Problem-solving and program design

1. Outline the steps in problem-solving

If you research the topic of Problem Solving by using either the internet or any reputable text book, most likely you will see the number of procedures range from probably four (4) to seven (7) steps. For example, you may come across the following seven (7) main steps when trying to solve a problem:

Step 1. Define the Problem

Step 2. Analyze the Problem

Step 3. Identifying Possible Solutions

Step 4. Selecting the Best Solutions

Step 5. Evaluating Solutions

Step 6. Develop an Action Plan

Step 7. Implement the Solution

The above seven steps above may be merged into four steps as follows:

Step 1. Define the Problem

State the problem clearly.

* Try to state the problem in your own words

* Write down what are you trying to find out or supposed to do

* Describe what data is input and what information is output

* Draw an INPUT-PROCESSING-OUTPUT (IPO) diagram

* Describe what calculations are needed that use the input to produce the output

Step 2. Identify and evaluate possible solutions

This involves selecting the best solutions

* Look for a pattern.

* Examine related problems, and determine if the same technique can be applied.

* Examine a simpler or special case of the problem to gain insight into the solution of the original problem.

Step 3. Develop and represent the algorithm

Develop a solution that is general in nature.

* Make a table.

* Make a diagram (such as an algorithm).

* Work backward.

* Work the problem by hand (with a calculator or a using a trace table) for a specific set of data.

Step 4. Test the solution with a variety of data to validate that you are solution correct to the problem given.

Ensure that you include error messages for data that is not expected on input or in processing that may cause the program to crash

Here is another set of steps on the phases. Can you place the following three steps into the four step above?

Problem-Solving Phase

1. Analysis and specification.(Define problem and what solution must do.)

2. General solution (algorithm).(Develop logical sequence of steps to solve problem.)

3. Verify.(follow steps - by hand.)

Implementation Phase

1. Specific solution (program).(Translate algorithm to code.)

2. Test.(Check computed results manually.)

Maintenance Phase

1. Use the program.

2. Maintain.(Modify to meet changed requirements or to correct errors.)

2. Decompose a simple problem into its significant parts

Input, Process, Output, Storage

Input

Entering data from an external source into the system can be achieved either:

1. Manually using keyboards which are often used to enter data into a system, or

2. Automatically, where data can be entered into the system using hardware such as OMR, microphone, or sensors.

Processing

Once this data is entered, it can be processed, meaning you can:

– Search

– Sort, and

– Perform Calculations on the data

Processing data is important as it leads to useful output from the system.

Output

After the data is processed into information, it can be presented in different ways for the user, such as:

– Printed to an output device such as a printer or fax (Hardcopy)

– Printed on the screen (softcopy)

– Sound (such as the sounds from ATMs when you are making a transaction or the sound from the cash register)

– Light, (flashing lights on an ambulance; a flashing light to take your money or card from the ATM)

– Can you think of any more ways to output information?

Output essentially deals with any data that is leaving the system.

Storage

If data doesn’t need to be output immediately after processing it may be saved for later use on a storage medium such as a hard drive.

Also, stored data may be accessed to be processed with new data to output more information.

Note the unique qualities of storage mediums such as CDs and Disk Drives. They can be input devices when data is being stored on them, but they become output devices when data is required from these devices.

3. Distinguish between variables and constants

Variables and Constants

The ability to manipulate variables is one of the most powerful features of a programming language.

Let's explain variables and constants by first understanding something that you are hopefully familiar with.

Suppose each week Janus and Kikiyo go to a particular room in a building for a meeting. The room is the same location  with the same furniture and may even have a name, such as Room LT4.

Each week when they go to the meeting, Janus may or may not sit in the same seat since Kikiyo or someone else who got there earlier may have taken that seat, so Janus sits somewhere else.

SO! The room LT4 is a constant. It is located in the same place for the duration of your meeting.

While, where Janus, Kikiyo and the other sit varies. Their seats are variable each week when they go to the meeting. Sometimes Janus or Kikiyo or someone else may miss the meeting. Janus, Kikiyo and the other people have names but their seat location in the weekly meeting is what varies.

In Pascal terms, the syllabus states that :

a variable is an area of storage whose value can change during processing. In our example the variable is the seat location

a constant can contain a value that never changes during the processing of the program. In our example the constant is the room.

Variables

Programmers generally choose names for the variables that are meaningful. That is, the programmer uses a variable that indicates what the variable is used for. It is good programming practice to use meaningful variable names that suggest what they represent, since this makes the program more readable and easier to understand.

For example:

Tax, FirstName, DeptCode, SeatNo, HoursWorked, and so on.

