Decision Maths

1
Decision Maths

• Lesson 10 Algorithms

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What is an algorithm?

• An algorithm is sequence of instructions that
  will solve a particular problem.
• There can be no ambiguity in the instructions.
• There must be a clause that can terminate the
  algorithm.
• Algorithms can be shown in a number of ways
• A list of instructions
• Pseudo code (computer language)
• Flow charts
• You will have been using Algorithms for years
  without even realising it. Addition, Subtraction,
  Long multiplication and division.

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Zellers Algorithm

• Do you know which day you were born on?
• Mondays child is fair of face,
• Tuesdays child is full of grace,
• Wednesdays child is full of woe,
• Thursdays child has far to go,
• Fridays child is loving and giving,
• Saturdays child works hard for a living,
• And a child that is born on the Sabbath day,
• Is fair and wise and good and gay.

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Zellers Algorithm

• Let day number D
• Month number M
• And year Y
• If M is 1 or 2 replace M with M add 12 and
  replace Y with Y subtract 1.
• Let C be the first two digits of Y and Y be the
  last to digits of Y.
• Add together the integer parts of (2.6M - 5.39),
  (Y/4), (C/4).
• Now add on D and Y and subtract 2C.
• Find the remainder when the answer is divided by
  7.
• 0 Sunday
• 1 Monday
• etc

• D 06
• M 02
• Y 1977
• D 06
• M 14
• Y 1976
• C 19 Y 76
• (2.6 x 14 5.39 31)
• 76/4 19
• 19/4 4
• 31 19 4 54
• 54 6 76 2 x 19 98
• 98/7 14 r 0
• Sunday
• Is fair and wise and good and gay.

Now try and see what day you were born on.
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Russian Peasants Algorithm

• Step 1 Write down the two numbers to be
  multiplied.
• Step 2 Beneath the left number write down
  double that number. Beneath the right number
  write down half that number
• Step 3 Repeat step 2 until the right number is
  1.
• Step 4 Delete those rows where the number in
  the right column is even.
• Step 5 Add up the remaining numbers in the left
  column.
• So 24 x 163 3912

• 24 163
• 48 81
• 96 40
• 192 20
• 384 10
• 768 5
• 1536 2
• 3072 1
• 3912

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Russian Peasants Algorithm

• The order of the sum is not important.

163 24 326 12 652 6 1304 3 2608 1
3912
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Russian Peasants Algorithm

• Now try these ones
• 76 x 53
• 76 53
• 152 26
• 304 13
• 608 6
• 1216 3
• 2432 1
• 4028

• 28 x 405
• 28 405
• 56 202
• 112 101
• 224 50
• 448 25
• 896 12
• 1792 6
• 3584 3
• 7168 1
• 11340

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Flow Charts

• A diamond indicates a question or choice to be
  made.
• A rectangle is an instruction box, it has one
  entrance and one exit.
• A circle (oval) indicates a start or stop.

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Flow diagrams

• Below is an algorithm expressed in psuedo code.
• Step 1 - Let M b2 4ac.
• Step 2 - If M lt 0 print no real solutions and
  go to step 5.
• Step 3 - Let x1 (-b vM)/(2a)
• Let x2 (-b vM)/(2a)
• Step 4 - Print x1 and x2.
• Step 5 - Stop
• What does this algorithm solve?
• Can you design a flow diagram to solve quadratic
  equations?

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Solving Quadratic Equations
Start
Read a, b, c
Let M b2 4ac
Yes
Is M lt 0
No
Let x1 (-b vM)/(2a)
Print no real Solutions
Let x2 (-b vM)/(2a)
Print x1, x2
Stop
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Euclids Method for finding HCF

• Pick two numbers and put them through this
  algorithm.
• What does the algorithm solve?

Read x, y
Subtract y from x to get new value of x
x gt y ?
Yes
No
Subtract x from y to get new value of y
x lt y ?
Yes
No
x y ?
No
Yes
Output x
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Homework

• Can you draw a flow chart that would represent an
  algorithm for making a cup of tea for your
  teacher?

Start
Boil the kettle
Place tea bag in teachers favourite mug
Fill mug to within 1 inch of the top
Add sachet of milk
Stir 4 times
Stop
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Tea Algorithm

• The example we have just looked at was a very
  simple (boring) algorithm.
• It was specific to one person and would apply to
  anyone else.
• Try to make your algorithm complex.
• Allow for more freedom and make a general
  algorithm that would suit all.
• Remember, there can be no ambiguity.

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Complexity

• There is often more than one way of solving a
  problem, and this can result in different
  algorithms.
• It is obvious that some algorithms will be more
  efficient than others.
• The more efficient, the fewer operations used to
  solve the problem.
• This can be useful in computing, as more
  efficient programs will use up less storage.
• Take the example of the Russian peasants
  algorithm (28 x 405) the work involved is more
  efficient is you switch the sum around to 405 x
  28.

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Complexity

• To understand complexity we are going to look at
  two examples that involve quadratic equations.
• Lets first consider the quadratic equation.
• In the algorithm we looked at earlier we
  calculated
• M b2 4ac once and then calculated
• x1 (-b vM)/(2a), and
• x2 (-b vM)/(2a)
• If you calculate the answers from the original
  equation then you can see that b2 4ac will be
  evaluated twice and hence is less efficient.

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Complexity

• Lets now consider the evaluation of the quadratic
  expression f(a) 3a2 2a 1
• f(4) 3 x 4 x 4 2 x 4 1 57
• There are 3 multiplications and 2 additions.
• We could however re-write f(a) in the form
• f(a) (3a 2)a 1
• Now f(4) (3 x 4 2) x 4 1 57
• Here there are 2 multiplications and 2 additions.

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Complexity

• Now let us extend this further to cubic
  polynomials.
• f(a) 4a3 3a2 2a 1
• f(4) 4 x 4 x 4 x 4 3 x 4 x 4 2 x 4 1
  313
• This has 6 multiplications and 3 additions
• Again we can re-arrange f(a)
• f(a) 4a3 3a2 2a 1
• (4a2 3a1 2)a 1
• ((4a 3)a 2)a 1
• Now ((4 x 4 3) x 4 2) x 4 1 313
• This has 3 multiplications and 3 additions.
• Hopefully you can see that the second way is more
  efficient.

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Complexity

• Lets generalise this for any polynomial
• anan an-1an-1 a1a a0
• We can use our previous results to help, and make
  a prediction for a polynomial with order 4.
• You can see from the table that the nested method
  for evaluating the quadratic is the most
  efficient.

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Complexity

• Because there is an n2 term in the number of
  multiplications required using the first
  approach, the complexity of that algorithm is
  quadratic.
• Note To appreciate the effect of this think
  about a polynomial with order 100.
• Using the straight forward method you would need
  to do (100/2)(100 1) 5050 multiplications and
  100 additions.
• Using nesting you would only need to do 100
  multiplications and 100 additions.