6.031 — Software Construction
Fall 2019

Problem Set 2: Multi-Startup Set

Alpha due
Monday, October 7, 2019, 10:00 pm
Code reviews due
Friday, October 11, 2019, 11:00 am
Beta due
Wednesday, October 16, 2019, 10:00 pm

The purpose of this problem set is to practice designing, testing, and implementing abstract data types. This problem set focuses on implementing mutable types where the specifications are provided to you; the next problem set will focus on immutable types where you must choose specifications.

Design Freedom and Restrictions

On several parts of this problem set, the classes and methods will be yours to specify and create, but you must pay attention to the PS2 instructions sections in the provided documentation.

You must satisfy the specifications of the provided interfaces and classes. On the last problem, you are permitted to strengthen some provided specifications or add new methods, but otherwise you are not permitted to change them at all.

On this problem set, Didit provides less feedback about the correctness of your code:

  • It is your responsibility to examine Didit feedback and make sure your code compiles and runs properly for grading.
  • However, correctness is your responsibility alone, and you must rely on your own careful specification, testing, and implementation to achieve it.

Please remember to push early: we cannot guarantee timely Didit builds, especially near the problem set deadline.

Get the code

To get started,

  1. Ask Didit to create a remote psets/ps2 repository for you on GitHub.mit.

  2. Clone the repo. Find the ssh:// URL at the top of your Didit assignment page and run: git clone «URL»

  3. Import into Eclipse. See Problem Set 0 for if you need a refresher on how to create, clone, or import your repository.


Java Collections Framework provides many useful data structures for working with collections of objects: lists, maps, queues, sets, and so on. Let’s use some of these building blocks to build a data structure of our own, and then build one small application of that ADT.

Interval sets · Multi-interval sets · Similarity between multi-interval sets

Interval sets

Problems 1-3: we will implement the interval set abstract type, a mutable set of unique labels where each label is associated with a non-overlapping interval on the number line.

For example, we can picture the interval set { A = [0,5), B = [10,30), C = [30,35) } as:

In this example, the labels are the unique String objects "A", "B", and "C"; and each is associated with an interval, which in our type will be an interval of long values. Every interval is half-open (also called half-closed), meaning that it includes its start point but not its end point, so [0,5) represents { x ∈ ℝ | 0 ≤ x < 5 }.

Read the Javadoc documentation for IntervalSet generated from src/​interval/​IntervalSet.java.

Labels do not have to have String type, but can have any immutable type. IntervalSet<L> is a generic type similar to List<E> or Map<K,V>. In the specification of List, E is a placeholder for the type of elements in the list. In Map, K and V are placeholders for the types of keys and values. Only later does the client of List or Map choose particular types for these placeholders by constructing, for example, a List<Clown> or a Map<Car,Integer>.

The specification of a generic type is written in terms of the placeholders. For example, the specification of Map<K,V> says that K must be an immutable type: if you want to make something the key in a Map, it shouldn’t be a mutable object because the Map may not work correctly.

In our specification for IntervalSet<L>, we make the same demand about the immutability of type L. Clients of IntervalSet who try to use mutable labels have violated the precondition. They cannot expect correct behavior.

For this problem set, we will implement IntervalSet twice, with two different reps, to practice choosing abstraction functions and rep invariants and preventing rep exposure. There are many good reasons for a library to provide multiple implementations of a type (for example, the ArrayList and LinkedList implementations of List satisfy clients with different performance requirements), and we’re following that model.

Multi-interval sets

Problem 4: using interval sets, we will implement the multi-interval set type, a mutable set of labels where each label is associated with one or more globally-non-overlapping intervals.

For example, { A = [[0,5),[20,25)], B = [[10,20),[25,30)], C = [[30,35)] }:

Once again, the labels are "A", "B", and "C". Label "C" is associated with one interval, labels "A" and "B" each with two.

Read the Javadoc documentation for MultiIntervalSet generated from src/​interval/​MultiIntervalSet.java.

Similarity between multi-interval sets

Problem 5: with our multi-interval set datatype in hand, the MVP for your new “ startup needs one crucial piece of proprietary technology — a way to measure the similarity between multi-interval sets.

For example, given { A = [[0,5),[20,25)], B = [[10,20),[25,30)] } and { A = [[20,35)], B = [[10,20)], C = [[0,5)] }:

"A" "B" "A" "B" 0 10 20 30 "C" "B" "A"

We might consider an interval on the number line completely similar (value 1) when the labels are equal (e.g. on [10,20) and [20,25)) and dissimilar (value 0) otherwise.

