Applications
1. Polynomial ADT:
A polynomial can be represented with primitive data structures. For
example, a polynomial represented as akxk ak-1xk-1
+ ... + a0 can be represented as a linked list. Each node is
a structure with two values: ai and i. Thus, the length of the
list will be k. The first node will have (ak, k), the second
node will have (ak-1, k-1) etc. The last node will be (a0,
0).
The polynomial 3x9 + 7x3 + 5 can be represented
in a list as follows: (3,9) --> (7,3) --> (5,0) where each pair
of integers represent a node, and the arrow represents a link to its neighbouring
node.
Derivatives of polynomials can be easily computed by proceeding node
by node. In our previous example the list after computing the derivative
would represented as follows: (27,8) --> (21,2). The specific polynomial
ADT will define various operations, such as multiplication, addition, subtraction,
derivative, integration etc. A polynomial ADT can be useful for symbolic
computation as well.
2. Large Integer ADT:
Large integers can also be implemented with primitive data structures.
To conform to our previous example, consider a large integer represented
as a linked list. If we represent the integer as successive powers of 10,
where the power of 10 increments by 3 and the coefficent is a three digit
number, we can make computations on such numbers easier. For example, we
can represent a very large number as follows:
513(106) + 899(103) + 722(100).
Using this notation, the number can be represented as follows:
(513) --> (899) --> (722).
The first number represents the coefficient of the 106 term,
the next number represents the coefficient of the 103 term and
so on. The arrows represent links to adjacent nodes.
The specific ADT will define operations on this representation, such as
addition, subtraction, multiplication, division, comparison, copy etc.
3. Window Manager ADT:
A window interface can be represented by lists. Consider an environment
with many windows. The fist node in the list could represent the current
active window. Subsequent windows are further along the list. In other
words, the nth window corresponds to the nth node
in the list.
The ADT can define several functions, such as Find_first_window which would
bring a window clicked upon to the front of the list (make it active).
Other functions could perform window deletion or creation.
4. Management of free space:
When memory is requested, a list of available blocks of memory might
be useful. Again, a list could represent blocks in memory available to
the user, with nodes containing pointers to these available blocks. The
list can be used like a stack (LIFO). The last freed memory becomes the
next available to the user. Such lists are called 'free space lists' or
'available space lists'. Since addition and deletion of nodes is at one
end, these lists behave like stacks. All operations on free space lists
can be done in O(1) time.
Stacks
1. Stack based languages:
Expressions can be evaluated using a stack. Given an expression in a
high-level language (for example, (a + b) * c) the compiler will transform
this expression to postfix form. The postfix form of the above example
is ab + c *, where a and b are operands, + and * are operators, and the
expression is scanned left to right. The expression is pushed on the stack
and evaluated as it is popped. The following algorithm illustrates the
process:
In the end the stack will hold one element (the result).
2. Text Editor:
A text editor can be implemented with a stack. Characters are pushed
on a stack as the user enters text. Commands to delete one character or
a command to delete a series of characters (for example, a sentence or
a word) would also push a character on a stack. However, the character
would be a unique identifier to know how many characters to delete. For
example, an identifier to delete one character would pop the stack once.
An identifier to delete a sentence would pop all characters until the stack
is empty or a period is encountered.
3. Postscript:
Postscript is a full-fledged interpreted computer language in which
all operations are done by accessing a stack. It is the language of choice
for laser printers. For example, the Postscript section of code
1 2 3 4 5 6 ADD MUL SUB 7 ADD MUL ADD
represents:
1 2 3 4 5 6 + * - 7 + * +
which in turn represents:
1 + ( 2 * ( ( 3 - ( 4 * ( 5 + 6 ) ) ) + ( 7 ) ) )
Very much as in the stack-based language example, the expression can be
evaluated from left to right. Expressions written in the form given above
are called postfix expressions. Their easy evaluation with the help of
a stack makes them natural candidates for the organization of expressions
by compilers.
