# CMPT 280– Intermediate Data Structures and Algoirthms Assignment 3

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2.1 Arrayed Trees
Question 1 is about a bounded binary tree implementation. You should remember binary trees from
CMPT 145 (or similar course) – they are trees in which each node has at most two children. What you
probably didn’t know is that binary trees can be stored using an array, rather than a linked structure. In
such an array, the contents of the root node are stored in offset 1 of the array (offset 0 is unused). The
contents of the children of the node whose contents are stored at offset i are stored at offset 2i and 2i + 1,
respectively. Thus, the left child of the root is at offset 2 × 1 = 2, the right child of the root is at offset
2 × 1 + 1 = 3, the left child of the left child of the root is at offset 2 × 2 = 4, and so on. The parent of the
node whose contents are at offset i, is at offset i/2 (integer division). Thus, the parent of node at offset 7
is at offset 3.
Example 1
Here is the array representation of a tree storing the elements 1 through 10, in no particular order. The array
– 7 1 4 3 9 10 2 8 5 6
is a representation of the tree:
7
1
3
8 5
9
6
4
10 2
Note that we do not allow any unused entries in the array between used ones. All the data items in
the array are stored contiguously. This means that we can represent only a particular subset of binary
trees with this representation. Namely, it is the set of trees where all levels except possibly the last level
are complete (full) and the nodes in the bottom level are all as far to the left as possible. You might be
thinking that this is too restrictive and not very useful because we can’t represent all binary trees with
this data structure. However, as we will see on future assignments, this array-based tree data structure is
highly useful and efficient as a basis for implementing certain other important data structures.
Also note that if we read off the items from left to right in each level of the tree, starting from the top
level, we get the items in the same order as they appear in the array.
2.2 m-ary Trees
An m-ary tree is one in which a node may have up to m children. Your lib280-asn3 library has
a class called BasicMAryTree280 which implements an m-ary tree. It has some similarities with
LinkedSimpleTree280 in that, like LinkedSimpleTree280, you have to build up larger trees from
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 2. More details and some examples on how to use this class
are/were provided in Tutorial 3.
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Question 1 (20 points):
Your task is to write a Java class called ArrayedBinaryTreeWithCursors280 which extends and implements the abstract class ArrayedBinaryTree280 (provided in the lib280-asn3.tree package
as part of lib280-asn3). This week’s lab will also talk more about array-based trees.
Methods to be Implemented
Some of the work of implementing ArrayedBinaryTreeWithCursors280 has already been done,
but there is more to do.
Firstly, there are several methods in ArrayedBinaryTreeWithCursors280 which are defined but
not implemented. You must implement these methods. These methods can be easily identified by
Secondly, since ArrayedBinaryTreeWithCursors280 implements some other interfaces, there are
several methods required by these interfaces that also need to be implemented but have not yet been
finished. The method headers for these methods have already been generated but the method bodies are empty. The definitions of these methods in their respective interfaces document what these
methods are supposed to do. These methods can be identified by their “TODO” commands as
well as the @Override directive above the method headers. Many of these methods are defined by
LinearIterator280, which is inherited through Dict280. These are the same linear iterator methods that you wrote for LinkedList280 on assignment 1. The rest of these methods are defined by
Cursor280.
implementation of the arrayed binary tree must pass the given regression test. If all the regression
tests are successful, the only output should be: Regression test complete.
You may not modify any of the existing code in the provided ArrayedBinaryTreeWithCursors280.java
file (including the regression test) but you can add to it. You may also not modify any other files
within lib280-asn3.
Implementation Hints
• You don’t need to declare an array instance variable in ArrayedBinaryTreeWithCursors280
to hold the data items. There is already one inherited from ArrayedBinaryTree280. You
should look at the other inherited instance variables too!
• One of your first tasks will be to start implementing the insert method and decide where the
new element should be inserted. If you think about it, there’s really only one place it can go…
• The algorithm for deleting an element is to replace the element to be deleted by the right-most
element in the bottom level of the tree, then delete the right-most element in the bottom level of
the tree.
• Not sure how a linear iterator works on a tree? If you think about it, there is only one reasonable
way to define a linear ordering on the elements of an array-based binary tree.
• Reminder: the items array (defined in the abstract class ArrayedBinaryTree280 ) represents
the nodes of the tree. You are storing the contents of nodes in the array. There is no node class.
It is very important that the contents of the root are stored in offset 1 and we don’t use offset 0 of
the array, otherwise, the given formulae for finding the child or parent of a node at offset i will
not work correctly.
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Question 2 (17 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
(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 which contains all the of the skills which
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. Tutorial 3 includes 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 you make use of the
previously implemented skillDependencies method.
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.
1
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 2, we used the term “skill trees” to mean skill trees that are, in fact, actual trees.
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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
4: –
4: –
4: –
3: –
3: –
2: Cleave , Cost : 2
3: Whirlwind , Cost : 3
4: Berzerk , Cost : 5
4: –
4: –
4: –
3: –
3: –
3: –
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 :
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 .