1. ADT description Implementations
Queues
ADT description
Implementations
October 03, 2017
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2. Cinda Heeren / Geoffrey Tien
Queues
Example: Print Queue
Assume that we want to store data for a print queue for a student printer
Student ID
Time
File name
The printer is to be assigned to file in the order in which they are received
A fair algorithm
October 03, 2017
Cinda Heeren / Geoffrey Tien

3. Classes for Print Queues
To maintain the print queue we would require at least two classes
Request class
Collection class to store requests
The collection class should support the desired behaviour
FIFO (First In First Out)
The ADT is a queue
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4. Cinda Heeren / Geoffrey Tien
Queues
In a queue items are inserted at the back (end/tail) and removed from the front (head)
Queues are FIFO (First In First Out) data structures – fair data structures
Applications include:
Server requests
Instant messaging servers queue up incoming messages
Database requests
Print queues
Operating systems often use queues to schedule CPU jobs
The waiting list for this course! (it’s presumably fair)
Various algorithm implementations
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Cinda Heeren / Geoffrey Tien

5. Cinda Heeren / Geoffrey Tien
Queue Operations
A queue should implement at least the first two of these operations:
enqueue – insert item at the back of the queue
dequeue – remove an item from the front
peek – return the item at the front of the queue without removing it
isEmpty – check if the queue does not contain any items
Like stacks, it is assumed that these operations will be implemented efficiently
That is, in constant time
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6. Cinda Heeren / Geoffrey Tien
Queue Implementation
Using an Array
Consider using an array as the underlying structure for a queue, we could
Make the back of the queue the current size of the array, much like the stack implementation
Initially make the front of the queue index 0
Inserting an item is easy
What to do when items are removed?
Either move all remaining items down – slow
Or increment the front index – wastes space
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7. Cinda Heeren / Geoffrey Tien
Circular Arrays
Trick: use a circular array to insert and remove items from a queue in constant time
The idea of a circular array is that the end of the array “wraps around” to the start of the array 1 3 2 4 5 6 7 1 2 3 4 5 6 7
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8. Cinda Heeren / Geoffrey Tien
The modulo Operator
The mod operator (%) calculates remainders:
1%5 = 1, 2%5 = 2, 5%5 = 0, 8%5 = 3
The mod operator can be used to calculate the front and back positions in a circular array
Thereby avoiding comparisons to the array size
The back of the queue is:
(front + num) % arrlength
where num is the number of items in the queue
After removing an item, the front of the queue is:
(front + 1) % arrlength
Member attributes:
int front;
int arrlength;
int* arr;
int num;
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9. Cinda Heeren / Geoffrey Tien
Array Queue Example 1
Queue q;
front
num
q.enqueue(6); 6 1 2 3 4 5
Insert item at (front + num) % queue.length, then increment num
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10. Cinda Heeren / Geoffrey Tien
Array Queue Example 1 2 5 4 3
Queue q;
front
num
q.enqueue(6);
q.enqueue(4);
q.enqueue(7);
q.enqueue(3);
q.enqueue(8); 6 4 7 3 8 1 2 3 4 5
q.dequeue();
Insert item at (front + num) % queue.length, then increment num
q.dequeue();
Remove item at front, then decrement num and make front = (front + 1) % queue.length
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Cinda Heeren / Geoffrey Tien

11. Cinda Heeren / Geoffrey Tien
Array Queue Example 2 5 4 3
Queue q;
front
num
q.enqueue(6);
q.enqueue(4);
q.enqueue(7);
q.enqueue(3);
q.enqueue(8); 5 7 3 8 9 1 2 3 4 5
q.dequeue();
Insert item at (front + num) % queue.length, then increment num
q.dequeue();
q.enqueue(9);
Remove item at front, then decrement num and make front = (front + 1) % queue.length
q.enqueue(5);
Need to check that the back of the queue does not overtake the front
October 03, 2017
Cinda Heeren / Geoffrey Tien

12. Cinda Heeren / Geoffrey Tien
Array queue resizing
Suppose we have an array-based queue and we have performed some enqueue and dequeue operations
Then we perform more enqueues to fill the array
How should we resize the array to allow for more enqueue operations? 12 5 76 33 2 41
front ?
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13. Cinda Heeren / Geoffrey Tien
Queue Implementation
Using a Linked List
Removing items from the front of the queue is straightforward
Items should be inserted at the back of the queue in constant time
So we must avoid traversing through the list
Use a second node pointer to keep track of the node at the back of the queue
Requires a little extra administration
Member attributes:
Node* front;
Node* back;
int num;
October 03, 2017
Cinda Heeren / Geoffrey Tien