Similarity will be defined as a ratio of the extent spanned by the two multi-interval sets, so the similarity between these two sets with the binary definition of label similarity is:

( 5×0 + 5×0 + 10×1 + 5×1 + 5×0 + 5×0 ) ÷ ( 35 - 0 ) = 15 / 35 ≈ 0.42857

(Hover or tap on individual terms to highlight that part of the figure above.)

However, our similarity function will support a client-supplied definition of label similarity, so for example the user might specify that "A" and "C" have similarity 0 but "A" and "B" are closer, similarity ½. Using that definition:

( 5×0 + 5×0 + 10×1 + 5×1 + 0.5 + 5×0 ) ÷ ( 35 - 0 ) = 17.5 / 35 = 0.5

Clients of similarity define label similarity by providing a list of pairs of labels with their numeric similarity between 0 and 1. The similarity between all other pairs of labels is 0.

Read the Javadoc documentation for Similarity generated from src/​startup/​Similarity.java.

Problem 1: Test IntervalSet​<String>

Devise, document, and implement tests for IntervalSet<String>.

For now, we’ll only test (and then implement) interval sets with String labels. Later, we’ll expand to other kinds of labels.

In order to accommodate running our tests on multiple implementations of the IntervalSet interface, here is the setup:

  • The testing strategy and tests for the static IntervalSet.empty() method are in EmptyTest.java. Since the method is static, there will be only one implementation, and we only need to run these tests once. We’ve provided these tests and you do not need to change them.

  • Write your testing strategy and your tests for the instance methods in IntervalSetTest.java. In these tests, you must use the emptyInstance() method to get fresh empty sets, not IntervalSet.empty()! See the provided testInitialLabelsEmpty() for an example.

Java note

IntervalSetTest is an abstract class. Abstract classes and subclassing have their uses, but in general should be avoided.

Unlike EmptyTest, which works like any other JUnit test class we’ve written, IntervalSetTest is different because you can’t run it directly. It has a blank waiting to be filled in: the emptyInstance() method that will provide empty IntervalSet objects. In the next problem, we’ll see two subclasses of IntervalSetTest that fill in the blank by returning empty sets of different types.

Your tests in IntervalSetTest must be legal clients of the IntervalSet spec. Any tests specific to your implementations will go in the subclasses in the next problem.

Refer to Abstract Data Types: Testing an abstract data type. When testing instance methods, remember to partition the state of input this.

reading exercises


Looking at IntervalSet.java and RepListIntervalSet.java, which of the following might you mention in your partitions for IntervalSet operation insert : IntervalSet<‌String> × long × long × String → void?

(missing explanation)

Commit to Git. Once you’re happy with your solution to this problem, commit and push!

Problem 2: Implement IntervalSet​<String>

Now we’ll implement interval sets with String labels — twice.

For all the classes you write in this problem set:

  • Document the abstraction function and representation invariant.
  • And along with the rep invariant, document how the type prevents rep exposure. See Documenting the AF, RI, & SRE.
  • Implement checkRep to check the rep invariant.
  • Implement toString with a useful human-readable representation of the abstract value.

All your classes must have clear documented specifications. This means every method will have a Javadoc comment, except when you use @Override:

  • When a method is specified in an interface and then implemented by a concrete class — for example, insert(..) in RepMapIntervalSet is specified in IntervalSetthe @Override annotation with no Javadoc comment indicates the method has the same spec. Unless the concrete class needs to strengthen the spec, don’t write the spec again (DRY).

You can choose either RepMapIntervalSet or RepListIntervalSet to write first. There should be no dependence or sharing of code between your two implementations.

reading exercises


Looking at RepListIntervalSet.java, what is the input space of RI : R → boolean for RepListIntervalSet?

(missing explanation)


What does an input to AF : R → A for RepListIntervalSet look like?

(missing explanation)

For an input that satisfies the rep invariant, what does the output look like?

(missing explanation)

Interval Set

Which describe exactly one abstract interval set?

(missing explanation)

2.1. Implement RepMapIntervalSet

For RepMapIntervalSet, you must use the rep provided:

private final Map<String, Long> startMap = new HashMap<>();
private final Map<String, Long> endMap = new HashMap<>();

You may not add fields to the rep or choose not to use one of the fields. You must use the names startMap and endMap, even if they are not the names you would choose. The no-argument constructor must create an empty interval set (you are free to add other constructors that take arguments).

Using the @Override annotation on toString ensures that you are correctly overriding the Object version of that method, not creating a new, different method.

All Java classes inherit Object.toString() with its very underdetermined spec. By using @Override and no Javadoc comment, RepMapIntervalSet keeps that same spec. The choice not to strengthen the spec is deliberate: since toString is intended for human consumption, keeping its spec weak helps prevent other code from depending on its particular format.

