Linear Structures

Variations on a Theme

Nathan Tenney

WSU Tri-Cities

Abstract Data Type

• Defined as A set of objects together with a set of operations.
• When we abstract something, we leave out certain details to simplify:
  • how the data structure is implemented
  • the type of data stored in the data structure

List ADT

• In general, we can consider a list to be a set of $N$ objects of the form $A_0,A_1,...,A_N$
• We define a special subset of a list called an empty list which is a list of size 0.
• List of objects may be of any type (most books use integers for simplicity)
• What type of operations might we want to do on a list?

List Implementations

• Two basic implementations of the List ADT:
  • Array based
  • Linked Structure (Linked List)
    • Singly
    • Doubly
    • Circularly

The STL Implementations

• STL is an implementation of common data structures
• Uses templates (as discussed previously) to provide an abstract implementation of these container objects
• Big Three functions: size, clear, and empty
• List and vector functions: push_back, pop_back, back, front
• List adds convenience methods for inserting and removing from front
• Vectors add convenience methods including using the [] operator.

Skip List

• Lists have one major drawback
• How can we improve this?
• We add the concept of a Skip List

Skip List

• Provides the ability to perform non-sequential searches on sequential lists.
• Given a skip list of n nodes, for each k and i S.T. $1 \le K \le \lfloor \log_2 n\rfloor$ and $1 \le i \le \lfloor n/2^{k-1} \rfloor$ the node in the $2^{k-1} * i$ position points to the node in the $2^{k-1} * (i+1)$ position.
• Every $2^{nd}$ node points to the node 2 positions ahead, every $4^{th}$ node points to the node 4 positions ahead…
• Number of pointers indicates the level of the node
• Max number of levels can be found by $\lfloor \log_2 n \rfloor + 1$

Skip List

• Skip list ideal positioning

Skip List

• What problems, if any, does the skip list have?
  • Insertions and deletions may change max level
  • Insertions and deletions require some manipulation of the pointers to maintain the structure of the list
  • Skip list uneven positioning
• To side step these issues, we randomly select the level of an item being inserted
• In worst case time, this can impact performance, but over the average case, it's close enough to $O\left(log_2 N\right)$

List Problems

• Searching is the Linked Lists Achilles Heel
• Skip lists were developed to combat this issue
• Skip lists alter the very structure of the list to improve searching. Are there other approaches?

Self-Organizing Lists

• Re-order list items
• New order depends on data
• We will explore 4 methods
  • Move-to-Front Method
  • Transpose Method
  • Count Method
  • Ordering Method

Move-To-Front

• Items searched for will be searched for again
• Move the found item to the front of the list
• Guarantees that the most recently found items are near the front of the list

Count Method

• Like the Move-To-Front method, assumes that items searched for will be searched for again
• Nodes in the list keep a count of how many times they've been accessed
• List is ordered by the count
• Ensures that the items most frequently used are near the front of the list

Transpose Method

• Least agressive of the access based methods
• Each time an item is accessed, it is moved forward one spot in the list
• The most used items bubble towards the front of the list.

Insertion of Items

• For each of these, where would it make sense to insert items?
• Move-To-Front
• Count Method
• Transpose Method

Ordering Method

• Data is ordered according to the natural ordering of the elements
• Numeric data
• Alphabetic data

Comparison of Methods

Comparison of Methods

Sparse Tables

• A Table often seems like a good choice to store data
• Consider the case of a university with 8000 students and 300 classes

Sparse Tables

Sparse Tables

Sparse Tables

Stacks

• How does a stack differ from a List?
• LIFO
• Methods available in the Stack ADT
  • Push - place an element on the stack
  • Pop - remove the topmost item from the stack
  • Top - examine the topmost element prior to removing it
• Note: pop or top on an empty stack is generally considered an error
• What is the complexity of stack operations?

