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General Algorithms (Edexcel International AS Maths: Decision 1)

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Paul

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Paul

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Introduction to Algorithms

What is an algorithm?

  • An algorithm is a set of precise instructions that, if strictly followed, will result in the solution to a problem
    • Algorithms are particularly useful for programming computers
      • Computers can process huge amounts of data and perform millions of calculations in very short time frames
    • Robots can be programmed to follow an algorithm in order to complete a task
      • E.g.  A robot vacuum cleaner or lawn mower
    • Algorithms can be performed by human beings too!
      • E.g.  The recipe and cooking instructions for a cake 

What does an algorithm look like?

  • Algorithms may be presented in a variety of ways
    • A list of instructions, in order, written in words
    • Flow charts are a visual way of presenting the steps of an algorithm
      • These clearly show the order of instructions and any parts of an algorithm that may need repetition

What else do I need to know about algorithms?

  • Some algorithms do not always provide an optimal (best) solution
    • However, they will give a solution that is sufficient for the purpose
  • Modern real-life situations can be complex to analyse
    • A compromise between the efficiency of an algorithm and its accuracy is often required
    • E.g. An algorithm may produce a 'shortest route' result that is inaccurate by 200 miles
      • For a very long journey, the time saved using this algorithm may be more important than this inaccuracy 
  • When using an algorithm with a small amount of data in an exam you must follow the algorithm precisely and accurately
    • This demonstrates understanding of the algorithm
    • Using common sense or intuition to 'see' the solution will not gain full marks

Examiner Tip

  • You must show that you have followed an algorithm precisely
    • Use checks built into the algorithm to know that an algorithm is complete
    • State (or 'print') any 'output' at the end if that is what an algorithm instruction requires
      • Do not assume the output is the last number in your working

Flow Charts

What is a flow chart?

  • A flow chart is a diagrammatic way of listing the instructions to an algorithm
  • Flow charts lend themselves to algorithms that have
    • conditional instructions
      • E.g.  Is a greater than b?
      • If the answer is yes, the flow chart directs the user to one instruction
      • If the answer is no, the flow chart directs the user to a different instruction
    • repetitive parts or stages

What do the different shaped boxes in a flow chart mean?

  • Each command in a flow chart is written inside a shape
    • The shape will depend on the type of command 
      • Ovals are used to represent the start and the end
      • Rectangles are used to represent instructions
      • Diamonds are used to represent questions

A diagram showing the different shaped boxes in a flow chart and their uses. A oval is used to show the start or end of a flow chart. A rectangular box is used for an instruction. A diamond shaped box is used for a question.

How do I work with a flow chart?

  • Following a flow chart is generally straightforward but make sure you follow the instructions carefully
  • Commands given in a flow chart are intuitive to follow
    • Examples include
      • 'Input ...' - The value(s)/data that the algorithm needs to be told
      • 'Let ...' - Assign or update a value for a variable
      • 'If ...' - A conditional statement, the outcome of which will lead to different parts of the algorithm
      • 'Output ...' or 'Print ...' -  Write down the answer
  • Write down any values when instructed to
    • This will often involve completing a table of values
    • Sometimes there is a specific instruction to 'output' the final answer
      • Make sure the answer is separate to the table
  • Some questions may ask for a description of what the flow chart/algorithm produces

Examiner Tip

  • If a question provides a table that is to be filled in with values that change, bear in mind that some values may not change
    • In such instances, only the first entry for a value may end up being entered
      • Not every cell in every row/column will necessarily need completing
      • Follow the instructions from the flow chart ('robot mode'!)

Worked example

An algorithm is presented as a flow chart, shown below.

flow-chart-we-qu

a)

Describe what the algorithm achieves.

The final instructions often give an idea of the purpose of an algorithm

The algorithm works out if the input value bold italic n is a square number or not

b)
For the input n equals 23, use the flow chart to perform the algorithm.
Complete the table.

n k p Is p greater than n? Is p equals n?
         
         
         
         
         
         
         
         
Output:  

The first instruction is to input n, so fill 23 under n in the first row of the table
The next two instructions are to let k equals 1 and to let p equals k squared, so also in the first row we can fill in 1 and 1 for k and p 

n k p Is p greater than n? Is p equals n?
23 1 1    
         
         

The next instruction is the first question, is p greater than n; the answer is no
For the answer no, the flow chart directs us to another question, is p equals n; the answer is no and so we can complete the top row of the table

n k p Is p greater than n? Is p equals n?
23 1 1 NO NO
         
         

The flow chart now updates the value of k; indicate this on the second row of the table
The flow chart then takes us back round the loop of finding and testing the value of p
n does not get updated so only appears in the first row of the table

n k p Is p greater than n? Is p equals n?
23 1 1 NO NO
  2 4 NO NO
         


The table can be completed by continuing to precisely follow the flow chart
The output needs stating at the end as per the flow chart instructions
Note how the question "is p equals n?" does not get considered as the algorithm outputs and stops following the answer 'yes' to the question "is p greater than n?"

n k p Is p greater than n? Is p equals n?
23 1 1 NO NO
  2 4 NO NO
  3 9 NO NO
  4 16 NO NO
  5 25 YES  
         
         
         
Output: 23 is not a square number

Text-Based Algorithms

What are text-based algorithms?

  • Text-based algorithms refer to instructions that are given in sentences
    • Each instruction may be labelled with 1, 2, 3, and so on
  • The following text-based algorithm is for a very basic single player dice game
    • The aim of the game is to minimise the number of rolls of the dice needed to reach 'Stop'

QfVompti_text-alg-dice

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.