General Algorithms (Edexcel IAL Maths)

Introduction to Algorithms

Spec point — 1.1

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

Exam Hint

• 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

Spec point — 1.1

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 $a>b$?
    • 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

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

Exam Hint

• 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.

a) Describe what the algorithm achieves.

Answer:

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

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

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

$n$	$k$	$p$	Is $p=n$?	Is $p>n$?
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 $k=1$ and to let $p=k^{2}$, so also in the first row we can fill in 1 and 1 for $k$ and $p$* *

$n$	$k$	$p$	Is $p=n$?	Is $p>n$?
23	1	1

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

$n$	$k$	$p$	Is $p=n$?	Is $p>n$?
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 $p=n$?	Is $p>n$?
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 In the $k=5$ row "is $p=n$?" still has a 'no' answer, but the 'yes' answer to the question "is $p>n$?" means that the algorithm gives its output and stops

$n$	$k$	$p$	Is $p=n$?	Is $p>n$?
23	1	1	NO	NO
	2	4	NO	NO
	3	9	NO	NO
	4	16	NO	NO
	5	25	NO	YES
Output:	Final answer: 23 is not a square number

Text-Based Algorithms

Spec point — 1.1

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'