CMPT 280– Intermediate Data Structures and Algoirthms
Total Marks: 55
1 Submission Instructions
• Assignments must be submitted using Moodle.
• Responses to written (non-programming) questions must be submitted in a PDF file, plain text file
(.txt), Rich Text file (.rtf), or MS Word’s .doc or .docx files. Digital images of handwritten pages
are also acceptable, provided that they are clearly legible.
• Programs must be written in Java.
• If you are using Eclipse (or similar development environment), do not submit the workspace (project).
Hand in only those files identified in Section 5. Export your .java source files from the workspace
and submit only the .java files.
• No late assignments will be accepted. See the course syllabus for the full late assignment policy for
A heap is a binary tree which has the following heap property: the item stored at a node must be at least as
large as any of its descendents (if it has any). In a heap, when an item is removed, it is always the largest
item (the one stored at the root) that gets removed. Also, the only item that is allowed to be inspected is
the top of the heap, in much the same way that the only item of a stack that may be inspected is the top
element. Stacks, queues, and heaps are all examples of collections of data items that we call dispensers. You
can put stuff into a dispenser, but the user doesn’t get to specify where – the collection decides according
to some rule(s). Likewise, you can take something out of a dispenser, but the dispenser decides what item
you get. Dispensers maintain a current item using an internal cursor, but the dispenser always decides
what is the current item, and thus the item that will next be dispensed when a user asks to remove or
inspect the current item. Dispensers do not have public methods to control the cursor position because
the user is not supposed to control this; it’s up to the dispenser. In a stack, the “current” item is always
the item at the top of the stack. In a queue it is the item at the front of the queue. In a heap it is the item
at the root of the heap.
In question 3 you will implement a heap by writing a class called ArrayedHeap280 that extends the
abstract class ArrayedBinaryTree280 pseudocode sketches of the insert and deleteItem algorithms:
Algoirthm insert (H , e )
Inserts the element e into the heap H .
Insert e into H normally , as in ArrayedBinaryTreeWithCursors280 <I
// ( put it in the left – most open position at the bottom level of the tree )
while e is larger than its parent and is not at the root :
swap e with its parent
Algorithm deleteItem ( H )
Removes the largest element from the heap H .
// Since the largest element in a heap is always at the root …
Remove the root from H normally , as in ArrayedBinaryTreeWithCursors280 <I
// ( copy the right – most element in the bottom level , e, into the root ,
// remove the original copy of e.)
while e is smaller than its largest child
swap e with its largest child
Additional background and examples on heaps are available in this week’s tutorial.
2.2 m-ary Trees
An m-ary tree is one in which a node may have up to m children. Your lib280-asn3 project has
a class called BasicMAryTree280 LinkedSimpleTree280 smaller trees, rather than inserting individual elements, because since m-ary trees have no defined structure in general and thus there is no obvious algorithm for automatically deciding where a new element
should go. You will use this class in Question 4. More details and some examples on how to use this class
are provided in this week’s tutorial.
3 Your Tasks
Question 1 (15 points):
A priority queue is a queue where a numeric priority is associated with each element. Access to
elements that have been inserted into the queue is limited to inspection and removal of the elements
with smallest and largest priority only. A priority queue may have multiple items that are of equal
Give the ADT specification for a bounded priority queue using the specification method described
in Topic 5 of the lecture notes. By “bounded”, it is meant that the priority queue has a maximum
capacity specified when it is created, and it can never contain more than that number of items.
Your specification must specify the following operations:
newPriorityQueue: make a new queue
insert: inserts an element with a certain priority
isEmpty: test if the queue is empty
isFull: test if the queue is full
maxItem: obtain the item in the queue with the highest priority
minItem: obtain the item in the queue with the lowest priority
deleteMax: remove from the queue the item with the highest priority
deleteAllMax: remove from the queue all items that are tied for the highest priority
deleteMin: remove from the queue the item with the lowest priority
frequency: obtain the number of times a certain item occurs in the queue (with any priority)
Question 2 (6 points):
Use the statement counting approach to show that the worst-case time complexity of the heap insertion and deletion algorithms given above in Section 2.1 are O(log n).
Question 3 (16 points):
Implement a heap by writing a class called ArrayedHeap280 that extends the existing abstract class
ArrayedBinaryTree280 should need to write are a constructor, and the insert and deleteItem methods required by Dispenser280<I,
but you can use additional private methods if you think it makes sense to do so.
Note that since you need to compare items in the heap to each other, you must write the generic type
parameter of your class’ header so that it is required to be a subclass of Java’s Comparable interface.
This is necessary to ensure that only items that implement the Comparable interface can be stored in
the heap. As a consequence, you’ll have to make sure that your constructor initializes the instance
variable items (inherited from ArrayedBinaryTree280<I) to an array of Comparable. If this is all
done properly, then any item x in the heap can be compared to another item y using x.compareTo(y).
Finally, you are required to write a main method for the ArrayedHeap280 class that performs a regression test to make sure the class functions properly. You need only test the methods in the
ArrayedHeap280 class, as one would normally assume that the regression test for the parent class
has already been run.
For this question, there will be marks awarded for suitable commenting of code, and for writing
appropriate javadoc comments.
