Algorithm Design and Problem-Solving: Trace Tables

This section focuses on creating and using trace tables to document the execution of algorithms. A trace table helps us understand how an algorithm works step-by-step, identify potential errors, and verify its correctness.

What is a Trace Table?

A trace table is a table used to show the values of variables and the operations performed during the execution of an algorithm. It provides a clear and organized way to follow the algorithm's flow.

How to Create a Trace Table

1. Identify the input values for the algorithm.
2. List all the variables used in the algorithm.
3. Create columns in the table for:
• Iteration/Step Number: Indicates the order of execution.
• Input: The input values provided to the algorithm.
• Variable A: The value of variable A after each step.
• Variable B: The value of variable B after each step.
• Operation: The action performed in that step (e.g., assignment, comparison, conditional statement).
• Result: The outcome of the operation.
4. Fill in the table by performing the algorithm step by step, recording the values of variables and the operations performed at each iteration.

Example: Simple Algorithm - Adding Two Numbers

Let's consider a simple algorithm that adds two numbers, X and Y, and stores the result in Z.

Algorithm:

1. Start
2. Set Z = X + Y
3. Output Z
4. Stop

Here's the trace table for this algorithm with input X = 5 and Y = 3:

Iteration/Step Number	Input	X	Y	Operation	Result
1	X = 5, Y = 3	5	3	Set Z = X + Y	Z = 5 + 3 = 8
2	Z = 8	5	3	Output Z	Output: 8
3				Stop

Example: More Complex Algorithm - Finding the Maximum of Two Numbers

Now, let's trace an algorithm to find the maximum of two numbers, A and B.

Algorithm:

1. Start
2. If A > B then
3. Set Maximum = A
4. Else
5. Set Maximum = B
6. Output Maximum
7. Stop

Trace table for A = 10 and B = 5:

Iteration/Step Number	Input	A	B	Operation	Result
1	A = 10, B = 5	10	5	If A > B then	10 > 5 is true
2	A = 10, B = 5	10	5	Set Maximum = A	Maximum = 10
3	Maximum = 10	10	5	Output Maximum	Output: 10
4				Stop

Benefits of Using Trace Tables

Using trace tables provides several benefits:

• Clarity: It clearly shows the flow of the algorithm.
• Debugging: Helps identify errors in the algorithm's logic.
• Verification: Allows us to verify that the algorithm produces the correct output for different inputs.
• Understanding: Improves understanding of how algorithms work.

Practice

Practice creating trace tables for various simple algorithms. This will help you develop strong problem-solving skills essential for computer science.