In the worst case we would have to draw an arbitrary graph of degree two as ASCII/Unicode-art which might be a bit of an overkill.

Therefore I returned to the first lesson and experimented with a simplified notation that would reveal such errors when both a forward and a backward traversal is shown. The result for the wrong link shown above was the following:

Forward:  [a ⇄ b ⇄ c → d ⇄ e]
Backward: [a ⇄ b ← d ⇄ e]

This takes a little more time to visualize in your head than the full “drawing”, but all the information is there: From the forward traversal you can see that node c links to d, but the back link is different. From the backward traversal you can infer that the back link that was missing in the forward traversal links to node d instead of c.

In fact, we can visualize all violations of the class invariant using this notation:

[a⇄b→c]

[a⇄b←c]

x←[a⇄b⇄c]

[a⇄b⇄c]→x

• Last element not reachable by forward links from first element

[a⇄b

• First element not reachable by backward links from last element

b⇄c]

• Infinite loop while following forward or backward links

[a⇄b⇄c⇄∞

The code to produce this representation is actually quite simple. Here is the implementation of the forward traversal:

if (asFirst == null) { return "[]"; }
StringBuilder sb = new StringBuilder();
while (cur != null && !visited.contains(cur)) {
if (cur == asFirst) { sb.append("["); }
sb.append(cur.content.toString());
if (cur == asLast) { sb.append("]"); }
if (cur.next != null) {
sb.append(cur.next.prev == cur ? ARROW_BOTH : ARROW_RIGHT);
}
cur = cur.next;
}
if (cur != null) { sb.append(INFINITY); }
return sb.toString();
}

The key idea is that we keep track of visited nodes with a HashSet to avoid running in an infinite loop. We only follow next links and use the prev links to check if the connection is bidirectional. To produce the error message, the resulting string is then checked for inconsistencies, such as [ and ] not being in the string or not being the first/last character or the occurrence of the characters , or . This allows us to give both the representation and a concrete list of error messages as feedback.

The whole check is available as a public method checkClassInvariant() in the doubly linked list implementation with a protected variant that allows to choose different nodes as substitute for first and last and to disable boundary checks, so that the students can also call the method from within their own methods to check at which point exactly the class invariant is broken.

### 4. Anticipate structural mistakes

We mostly addressed this point by checking the class invariant of our doubly linked list after sorting. However, we can also already anticipate some additional mistakes. Students may…

• … accidentally produce an algorithm that does not run in O(n log n).
• … allocate more heap space than allowed.
• … write an entirely different sorting algorithm.

We will address all these points in the next section.

### 5. Provide guiding restrictions

We want the algorithm to run in O(n log n). Time measurements in Java are a can of worms, but since we are only concerned with big O notation, we do not need accurate measurements. Sorting a list of 100,000 elements should take less than a second with an O(n log n) algorithm, while any O(n²) algorithm should take multiple minutes. This distinction is good enough and it also allows us to find the students that copied the recursive solutions found online and therefore allocate O(n) stack space.

Providing a restriction of the heap space is a little more tricky. One thing we can do is to keep track of the number of node objects that are created by adding a static counter variable to the node class that is incremented each time the constructor is called.

Side note: If the tests are in the same package as the list implementation but the student solution is not, you can use the package private visibility to make the counter accessible to the tests but not to the student code.

This however only captures space allocated by node objects. A student could still store some information in a LinkedList or ArrayList or any other data structure. If this structure is only accessible via the stack, it is not possible to find them via the reflection API. Fortunately, we again only need a very rough solution. We can simply use Runtime.getRuntime().freeMemory() to get an estimate of the unused heap space before and after the sort operation.

But wait, what about garbage collection‽ Shouldn’t objects that only live on the stack be garbage collected when the method ends and its stack frame is deleted? Before Java 9 this would be a real concern. If the garbage collector really decides to run right after the sort method we would not be able to detect that there ever where additional objects on the heap. Fortunately for us, since Java 9 the default garbage collector is G1 which is optimized for low latency and does not run in big chunks. So even if the memory can be freed right after the sort method is called, there will still be some objects left when we do our second call to Runtime.getRuntime().freeMemory().

With this technique the unit test was able to detect the O(n) dummy objects that I initially created, even if most of them where created in the first calls to the merge function on very small sub-lists and could have been garbage collected while the merge function worked upwards to larger chunks of the list. One can argue that this is a bit overzealous, because if there is at any time only one dummy object with active references the algorithm still operates in-place. However, I would argue that while such an algorithm meets the theoretical conditions for an in-place algorithm, it fails to meet the practical expectation that it will not lead to large amounts of memory allocation.