Question 4 (18 points):
In video games, especially those in the roleplaying genre, it is common that characters in the game are
advanced in power through the use of a skill tree. Generally, a skill tree defines the prerequisite for the
various skills that your character in the game might acquire. For example, in a hypothetical game, if
the Shield Bash, Defensive Stance, and Shield Ally skills all require that your character first have the skill
Shield Proficiency, then this might be represented by the following skill tree:
Shield Proficiency, Cost: 1
Shield Bash, Cost: 2 Defensive Stance, Cost: 1 Shield Ally, Cost : 3
More formally, a skill in the skill tree can only be gained if the character first gains all of the skills
which are ancestors of that skill in the tree.1
Your task in this question is to write a class called SkillTree which extends BasicMAryTree280<Skill
(an m-ary tree of Skill objects; a complete Skill.java is provided). A template for the SkillTree
class is provided. It contains a constructor and a couple of useful methods. You will add additional
methods to this class in the following steps, which you should complete in order:
(a) Write a main() method in the SkillTree class in which you construct your own skill tree for your
own hypothetical video game. Your tree must contain at least 10 skills. However, for the sanity
of everyone involved, try to keep it under 15 skills. Be creative! There is no reason why any
two students should hand in exactly the same (or even very similar) skill trees, nor should you
just duplicate the skill tree shown in the sample output. Print your tree to the console using the
toStringByLevel() method inherited from BasicMAryTree280.
(b) Write a method in the SkillTree class called skillDependencies which takes a skill name as
input and returns an instance of LinkedList280 are prerequisites for obtaining the input skill (including the input skill itself!). A RuntimeException
exception should be thrown if the tree does not contain the given skill. A good implementation
approach for this method is to use a recursive traversal of the tree to find the named skill, and
then add skills to the output list as the recursion unwinds. This week’s tutorial will include some
discussion of recursive traversal of m-ary trees. Add to your main() program a few tests of this
method, and print out the lists that is returned (you can use the list’s toString() method for
this). Be sure to test the case where the named skill does not exist in the tree.
(c) Write a method in the SkillTree class called skillTotalCost which takes a skill name as input and returns the total number of skill points that a player must invest to obtain the given
skill. If the named skill is not in the skill tree, then the skillTotalCost method should throw
a RuntimeException exception. Hint: this method is quite easy to implement if it makes use of the
For example, in the above skill tree, if a character wants the Shield Ally skill they would need to
spend 1 skill point to get Shield Proficiency, and then spend 3 skill points to get Shield Ally for an
overall investment of 1 + 3 = 4 points, so for the above tree, skillTotalCost(“Shield Ally”)
should return 4. Note that the Skill object contains the cost of the skill.
Add to your main() program a few tests of skillTotalCost, and print out the total costs returned.
Be sure to test the case where the named skill does not exist in the tree.
(d) Run your main() program. Cut and paste the console output to a text file and submit it with your
assignment. See the sample output on the next page.
In the video game world, the term “skill tree” sometimes refers to things that actually aren’t trees; a noteworthy example is the
skill tree in the ARPG Path Of Exile, which, if you click the link, can see is clearly not a tree, even though they call it that. Here in
question 4, we used the term “skill trees” to mean skill trees that are, in fact, actual trees.
3.1 Sample Output
Here is an example of what the output of your program might look like. Remember, you are expected to be creative in designing your skill tree, and your submission should not attempt to duplicate
what you see here aside from the general formatting (the formatting can be the same, but the data
should be different!). Note that the formatting of output of the skill tree contents is done by the
toStringByLevel() method of BasicMAryTree280.
My Skill Tree :
1: Slash , Cost : 1
2: Mighty Blow , Cost : 2
2: Shield Bash , Cost : 1
3: Shield Charge , Cost : 2
3: Parry , Cost : 2
4: Shield Wall , Cost : 4
2: Cleave , Cost : 2
3: Whirlwind , Cost : 3
4: Berzerk , Cost : 5
2: Mobility , Cost : 1
Dependencies for Shield Wall :
Slash , Cost : 1 , Shield Bash , Cost : 1 , Parry , Cost : 2 , Shield Wall , Cost : 4 ,
Dependencies for Mobility :
Slash , Cost : 1 , Mobility , Cost : 1 ,
Dependencies for Slash :
Slash , Cost : 1 ,
Dependencies for FakeSkill :
FakeSkill not found .
To get Whirlwind you must invest 6 points .
To get Mighty Blow you must invest 3 points .
To get Slash you must invest 1 points .
FakeSkill not found .
4 Files Provided
lib280-asn3: A copy of lib280 which includes the ArrayedBinaryTree280 and the BasicMAryTree280 class which is needed for Question 4.
Skill.java : A complete implementation of the Skill class needed for Question 4.
SkillTree.java : A template for your implementation of the SkillTree class in Question 4.
5 What to Hand In
You must submit the following files:
Q1-2.doc/docx/rtf/pdf/txt – your answers to questions 1 and 2. Digital images of handwritten pages are
also acceptable, provided that they are clearly legible.
ArrayedHeap280.java – Your implementation of an arrayed heap.
SkillTree.java – Your finished implementation of the skill tree.