The assignment statement creates a new variable and assigns them values. For example, in Pascal the following assignment statements are valid:

SeatNo := 'Janus';

HoursWorked := 35;

Please note the following in every assignment statement:

the variable to be assigned a value must appear on the left of the assignment operator the assignment operator is := a legal expression (value to be assigned) must appear on the right

 Constants

A constant represents a value that does not change for the duration of the program. So, for example, in Pascal , a value of 25% or .25 will be assigned to the variable Tax. However, Tax cannot change to any other percentage while the program is being run.

Const Tax = 0.25; {to represent 25 %}

Note the use of the equal sign for the Constant Declaration.

4. Use appropriate data types

Problem 1 Write an algorithm and draw a flowchart that reads two values, determines the largest value and prints the largest value with an identifying message.ALGORITHM

Step 1:    Input VALUE1, VALUE2

Step 2:   if (VALUE1 > VALUE2) then

                 MAX = VALUE1

               else 

                   MAX = VALUE2

             endif

Step 3:   Print “The largest value is”, MAX

Flowchart

 Problem 2

Write an algorithm and draw a flowchart to convert the length in feet to centimeter.  Algorithm

Input the length in feet (Len_FT)Calculate the length in cm (Lcm) by multiplying Len_FT by 30Print length in cm (Len_CM)  Pseudocode: 

Step 1:  Input Len_FTStep 2:   Len_CM Step 3:   Print Len_CM  Flowchart

Problem 3Write an algorithm and draw a flowchart that will read the two sides of a rectangle and calculate its area.Algorithm

Input the width (W) and Length (L) of a rectangleCalculate the area (Area) by multiplying L with WPrint Area Pseudocode

Step 1:   Input W, LStep 2:   Area = W * LStep 3:   Print Area

Flowchart

5. Explain the concept of algorithms

Algorithms and flowcharts 

Using our steps in problem solving, a typical programming task can be divided into two phases: 1. Problem solving phase

This phase is where you produce an algorithm ( an ordered sequence of steps) that describe solution of problem

 2. Implementation phase

This phase is where you implement the program in some programming language, in this case, Pascal.

 Steps in Problem Solving First produce a general algorithm An algorithm as a set or list of instructions for carrying out some process step by step until it reaches the end of the algorithm. A recipe in a cookbook is an excellent example of an algorithm. So, the recipe includes the ingredients and the method of cooking until you (hopefully) end up with a nice dish! 

An example of an algorithm people use would be a recipe:

4 extra large eggs, beaten  1&1/2 C. stock  1/2 teaspoon salt  1 onion, minced  1 C. tuna flakes  1 t. soy sauce  1 Tablespoon oil 

1. Mix all the ingredients, except the oil, in a deep bowl2. Put 1" water in wide pot, then place deep bowl of batter inside the pot3. Cover pot tightly and steam 15 min.4. Heat oil very hot and pour over custard5. Steam 5 more min.

Serves 4 people Examples of Algorithms

Algorithms can be represented as:

1. Pseudocode - English like words that specify the sequence of steps in an algorithm2. Flowcharts - a graphical tool using standard symbols to show the sequence of steps in an

algorithm

 So whether you use pseudocode or flowcharts to expand your algorithm, here is why algorithm are so useful:

Every algorithm must have an end, so there must be a set of steps followed in that algortihm to reach the end. For example, you follow the steps of the recipe to reach the end which is your dish that is ready to eat!

Every instruction should therefore be clear, with short instructions and easily understood The instructions should be in a sequence from top to bottom. Using the example recipe

above, you follow the instructions from step 1 to step 5. 

To illustrate how we use algorithms (pseudocode and flowcharts) to a programming problem, let us use an example:

 Example 1:Write an algorithm to determine a student’s final grade and indicate whether the student has passed or failed. The final grade is calculated as the average of four marks. Pseudocode

Remember, pseudocode is very similar to everyday English, which helps programmers develop their algorithms.

Refine the pseudocode successively to get step by step detailed statements that are easy to understand

Pseudocode: First try List the tasks that need to be performed. You can also use an IPO chart to help you:

INPUT PROCESSING OUTPUT

M1, M2, M3, M4

calculate GRADE (average)

as (M1+M2+M3+M4)/4

Compare GRADE (less than 50)

Print Message FAIL or PASS

Input a set of 4 marks Calculate their average by summing and dividing by 4 if average is below 50

  Print “FAIL”

else

  Print “PASS”Detailed Pseudocode

Input M1,M2,M3,M4

GRADE = (M1+M2+M3+M4)/4

If (GRADE < 50)

Then

         Print “FAIL”

 Else

         Print “PASS”

Endif

Flowcharts Now we can also transform the pseudocode to a flowchart. A flowchart is a graphical representation of the algorithm (again, a sequence of instructions that reach an end). Different symbols are used to draw each type of flowchart. Flowcharts show how data flows from the input of data through the computer to final output or storage (note the IPOS diagram in a previous section).