Finally, the last restriction that we have to put into place concerns the type of algorithm used. A sort() method without any parameters does not leave any possibility to test what happens inside, but on the other hand, specifying an exact call structure for the merge function could be a little too restricting. I therefore chose to overload the sort method with an additional listener parameter that had the following structure.

public interface MergeSortListener<E> {
default void split(

default void startMerge(
) {}

default void finishMerge(
) {}
}

With this it was possible to check the internal call structure - and thus provide additional feedback for students.

I implemented a simple listener that visualizes the structure of the split and merge operations by collecting information in the split() and finishMerge() steps in an internal list of StringBuilders. The idea is that for each call to split() you will go one level deeper in the call hierarchy while each finishMerge() takes you one step back up. This allows the listener to keep track of the level where an operation takes place leading to the following string representation when the list [c, a, f, e] is sorted.

[c, a][f, e]
[c][a][f][e]
[a, c][e, f]
[a, c, e, f]

### 6. Only test what is essential for a correct solution

I already said that I refrained from restricting the signature of the merge function, because I wanted to leave some design space for my students. This is also critical for the test for the right call structure in the previous methods. Do you split the list [a, b, c, d, e] into [a, b, c] [d, e] or into [a, b] [c, d, e]? Both choices result in valid merge sort implementations and I would not want my students to get an error message just because they chose a different path than my sample solution. Therefore the structural test is only performed for lists of length 2n where no uneven split should ever occur.

Another point where I consciously allowed for different solutions was the test for the in-place property. I could have restricted the calls to the constructor of the node object to zero, but allowed one object instead, so that implementations that use a dummy object to save some lines of code still remain valid.

### 7. Fix the order of your unit tests

My unit tests used the following order:

1. Empty list
2. List with one string
3. List with two strings
4. Check call structure for list with four strings (without class invariant check)
5. Check result for the same list again with class invariant check but without listener
6. List with four times the same string
7. List with five strings
8. List with ten integers
9. List with 40 random integers
10. List with 1000 random integers
11. List with 100,000 random integers with timeout of 1s and without correctness check
12. List with 100,000 random integers that only checks number of nodes allocated and free heap space

As you can see the golden thread runs from fast and easy tests to slow and hard ones. Unless otherwise stated tests 1-9 use the class invariant checks for additional guidance, but for tests 10-12 these checks are disabled for performance reasons. The lists in 1-9 also have in common that they can be viewed in total when printed to a console window of an IDE on a widescreen display. Therefore the checks for list equality also use the string representation of the list. Test 10, however, is to large for the string representation to be meaningful, so the test is done in the same style as assertArrayEquals reporting the index of the first difference in the actual and expected lists.

### 8. Avoid dependencies between tests if possible

The last section already gives an impression of how I tried to separate different concerns. For example test 4 is all about call structure and not the class invariant, test 6 is only about duplicate values, test 8 about integers.

Specifically I chose not to test the correctness of the sorted lists in test 11 and 12, because that is not the point of the tests. I also separated the speed and heap space tests to ensure that a student whose code is too slow but runs in-place also gets the right feedback.

## Critical Evaluation

I asked my students what they thought of the tests for this exercise and the results where mixed. Here are two representative entries from my feedback box:

• MergeSort was posed well. Despite the start and the adjustment being relatively hard, the exercise taught the handling of packages. Utterly pleased!

• The implementation through the interface and the restrictions established by the tests in my case posed more problems than the exercise itself.

On the one hand the tests seemed to be helpful for some students and indeed I had no problem with plagiarism this time. On the other hand it seemed that some students had a hard time adjusting to the code framework that they had to use for this exercise. In particular they where confused by the concept of the listener and by the option to use a dummy element which was mentioned in the exercise text. In hindsight I also have to admit that my unit tests where still lacking some crucial examples such as lists that are sorted in reverse order. It seemed, however, that the tutors found the tests sufficient for grading as solutions that passed all tests also passed the manual inspection.

## Conclusion

Again this post has become larger than originally anticipated. I hope it can give some inspiration how to implement the ideas from my previous post. I see the critique from my students mostly not as a critique of the tests per se but more as a critique of previous exercises and programming courses that did not teach them the concept of listeners or how to program against a given code framework. All in all I am quite happy with the new version of the exercise and I will try to build more exercises of this type in the future.