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The Assignment
This assignment contains two main parts:

Part 1 [40 marks] – extension of the SSet interface that supports two extra operations
The IndexedSSet interface is an extension of the SSet interface described in the textbook that
supports two extra operations
• get(i): return the value at index i, in other words, return the value x in the set that is larger
than exactly i other elements in the set.
• rangecount(x, y): return the number of elements in the set that are between x and y
inclusive (x is not necessarily smaller than y).

Currently there are two identical implementations of the IndexedSSet interface using skiplists
included in the assignment called SkippitySlow and SkippityFast. Both of them are slow: each of
them has an implementation of get(i) and rangecount(x, y) that runs in 𝛺(𝑛) time, in the worst
case.

Modify the SkippityFast implementation so that the get(i) and rangecount(x, y)
implementations each run in 𝑂(𝑙𝑜𝑔𝑛) expected time and none of the other operations (add(x),
remove(x), find(x), etc.) have their running-time increased by more than a small constant factor.
Look at the SkiplistList implementation discussed in class for inspiration on how to achieve this.

Testing: For this assignment, you will have to test your own code thoroughly. The submission server will
perform a range of tests on your implementation. In case your code throws an exception, the server will
tell you which method threw the exception and what kind of exception it was. Other than that, the
server will consume anything that you try to print to System.out or System.err, so the testing done
on the server will be almost completely opaque.

It will only give you a generic error of the form “get(i)/
rangecount()/… returns incorrect value”, or “Program timed out”. On the positive side, testing is
something you can do easily because SkippitySlow is a correct implementation that you can use to
test against. Be sure to test a good mix of operations that includes and interleaves add(x),
remove(x), get(i), find(x), size(), and rangecount(x, y).

Part 2 [60 marks] – Binary Tree Traversals and extra operations

For this part of the assignment, you will work with the BinaryTree class provided in the zip file. It
contains four uncompleted functions, which you are supposed to complete. For full marks, each of these
functions should run in 𝑂(𝑛) time and must not use recursion. See the function traverse2() in
BinaryTree.java, which we also discussed in class, for an example of how to do tree traversal
without recursion. Note that this is just one example, and there could be other functions that you may
take inspiration from.

You need to implement the following:

1. The method leafAndOnlyLeaf() should return the number of leaves in the tree, or 0 if the
tree has no nodes at all. (A leaf is a node that has no left child and no right child.)
2. The method dawnOfSpring() should return the smallest depth in the tree that has leaves in it,
or -1 if the tree has no nodes.
3. The method monkeyLand() should return the number of nodes in the most populated depth
(the depth where there is the largest number of nodes). If there are multiple depths with the
maximum number of nodes, it should return the smallest of them.

4. The method bracketSequence() should return a string that gives the dot-bracket
representation of the binary tree. The dot-bracket representation of a binary tree can be
defined as follows:
• the dot-bracket representation of a tree with no nodes is the string “.”
• the dot-bracket representation of a binary tree with root node r consists of an open
bracket ( followed by the dot-bracket representation of r.left followed by the dotbracket representation of r.right followed by a closing bracket )

Some examples:

• the dot-bracket representation of the binary tree with only one node is (..)
• the dot-bracket representation of a 2-node binary tree is either ((..).) or (.(..))
depending on whether the root has no right child or no left child
• the dot-bracket representation of the height-1 binary tree with two leaves is ((..)(..))
Testing: The BinaryTree class implements a static method called randomBST(n) that returns a
random-looking binary tree.

BinaryTree also implements the toString() method, so you can use
System.out.println(t) to view a representation of a BinaryTree that is closely related to the
method bracketSequence(). You can use this for testing your functions.
You should test your own code thoroughly since the testing done on the server will be mostly opaque.
On the server, your code will be tested on a Java virtual machine with a limited stack size. You can do
similar testing yourself by using the java -Xss argument (although you won’t have to worry about
this if you don’t use recursion).

Tips, Tricks, and FAQs

How should I approach each problem?

• For part 1, it’s best first to study how random access can be performed on skiplists (specifically,
ODS section 4.3) and then attempt to solve it. Also, look at the skeleton code and try to
understand how various operations work. It’s okay if you do not understand every detail of the
code as long as you understand enough to do what you need to do.

• For part 2, you can look up the ODS textbook on how binary trees can be traversed in various
ways without recursion (specifically, ODS section 6.1.2). You can find these traversals already
implemented in the skeleton code as well. If you are unsure what the operations are supposed
to do, the following figure might help you understand them better.
• Make sure you understand what the operations do. Construct small examples and compute (by
hand) the expected output. If you aren’t sure what the output should be, go no further until you
get clarification.

• Now that you understand what you are supposed to output, and you’ve been able to solve it by
hand, think about how you solved it and whether you could explain it to someone. How about
explaining it to a computer?
• If it still seems challenging, what about a simpler case? Can you solve a similar or simplified
problem? Maybe a special case? If you were allowed to make certain assumptions, could you do
it then? Try constructing your code incrementally, solving the smaller or simpler problems, and
then only expanding the scope once you’re sure your simplified problems are solved.

How should I test my code?

• You can modify the Tester class. For example, you can change the “20” in
skippityTest(20); to a smaller/bigger number to test less/more operations.
• The Tester class provided with the assignment does a sequence of adds, followed by a
sequence of gets and rangecounts. This is a very basic test and far from an exhaustive one. In
fact, the provided tester is meant to be more of a user guide to get started with the skeleton
code and by no means, a representative of the tests performed on the server side.

Design your
own test cases, and throw in a good mix of add, remove, get, find, rangecount, and size.
Try to break your solution with the tester in every way you can think of. And, of course, don’t
forget to test with large test sizes.

• You should be testing your code as you go along. The slow (but correct) solutions can always be
taken as a reference when testing.
• Use small tests first so that you can compute the correct solution by hand.
• You will be dealing with pointer manipulation. Beware of null pointer exceptions.

• If you have print statements in your program that use System.out, the server will execute
those statements but ultimately will remove the output generated by those statements from
the response. You probably already know by now how console output can be painfully slow.

Even worse, it could cause the testing program to crash if the output is large. So, be sure to
remove such statements you may have used for debugging or any other reason before
submitting your work.
• Test for speed.