A Flowchart

shows logic of an algorithm emphasizes individual steps and flow of data from one action to the next

 Flowcharting guidelines 

The flowchart should flow from top to bottom Each symbol is connecting by lines that have arrows (not lines alone) Avoid having flow lines crossing each other Each symbol has at least one flow line entering it and another one leaving it, except for

the START and STOP symbols. The decision symbol has two exit points; these can be on the sides or the bottom and one

side (See the decision above for 'Is Grade < 50?') Use meaningful phrases in the symbols

6. Identify ways of representing algorithms

Representation of algorithms

Represent of Algorithms as Pseudocode and Flow Charts

Design an algorithm and the corresponding flowchart for adding the test scores as given below:

29, 38, 97, 85, 52, 55

Pseudocode

1. Start

2. Total = 0

3. Get the first testscore

4. Add first testscore to Total

5. Get the second testscore

6. Add to Total

7. Get the third testscore

8. Add to Total

9. Get the Forth testscore

10. Add to Total

11. Get the fifth testscore

12. Add to Total

13. Get the sixth testscore

14. Add to Total

15. Output the Total

16. Stop

Related Flow chart

The example 1, algorithms (pseudocode and flowchart) illustrate the steps for solving the problem of adding six testscores. Where one testscore is added to Total at a time.

Note that the flowchart should have a Start step at the beginning and at least one stop step at the end. 

Since we want the sum of six testscores, then we should have a variable to store the resulting sum after each addition. In this example, the container is called Total and we make sure that Total should start with a zero value by step 2.

LOOPS For loop

Since we know that there are six testscores, we can can use a FOR loop. Here is the flowchart that uses a FOR loop:

Pseudocode

                         1. Start 

                         2. Total = 0 

                        3. Count = 0 (to count the six testscores)

                        4. If the Count = 6, go to step 7 

                        5. Get a testscore

                       6. Add the testscore to Total (Total = Total + testscore) 

                       7. Add 1 to the Count 

                       8. Go to step 4 to get next testscore

                       7. Output the Total

                       8. Stop

 Flowchart

While Loop

Here is another way, we need to add a last number to the list of numbers given. This number should be special, so unique so that each time we get a testscore, we test it to see if we have reached that special, unique number. Common numbers are -1, or 99. In this example, since we are using testscores, it may not be wise to use 99 as the unique number to stop the program.

Pseudocode

                     1. Start 

                     2. Total = 0 

                    3. Get a testscore

                    4. If the testscore is equal to -1, go to step 7 

                    5. Add to testscore ( Total = Total + testscore) 

                    6. Go to step 3 to get next testscore

                    7. Output the Total

                    8. Stop

Flowchart

In this flowchart we use a While loop. While we have not entered that special unique testscore of -1, the user is prompted to enter another testscore.

TRUTH TABLES

Boolean Operators:   AND

When using the AND Operator, both expressions must be TRUE for the compound expression to be true.

If (expression1) AND (expression2)

THEN  Output (‘Both expressions are true’);

So if a is 4 and b is 2

Then 

(a <= 2) AND (b = 2) 

 Evaluates to

(4 <= 2) AND (2 = 2)

FALSE AND TRUE

FALSE

Boolean Operators:   OR

When using the OR Operator, either one expression or both expressions can be TRUE for the compound expression to be true.

If (expression1) OR (expression2)

THEN  Output (‘One or both expressions are true’);

So if a is 4 and b is 2

Then

(a <= 2) OR (b = 2) 

(4 <= 2) OR (2 = 2)

FALSE OR TRUE

TRUE

Boolean Operators: NOTThe NOT Operator simply reverses the boolean value of the boolean expression that appears immediately to its right.

If NOT (expression)

THEN  writeln(‘This expression is FALSE’);

So if a is 4 and b is 2

Then

NOT ((a <= 2) OR (b = 2) ) will evaluate to

NOT (4 <= 2) OR (2 = 2))

NOT (FALSE OR TRUE)

NOT (TRUE)

to FALSE.

  Precedence of Arithmetic OperatorsPriority                   Operator(s)

1st                             NOT

2nd                            AND, *, /

3rd                           OR, +, -

4th                            =, <>, >, <, >=, <=

Truth Tables

Negation

~ p

F T

T F

 AND

Here both statements are true

p AND q p AND q

T   T T

T   F F

F   T F

F   F F

Conditional Statement

Relational Operators compare two values. Let us assume that the values A and B contain two numbers. We can compare the two values as follows:

NAME SYMBOL  MEANINGGreater than A > B "A" is greater than BGreater-than or equal A >= B "A" is Greater-than or equal to "B" Equals A = B "A" equals Bnot-equals "A" <> "B" "A" does not equal "B" Less-than "A" < "B"  "A" is Less than "B"Less-equal "A" <= "B"  "A" is less than or equal to "B" 