14. Cinda Heeren / Geoffrey Tien
List Queue Example
Queue q;
q.enqueue(6);
front 6
q.enqueue(4);
q.enqueue(7); 4
back
q.enqueue(3);
q.dequeue(); 7 3
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15. Cinda Heeren / Geoffrey Tien
Exercise
Shallow Copy, Deep Copy
Given a Queue class implemented with a (dynamic) circular array of 8 int elements and an integer size member, the following block of code is executed:
Queue qA;
qA.Enqueue(34);
qA.Enqueue(29);
qA.Enqueue(11);
qA.Dequeue();
qA.Enqueue(47);
Queue qB = qA;
qA.Enqueue(6);
qB.Enqueue(52);
qB.Dequeue();
What will be output as the contents of each queue qA and qB if:
qB is created as a shallow copy of qA?
qB is created as a deep copy of qB?
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16. Introduction & terminology
Trees
Introduction & terminology
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17. Cinda Heeren / Geoffrey Tien
Review: Linked Lists
Linked lists are constructed out of nodes, consisting of
a data element
a pointer to another node
Lists are constructed as chains of such nodes
List
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18. Cinda Heeren / Geoffrey Tien
Trees
Trees are also constructed from nodes
Nodes may now have pointers to one or more other nodes
A set of nodes with a single starting point
called the root of the tree (root node)
Each node is connected by an edge to another node
A tree is a connected graph
There is a path to every node in the tree
A tree has one less edge than the number of nodes
Root
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19. Cinda Heeren / Geoffrey Tien
Is it a tree?
No!
Nodes are not all connected
Yes!
Yes!
Although not a binary tree
No!
There is an extra edge
(5 nodes, 5 edges)
Yes!
Actually equivalent to the blue tree
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20. Cinda Heeren / Geoffrey Tien
Tree Relationships
Node v is said to be a child of u, and u the parent of v if
There is an edge between the nodes u and v, and
u is above v in the tree,
This relationship can be generalized
E and F are descendants of A
D and A are ancestors of G
B, C and D are siblings
F and G are?
root
Parent of B, C, D A
edge B C D E F G
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21. Cinda Heeren / Geoffrey Tien
More Tree Terminology
A leaf is a node with no children
A path is a sequence of nodes v1 … vn
where vi is a parent of vi+1 (1  i  n)
A subtree is any node in the tree along with all of its descendants
A binary tree is a tree with at most two children per node
The children are referred to as left and right
We can also refer to left and right subtrees
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22. Tree Terminology Example
A path from A to D to G B C D
A subtree rooted at B E F G
C, E, F, G are leaf nodes
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23. Binary Tree TerminologyC
Right child of A
Left subtree of A D E F G
Right subtree of C H I J
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24. Cinda Heeren / Geoffrey Tien
Measuring Trees
The height of a node v is the length of the longest path from v to a leaf
The height of the tree is the height of the root
The depth of a node v is the length of the path from v to the root
This is also referred to as the level of a node
Note that there is a slightly different formulation of the height of a tree
Where the height of a tree is said to be the number of different levels of nodes in the tree (including the root)
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25. Tree Measurements Explained
Height of tree is 3
Height of node B is 2 B C
Level 1 D E F G
Level 2
Depth of node E is 2 H I J
Level 3
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26. Cinda Heeren / Geoffrey Tien
Perfect Binary Trees
A binary tree is perfect, if
No node has only one child
And all the leaves have the same depth
A perfect binary tree of height h has
2h+1 – 1 nodes, of which 2h are leaves
Perfect trees are also complete
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27. Height of a Perfect Tree
Each level doubles the number of nodes
Level 1 has 2 nodes (21)
Level 2 has 4 nodes (22) or 2 times the number in Level 1
Therefore a tree with ℎ levels has 2 ℎ+1 −1 nodes
The root level has 1 node 01
Bottom level has 2h nodes, i.e. just over half of the nodes are leaves 12 11 21 22 23 24 31 32 33 34 35 36 37 38
October 03, 2017
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28. Cinda Heeren / Geoffrey Tien
Complete Binary Trees
A binary tree is complete if
The leaves are on at most two different levels,
The second to bottom level is completely filled in and
The leaves on the bottom level are as far to the left as possible A B C D E F
October 03, 2017
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29. Cinda Heeren / Geoffrey Tien
Balanced Binary Trees
A binary tree is balanced if
Leaves are all about the same distance from the root
The exact specification varies
Sometimes trees are balanced by comparing the height of nodes
e.g. the height of a node’s right subtree is at most one different from the height of its left subtree (e.g. AVL trees)
Sometimes a tree's height is compared to the number of nodes
e.g. red-black trees
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30. Cinda Heeren / Geoffrey Tien
Balanced Binary Trees A A B C B C D E F D E F G
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31. Unbalanced Binary TreesF
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32. Readings for this lesson
Koffman
Chapters 6.1 – 6.3 (Queues)
Chapter 8.1 (Tree terminology)
October 03, 2017
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