Because of this very weak spec, you do not need to test toString.

Run your IntervalSetTest tests on RepMapIntervalSet by right-clicking on RepMapIntervalSetTest.java and selecting Run As → JUnit Test.

Commit to Git. Once you’re happy with your solution to this problem, commit and push!

2.2. Implement RepListIntervalSet

For RepListIntervalSet, you must use the rep provided:

private final List<String> labelList = new ArrayList<>();
private final List<Long> valueList = new ArrayList<>();

You may not add fields to the rep or choose not to use one of the fields. You must use the names labelList and valueList, even if they are not the names you would choose. The no-argument constructor must create an empty interval set (you are free to add other constructors that take arguments).

Run the tests by right-clicking on RepListIntervalSetTest.java and selecting Run As → JUnit Test.

Commit to Git. Once you’re happy with your solution to this problem, commit and push!

Problem 3: Implement generic IntervalSet<L>

Nothing in either of your implementations of IntervalSet should rely on the fact that labels are of type String in particular.

The spec says that labels “must be immutable” and are “compared using equals.” Let’s change both of our implementations to support labels of any type that meets these conditions.

3.1. Make the implementations generic

  1. Change the declarations of your concrete classes to read:

    public class RepMapIntervalSet<L> implements IntervalSet<L> { ... }


    public class RepListIntervalSet<L> implements IntervalSet<L> { ... }
  2. Update both of your implementations to support any type of label, using placeholder L instead of String. Roughly speaking, you should be able to find-and-replace all the instances of String in your code with L!


    The RepMap- and RepList­Interval­Set­Test test files say “you should not modify this file.” That’s not quite correct: you should fix the constructor calls there to use generic syntax.

    This refactoring will make RepMapIntervalSet and RepListIntervalSet generic types.

    Any place you previously referred to these types without a generic parameter, you will need to add it. For example, if you called a constructor like new RepMapIntervalSet(), that will need to become new RepMapIntervalSet<String>(). Depending on context, you can use diamond notation <> to avoid writing type parameters twice.

    When you’re done with the conversion, all your instance method tests should pass.

Commit to Git. As always, once you’re happy with your solution to this problem, commit and push!

3.2. Other types of labels

Because the implementation of IntervalSet does not know or care about the actual type of the labels, and only compares them according to the IntervalSet specification, your test suite with String labels is sufficient…

Except that our choice of String for the test suite could prevent us from finding two Java-String-specific bugs:

  1. If the implementation incorrectly compares labels using == instead of equals, identical String literals are very likely to return true when compared with ==, masking the bug. See Basic Java: == vs. equals(), exercise “Optimation hides a bug.”
  2. If the implementation incorrectly compares labels using their toString, it will break for label types where toString can return the same string for unequal instances (toString doesn’t guarantee any uniqueness), but we cannot find this bug by testing String labels.

Review your RepMap- and RepListIntervalSet implementations to see that you:

  1. Never compare label objects with ==.
  2. Never convert labels to strings with toString, except in the RepMap- or RepListIntervalSet class’ own toString.

Note: you cannot write tests with other types of labels in IntervalSetTest, because those tests must use the emptyInstance() method to get new sets. That method always returns IntervalSet<String>.

3.3. Implement IntervalSet.empty()

Pick one of your IntervalSet implementations for clients to use, and implement IntervalSet.empty(). Unlike ArrayList and LinkedList, where clients who only want a List are forced to break the abstraction barrier by calling a concrete constructor, clients of IntervalSet should not be aware of how we’ve implemented it.

Obscure Java note

You may use @SafeVarargs where necessary, for example if you write testing helper functions that take a variable number of arguments.

At this point, all of your interval set code (implementations and tests) must have no warnings from the compiler (warnings have a symbol, as opposed to the for errors) and no @SuppressWarnings annotations (which would disable the warning, nice try). Please see the correction in 3.1 about fixing warnings in RepMap- and RepList­Interval­Set­Test.

If you cannot figure out how to make your implementations generic while satisfying the Java compiler’s static type checking, please visit lab or office hours, or ask a question on Piazza.

At this point in the problem set, we’ve completed the test-first implementation of IntervalSet.

  • RepMap- and RepListIntervalSetTest should have a green bar from JUnit and excellent code coverage from the coverage tool.

  • This is a good opportunity for self code review: remove dead code and debugging printlns, DRY up duplication, and so on.

  • And of course, commit and push.

Problem 4: Multi­IntervalSet<L>

Just as we implemented the IntervalSet ADT by relying on other ADTs as building blocks, we can use IntervalSet as part of the specification and as part of the representation of another ADT.