Making decisions

Conditional Statements are used to help with decisions that can be made based on the results of comparing two values. For example, the expression A>B is a logical expression. It describes a condition we want to test. So:if A>B is true (if A is greater than B) we take one action, such as print the value of Aif A>B is false (if A is not greater than B) we take another action, such as print the value of BThe algorithm for conditional statements are shown below:

If condition  then

  true alternative

  else

  false alternative

endif

An algorithm for the conditional statement is:

                                              If A>B then

                                              print A

                                                     else

                                               print B

                                               endif

 IF-THEN Statement

IF (Boolean expression) THEN  STATEMENT;

•In evaluating the Boolean expression,  If it is TRUE then the STATEMENT is

executed.  If it is FALSE then the STATEMENT is skipped.  IF-THEN-ELSE StatementIF (Boolean expression) THEN  STATEMENT1

ELSE

   STATEMENT2;

In evaluating the Boolean expression, If it is TRUE then  STATEMENT1 is executed.  If it is FALSE then STATEMENT2 is executed.

7. Develop algorithms to solve simple problems; Solving Simple problems

Problem 1 Write an algorithm and draw a flowchart that reads two values, determines the largest value and prints the largest value with an identifying message.ALGORITHM

Step 1:    Input VALUE1, VALUE2

Step 2:   if (VALUE1 > VALUE2) then

                 MAX = VALUE1

               else 

                   MAX = VALUE2

             endif

Step 3:   Print “The largest value is”, MAX

Flowchart

 Problem 2

Write an algorithm and draw a flowchart to convert the length in feet to centimeter.  Algorithm

Input the length in feet (Len_FT)Calculate the length in cm (Lcm) by multiplying Len_FT by 30Print length in cm (Len_CM)  Pseudocode: Step 1:  Input Len_FTStep 2:   Len_CM Step 3:   Print Len_CM  Flowchart

Problem 3Write an algorithm and draw a flowchart that will read the two sides of a rectangle and calculate its area.Algorithm

Input the width (W) and Length (L) of a rectangleCalculate the area (Area) by multiplying L with WPrint Area Pseudocode

Step 1:   Input W, LStep 2:   Area = W * LStep 3:   Print Area

Flowchart

8. Test Algorithm for correctness

Trace tables

A trace table is a useful technique used to test an algorithm with data to make sure that program does what it is supposed to do and no logical errors occur

The best way to use a trace table is through manual tracing or a dry run of the program statements. 

Advantages of using a trace table include:

Checking the sequence of steps of the program algorithm or code before you execute the code

locate errors in your algorithm or program code 

Let us use some program code below along with a trace table for all the variables and outputs.

Program Code

y = 3

For x =1to4

        y = y + x

Loop

Print (y)

Trace Table

x y output

1 3  

2 4  

3 6  

4 9  

4 13 13Many times a question on trace tables will ask you to complete the table which will include headings and some values to use as input with sometimes a few values as output.

Note the variables in the headings will also be found in the fragment of code given

Fill in all values that are input to the variables in the columns

Fill in all outputs from the statements

Go through EACH LINE of the program

9. Use the top down approach to problem solving.

Top-down Design Approach

A problem may seem impossible to solve because at first glance it can be very complex.

Complex problems can be solved using top-down design, also called stepwise refinement, where 

We divide the problem into sub-problems (smaller parts) Then break those sub-problems into smaller parts Soon, it is easy to solve the small easy problems Then the program as a whole will be solved

 

Advantages of Top-Down Design

Sub-dividing the problem into parts helps clarify what needs to be done. At each step of refinement, the new sub-tasks become less complicated and, therefore, easier to

program. Parts of the solution may turn out to be used in other parts of the program Subdividing the problem into sub-parts allows more than one person to work on the solution.

 

 Example of Top-Down Design

Problem:

Make some bakes

Unfortunately, stating the problem isn't enough to actually make any bakes, we need to break the task down:

Make some bakes

1. Organise Kitchen2. Make bakes3. Serve

Each of these tasks can then be broken down further:

1. Organise Kitchen1. Clean surfaces2. Get out mixing bowl, whisk, spoon, sieve3. Get out flour, sugar, salt, eggs, milk, butter4. Put on apron

2. Make bakes1. Sift salt and flour into bowl

2. Add sugar3. Mix dry ingredients4. cream butter into dry ingredients until the mixture resembles breadcrumbs5. Add water (milk is optional)6. Mix well. Add flour so that the batter is not too sticky7. Cook8. Serve

And each of these tasks can be broken down further, let us take a look at the Cook:

Fry Bakes

1. Heat pan to right temperature2. Scoop tablespoon of batter3. Spoon scoops of batter into pan4. When brown, flip bakes5. Remove, when both sides are brown 6. Place bakes on paper towels or on rack

We can break down some of these tasks even further.

So starting at a single point, making bakes, we have broken down the task into it individual parts.