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Properties of Algorithms

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  • 0:02 Algorithms
  • 1:45 Properties
  • 2:37 Sample Problem
  • 4:13 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Algorithms are a set of step-by-step instructions that satisfy a certain set of properties. In this lesson, we'll explore the properties an algorithm must satisfy in order to be useful using an example.

Algorithms

Have you ever tried to assemble a piece of furniture by yourself? If so, you probably used a set of step-by-step instructions to assist you in your endeavor. Wouldn't it be nice if math problems came with a set of instructions like this? Oh wait, they do!

When solving a math problem, we usually use an algorithm, or a set of step-by-step instructions. For example, suppose we're trying to figure out what the perimeter of a rectangle with length 5 units would be for various widths. Since the formula for the perimeter of a rectangle is:

  • P = 2l + 2w, where l = length and w = width

we can plug in 5 for l to get P = 2(5) + 2w = 10 + 2w. Ultimately, we're trying to find the different values of P for various values of w, where

  • P = 10 + 2w

To figure out the perimeter for this rectangle, for some width w, we follow these steps:

  1. Multiply w by 2.
  2. Add 10 to the result. This is the perimeter.


algoprop1


This is an example of an algorithm. It is a set of steps that we can follow in order to find the perimeter of the rectangle for a given width, w. Now suppose we want to know what the perimeter of this rectangle would be if it had a width of 8 units. Again, we can use our algorithm.

  1. Multiply 8 by 2: 8 ⋅ 2 = 16
  2. Add 10 to the result: 16 + 10 = 26

Here we end up with a perimeter of 26 units. Now let's discuss some of the properties of algorithms.

Properties

In order for an algorithm to be useful, it must help us find a solution to a specific problem. For that to happen, an algorithm must satisfy five properties.

  1. Input: The inputs used in an algorithm must come from a specified set of elements, where the amount and type of inputs are specified.
  2. Output: The algorithm must specify the output and how it is related to the input.
  3. Definiteness: The steps in the algorithm must be clearly defined and detailed.
  4. Effectiveness: The steps in the algorithm must be doable and effective.
  5. Finiteness: The algorithm must come to an end after a specific number of steps.

When an algorithm satisfies these five properties, it is a fail-proof way to solve the problem for which it was written.

Sample Problem

To further our understanding of these five properties, let's take a look at our opening example where we used an algorithm to find the perimeter of a rectangle with length 5 units for various widths and see how it satisfies each of the properties.

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