Stacks

• In general linked lists are the best choice for implementing a stack.
• For special cases, we may need a specific implementation of a stack.
  • Linked structure - singly linked list
  • Array based - Vector implementation.

Application: Symbol Balancing

• One misplaced symbol can cause pages of errors
• Balance checking program (for simplicity sake we look at balancing parens, brackets and braces)
• The solution turns out to be so simple that most editors/IDE's do this now.

Application: Symbol Balancing

• Each time we encounter an opening symbol, we push that symbol onto a stack
• If the symbol is a closing symbol and the stack is empty, report an error
• If the symbol is a closing symbol, pop a symbol off the stack and compare it to the closing symbol found.
  • If they don't match, report an error
• If the stack is not empty at the end of processing, report an error

Application: Postfix Math

Calculating the cost of a shopping trip

• $4.99 + 5.99 + 6.99 * 1.083$
• 4 function calculator result: $19.46$
• scientific calculator result: $18.55$

Application: Postfix Math

• Reason for this is operator precedence
• If we wanted to apply tax only to specific items, first and last for example
  • $4.99 * 1.083 + 5.99 + 6.99 * 1.083$
    • Scientific calculator result: $18.96$
    • 4 function calculator result: $19.91$

Application: Postfix Math

• Essentially, to get the correct result on a normal calculator, we need to keep track of intermediate results
• Alternatively, we can write out the proper order of the sequence of events as follows:
  • $4.99 1.083 * 5.99 + 6.99 1.083 *$
  • this is known as Postfix Notation or RPN (Reverse Polish Notation)
  • easiest way to evaluate is using a stack
• Evaluate the postfix expression: $6 5 2 3 + 8 * + 3 + *$

Application: Postfix Math

Read 6

Application: Postfix Math

Read 5

Application: Postfix Math

Read 2

Application: Postfix Math

Read 3

Application: Postfix Math

Read +

Application: Postfix Math

Read 8

Application: Postfix Math

Read *

Application: Postfix Math

Read +

Application: Postfix Math

Read 3

Application: Postfix Math

Read +

Application: Postfix Math

Read *

Application: Function Calls

• Two basic things we need to handle when making a function call
• Activation record or stack frame is created an placed on a stack
• As you return from a function call, a stack frame is popped off the stack, and state is restored
• Program continues execution.
• Used in most modern programming languages

Queues

• FIFO - First in First out
• Insertion at the rear of the structure, removal from the front
• $O\left(1\right)$

  operations, why?

• May use either a list or array implementation
• Circular Array
  • Be careful in how you maintain the size of the queue

Queues

• Many different real world situations can be managed by queues
• Hold systems at large businesses
• A whole branch of mathematics, called queueing theory, deals with modeling real world situations utilizing queues

Queues

• Recorded over a period of 3 months
  • Number of customers in the bank
  • Time it takes to serve them
• Six clerks were employed
• No lines observed
• How to determine the appropriate number of clerks?

Queues

Queues

• Wrote a simulation of customers and tellers
• For each minute
  • Number of arriving customers is randomly chosen
  • Transaction time for each customer is chosen at arrival time
• As a clerk becomes available, a customer is removed from the queue, and the tellers timer is incremented.
• Run Demo

Priority Queues

• There are instances where a queue is impractical due to priority constraints
  • VIP Constraints
  • Process scheduling
  • Emergency room
• Can be difficult to implement
  • Need to find an implementation that keeps enqueue and dequeue operations fast

Priority Queues

• Basic Linked List (2 ways)
  • Entry Ordered
  • Priority Ordered
  • Which is better?
• Two list implementation
  • Short, ordered list
  • Unordered list

Priority Queues

• Hendriksen's method
  • Ordered linked list
  • Array of pointers that index into the list
• Comparisons
  • Linked list implementations work best on queues of 10 items or less
  • Depending on distribution of priorities, 2 list method, comparable to Linked List implementations
  • Hendriksen's method performs well on queues of any size.