Lecture 2: Problem Solving Concepts.
Overview:
q Problem Solving is a creative act and cannot be
completely explained
q There are many methods that help structure our
thoughts as we solve problems
q The Prominent are:
Analytic, Algorithmic & Software engineering
Objective of this Lecture:
q Discuss two problem solving methods – Analytic
& Algorithmic
q Solve example
problems using each of these methods
Common Components (steps)
Each of these two methods consists of four
basic components
q Problem:-
Specification of the problem to be solved
q Reasoning:- Comprehension of the problem (Analysis)
q Solution:- Specification
of the solution (Design)
q Test: Checking
that the solution is correct
The Analytic approach
q It is used to solve mathematic, physics and similar sciences
& engineering problems
q The method (solution component) involves :
Ø Isolating the given
quantities, denoting them by some variables
Ø Identifying the relation
(equation) connecting the quantities
Ø Solving the equation by
applying some algebraic rules and/or applying existing formulae
Example :
Problem: A school needs 12 identical computers for a
laboratory. Carpeting & furniture
is budgeted at $6,000.00. If the total
budget is 90,000.00, write an algorithm that finds the highest cost of the most
affordable computer the school can buy.
Reasoning: To find the highest cost of the most
affordable computer, first we subtract the cost of carpeting & furniture
from the total budget and then divide the result by 12.
Solution: Let x be the cost of one such computer. Then the cost of 12 such computers is
12x. The budget for the 12 computers is
the total budget less the cost of carpeting and furniture.
i.e.
12x = total – cost of carpeting &
furniture
i.e. 12x = 90000 – 6000
i.e. 12x = 84000
so
that x = 84000/12
= 7000
Test: To test the solution, we Substitute in the
first equation:
LHS = 12 x 7000 = 84000 = RHS
Note: Many problems cannot be solved analytically,
for example, the polynomial :
Ax5 + bx4 + cx3 + dx2 + ex + f = 0 , f¹0.
The Algorithmic Approach
q This was first developed by an Arab mathematician
called Al-Khuwarizmi.
q It involves breaking down a solution into a sequence
of executable instruction with the following properties:
Ø There is no ambiguity in any
instruction.
Ø There is no ambiguity in the
order of execution.
Ø The sequence is finite
Ø Execution terminates after a
finite number of steps.
q Algorithmic solutions are not unique and are not
restricted to mathematical problems.
q In the reasoning step, the problem is summarized by generating
a formal statement. This specifies:
Ø The input needed
Ø The output to be produced
&
Ø The process to be performed
to get the output
q In the solution step, the actions needed to solve the
problems are listed in sequence. Some
of these actions may be skipped depending on certain condition (selection) and
the entire sequence of actions of part of the sequence may be repeated
(repetition).
q The test step involved applying the algorithm to
various input data to ensure that it works.
Elements of an Algorithmic Language.
The
sequence of instruction in an algorithm are written using pseudocodes – words
that resemble high label programming language.
The table below summarises these words:
|
LANGUAGE
ELMTS.
|
MEANING
|
|
INPUT
|
Accept
data from user
|
|
OUTPUT
|
Send
results to the user
|
|
=
|
Assign
the result of the expression on the right to the variable on the left
|
|
STOP
|
End
the algorithm
|
|
IF
ELSE
|
For
selection
|
|
LOOP
|
For
repetition involving a specified number of times
|
|
WHILE
|
For
repetition that terminates based on a condition (unspecified number of times)
|
Example 1:
Problem: Write an algorithm to compute the volume of
a sphere. The algorithm should check
that the radius is greater than 0. (Take p = 3.14159)
Reasoning: We need to
use the formula for computing the volume of a sphere. We need to use
selection so that we only compute the volume for positive radius
Formal
Statement: Volume of a sphere problem
Input:
radius
Ouput: volume
Process: Volume =
(4/3)*3.14159*radius*radius*radius
Solution:
Step1: INPUT radius
Step2: IF radius > 0
a. volume = (4/3)*3.14159*radius*radius*radius
b. OUTPUT volume
Step3: STOP
Test: The algorithm can be checked by trying it
with positive and negative values of the radius.
Example 2:
Problem : Modify example 1 to allow the user to
repeatedly enter values for the radius and print the volume of the
corresponding sphere, and terminates when the radius is less than or equal to
zero.
Reasoning: Same as example 1
except that we need to use repetition.
Solution:
Step1: INPUT radius
Step2: WHILE radius > 0
a. volume = (4/3)*3.14159*radius*radius*radius
b. OUTPUT volume
Step3: STOP
Test: The algorithm can be checked by trying it
with a number of positive and negative values of the radius.