Discrete Mathematics

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Discrete Mathematics

• Algorithms

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Introduction

• Discrete maths has developed relatively recently.
  Its importance and application have arisen along
  with the development of computing.
• Discrete maths deals with discrete rather than
  continuous data and does not employ the
  continuous methods of calculus.
• Computers deal with procedures or
• algorithms to solve problems and
• algorithms form a substantial part
• of discrete maths.

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Algorithms

• An algorithm is a procedure or set of
  instructions used to solve a problem.
• A computer programmes are algorithms written in a
  language which computers can interpret.
• The algorithm enables a person (or computer) to
  solve the problem without understanding the whole
  process

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Sorting algorithms

• A frequently needed operation on a computer is a
  sort
• Such as sorting data into numerical order

e.g. Sort the following list into numerical
order starting with the smallest. (time how long
it takes you) 3, 9, 10, 24, 2, 7, 1, 56,
43, 29, 36, 17, 4, 12, 77, 21, 100.
A computer will do this in much less than a
second. How do you know you have not made a
mistake? How would you cope with a list of 100
numbers or more?
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Sorting algorithms

• Bubble sort.
• This algorithm depends on successive
  comparisons of pairs of numbers.
• Compare the 1st and 2nd numbers in a list and
  swap them if the 2nd number is smaller than the
  1st.
• Compare the 2nd and 3rd numbers and swap them if
  the 3rd is smaller.
• Continue in this way to the end of the list.
• This procedure is called a pass

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Bubble sort

• Example
• Use a bubble sort to place the numbers in the
  list in order.
• 5, 1, 2, 6, 9, 4, 3.

This represents one pass and required 6
comparisons and 4 swaps
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Bubble sort

• The complete algorithm is written as
• Step 1 If there is only one number in the
  list then stop.
• Step 2 Make one pass down the list comparing
  numbers in pairs and swapping if
  necessary.
• Step 3 If no swaps have occurred then stop.
  Otherwise ignore the last element in
  the list and return to step 1.

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Example

• The table shows the complete bubble sort
  carried out on the list given on the previous
  slide.

Original list 1st pass 2nd pass 3rd pass 4th pass 5th pass
5 1 1 1 1 1
1 2 2 2 2 2
2 5 5 4 3 3
6 6 4 3 4 4
9 4 3 5 5 5
4 3 6 6 6 6
3 9 9 9 9 9
(The numbers in purple are the ones which are
ignored.)
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Shuttle Sort

• One disadvantage of the bubble sort is that
  you have to do a final pass after the list is
  sorted to ensure the sort is complete.
• The shuttle sort partially overcomes this
  problem.

Ist pass Compare the 1st and 2nd numbers in
the list and swap if necessary. 2nd pass
Compare the 2nd and 3rd numbers in the list
and swap if necessary. If a swap has
occurred, compare the 1st and 2nd numbers
and swap if necessary. 3rd pass Compare the
3rd and 4th numbers in the list and swap if
necessary. If a swap has occurred, compare the
2nd and 3rd numbers, and so on up the list.
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Shuttle sort

• Using the same list of numbers the table
  below shows a shuttle sort

Original list 1st pass 2nd pass 3rd pass 4th pass 5th pass 6th pass
5 1 1 1 1 1 1
1 5 2 2 2 2 2
2 2 5 5 5 4 3
6 6 6 6 6 5 4
9 9 9 9 9 6 5
4 4 4 4 4 9 6
3 3 3 3 3 3 9
The shuttle sort has involved the same number of
swaps as the bubble sort but 14 comparisons
instead of 20.
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Practice Questions

• Exercise 1A page 5

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The order of an algorithm

• The efficiency of an algorithm is a measure of
  the run-time for the algorithm. This will
  often be proportional to the number of operations
  which have to be carried out.
• The size of the problem is a measure of its
  complexity. E.g. in a sorting algorithm it is
  likely to be related to the number of numbers in
  the list
• The order of an algorithm is a measure of the
  efficiency of the algorithm as a function of the
  size of the problem.
• Examples of different orders of algorithms

Algorithm Size Efficiency Order
A n 5n n or linear
B n n2 7n n2 or quadratic
C n 2n3 3n n3 or cubic
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Packing algorithms

• In business and industry, efficient packing to
  make best use of space is important and, for
  example, computerised systems are used to
  organise storage.

First-Fit Algorithm Place each object in turn in
the first available space in which it will fit.
This is the simplest algorithm but rarely lead to
the most efficient solution
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Examples

• Question
• A small ferry has 3 lanes, each 25 metres long.
  The lengths of the vehicles in the queue, in the
  order in which they are waiting are
• 3 5 4 3 14 5 9 3 4 4 4
  3 11
• Using the first-fit algorithm

The final 11 m vehicle does not fit. 16 m of
space is unused. The solution could be improved
by putting the 9 m vehicle (no.7) in lane 3 and
then the 11m vehicle (no13) will fit in lane 2.
5 m of space is unused.
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Increasing efficiency

• First-Fit Decreasing Algorithm
• Order all objects in decreasing size and then
  apply the first-fit algorithm

Using the First-Fit Decreasing Algorithm First
place the vehicles in order of decreasing
size. 14 11 9 5 5 4 4
4 4 3 3 3 3 Then apply the First-Fit
Algorithm
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Example

•    
•    
•  

This is more efficient than the first-fit
algorithm and accommodates all vehicles and
leaving 3 m space.
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Flow diagrams

• A flow diagram is pictorial representation of an
  algorithm.
• The shape of the box indicates the type of
  instruction.

Oval boxes are used for starting and stopping,
inputting and outputting data
Rectangles are used for calculations or
instructions.
Diamond shapes are used for questions and
decisions
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Notation

• means take the number in
  pigeon hole n and put it in pigeon hole m.
  The notation for this is m n
• Similarly
• m 2 means put the number 2 in pigeon hole
  m.
• m m 1 means take the number already in
  pigeon hole m, subtract 1 and put the result back
  into pigeon hole m.
• Pigeon holes are usually called stores.

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Example

• An algorithm has a flow diagram below.

1. What is the output if N 57?
2. What has the algorithm been designed to do?

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Solution

1. After 6 successive passes around the flow
   diagram, the values of N and R are as follows.

Pass N R Written down
1 28 1 1
2 14 0 01
3 7 0 001
4 3 1 1001
5 1 1 11001
6 0 1 111001
The algorithm converts N into a binary number.
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Practice questions

• Cambridge Advanced Level Maths
• Discrete Mathematics 1
• Chapter 1
• Exercise 1 A
• Exercise 1 B
• Miscellaneous Exercise 1