4. Scratch pad:
Stacks are used to write down instructions that you can not act on immediately.
For example, future work to be done by the program, information that may
be useful later, and so forth (just as with a scratch pad). An example
of this is the rat-in-maze problem (see below). A stack can be used to
solve the problem of traversing a maze. One must keep track of previously
explored routes, or else an infinite loop could occur. For example, with
no previous knowledge of exploring a specific route unsuccessfully, one
can enter a path, find no solution to the maze, exit the path through the
same route as entrance, then enter the same unsuccessful path all over
again. This problem can be solved with the help of a stack.
5. Recursion:
Stacks are used in recursions. Every recursive program can be rewritten
iteratively using a stack. One related problem is the knapsack problem:
Consider a knapsack with volume represented as a fixed integer. One is
given a series of items of varying size (the size of the objects is represented
as an integer). The knapsack problem is to find a combination of items
that will fit exactly into the knapsack (i.e. no unused space). The function
call is written as 'knapsack(target: , candidate: )' where 'target' is
the amount of space left in the sack, and 'candidate' is the reference
to the item being considered to be added. The function returns a boolean
result; 'true' if target can be filled exactly using a subest of the items
numbered "candidate, ..., n". Here 'n' is the total number of
items. Define size[.] as an array of sizes of the items. The following
is a recursive solution to the problem:
knapsack(target,candidate)
if target = 0 then return "true"
if candidate > n or target < 0 then return "false"
if knapsack(target - size(candidate) , candidate + 1) then
- return "true"
else return knapsack(target, candidate + 1)
A knapsack of size 26 can be filled with items of size 15 and 11.
...
JAVA APPLET
This applet demonstrates an excellent application of the Stack ADT.
In order to find its way out of the maze, the algorithm pushes its moves
onto a stack until it either reaches the end, or becomes blocked. If it
becomes blocked, the algorithm will backtrack its moves by popping them
from the stack until it reaches a square with at least one free neighbor.
It will then follow this new path until it again becomes blocked or it
finishes. This process is repeated until the end of the maze is reached.
To see this algorithm in action, press the "Next ->" button
and watch the green line advance one move. Press "<- Back"
to go back one iteration. When the green line has to backtrack, it leaves
behind a grey line to indicate the squares it has already visited.
References
For More Information on Data Structures, consider these books:
- Aho, Alfred V. Data Structures and Algorithms. Addison-Wesley, 1983:
10-3. Describes the role of primitive data structures in overall program
design and provides an example of those implementations.
- Feldman, M.B. Data Structures with Modula-2. Prentice-Hall, 1988: 6-8.
A basic discussion of how primitive data structures, together with operations,
permit the creation and manipulation of objects of that type.
- Horowitz, Ellis. Fundamentals of Programming Languages, Second ed.
Computer Science Press, 1984: 256-60. Covers abstract specification of
queue, stack and linked lists. Considers definitions of a data type in
its most general form.
- Sedgewick, Robert. Algorithms in C, 2nd ed. Addison-Wesley, 1990: 31-32.
- Standish, T.A. Data Structures, Algorithms and Software Principles.
Addison-Wesley, 1994: 20-23. Basic concepts and principles, ideas of the
stacks, queues and linked lists.
- Weiss, Mark Allan. Algorithms, Data Structures, and Problem Solving
with C++. Addison-Wesley, 1996: 41-43. Explains the concept of a primitive
data structure.
This Web Page was Created by:
The text is based on class notes taken by Daniel Taranovsky. Web Page
text was authored by Damian Orfamoudakis and Daniel Taranovsky. Daniel
Levin created the Java applet and accompanying text. Graphics designed
and drawn by Danny Wong.
Copyright © 1997, "the D4 production".
All rights reserved. Reproduction of all or part of this work is permitted
for educational or research use provided that this copyright notice is
included in any copy. Disclaimer: this collection of notes is experimental,
and does not serve as-is as a substitute for attendance in the actual class.