4.1 Test MultiIntervalSet

Devise, document, and implement tests for MultiIntervalSet in Multi­IntervalSetTest.java.

Your tests in MultiIntervalSetTest must be legal clients of the MultiIntervalSet spec. As before, since the spec of toString is very weak, you do not need to test toString.

Java note

If you define a helper type to use in the rep of MultiIntervalSet, we require that class be declared in Multi­Interval­Set.java.

And if your helper type is immutable, be careful: not implementing equality as described in class 15 will mean you cannot use equals to compare objects of your helper type.

4.2 Implement MultiIntervalSet

Implement MultiIntervalSet in MultiIntervalSet.java.

You must use IntervalSet in the rep of MultiIntervalSet. For example, IntervalSet instances might be elements in a list or values in a map. Your MultiIntervalSet should rely only on the specs of IntervalSet, not on the details of a particular IntervalSet implementation, and it should use the static empty() method to obtain interval set instances. The implementation of MultiIntervalSet is otherwise entirely up to you.

Run all the IntervalSet and MultiIntervalSet tests by right-clicking on the interval package in the test folder and selecting Run As → JUnit Test.

Problem 5: Fame and fortune with IntervalSet

This problem description was randomly generated

If you don’t like it, please reload the page to receive a different problem.

You are the founder and CTO of , the hottest new startup that just called “ for , but .” In your MVP, you’re leveraging the most advanced artificial intelligence algorithms and machine learning architecture to deliver unprecedented breakthrough results in the world of .

Of course, that’s all a bunch of . As CTO, you realize it all comes down to managing multi-interval sets that represent . works by comparing . You need to develop tools for measuring the similarity of multi-interval sets: completely different will score 0, with the score approaching 1 as they become more alike.

And best of all, when you pivot your startup to and rebrand as , the startup in , you’ll use those multi-interval sets to represent ! In the new startup, you’re going to compare — so your success in the startup world all comes down to this one problem.

5.1 Specify and test similarity

Devise, document, and implement tests for similarity(..) in SimilarityTest.java.

You are free to add additional methods to the Similarity class, but you may not change the signature of the required method. You are free to strengthen the required specification, but you may not weaken it. Note how this might mean your tests for Similarity are not usable against someone else’s implementation, because you’ve strengthened or added to the spec.

You are even free to make Similarity into an abstract data type, in which case you should decide if it is mutable or immutable, document its rep invariant and abstraction function, specify its operations, and so on. Similarity must still provide the static similiarity(..) method.

5.2 Implement similarity

Implement similarity(..) in Similarity.java.

Your Similarity code should rely only on the specs of MultiIntervalSet, not on the details of any particular implementation.

The implementation of Similarity is entirely up to you.

Run all the tests in the project by right-clicking on the test folder and selecting Run As → JUnit Test.

Before you’re done

  • Make sure you have documented specifications, in the form of properly-formatted Javadoc comments, for all your types and operations.

  • Make sure you have documented abstraction functions and representation invariants, in the form of a comment near the field declarations, for all your implementations.

    With the rep invariant, also say how the type prevents rep exposure.

    Make sure all types use checkRep to check the rep invariant and implement toString with a useful human-readable representation of the abstract value.

  • To gain static checking of the correct signature, use @Override when you override toString.

    Also use @Override when a class implements an interface method, to remind readers where they can find the spec.

  • Make sure you have a thorough, principled test suite for every type.


Make sure you commit AND push your work to your repository on Athena. We will use the state of your repository on Athena as of 10:00pm on the deadline date. When you git push, the continuous build system attempts to compile your code and run some basic tests. You can always review your build results at didit.csail.mit.edu/6.031/fa19.

Didit feedback is provided on a best-effort basis:

  • There is no guarantee that Didit tests will run within any particular timeframe, or at all. If you push code close to the deadline, the large number of submissions will slow the turnaround time before your code is examined.
  • If you commit and push right before the deadline, the Didit build does not have to complete in order for that commit to be graded.
  • Passing some or all of the public tests on Didit is no guarantee that you will pass the full battery of autograding tests — but failing them is almost sure to mean lost points on the problem set.


Your overall ps2 grade will be computed as approximately:
~35% alpha autograde + ~5% alpha manual grade + ~45% beta autograde + ~15% beta manual grade

The autograder tests will not change from alpha to beta.

Manual grading of the alpha will only examine the internal documentation (AF, RI, etc.) and implementation of your ADTs. Manual grading of the beta may examine any part, including re-examining ADT implementations, and how you addressed code review feedback.