Lesson Objective: Understand what an algorithm is and be able to use them to solve a simple problem

Lesson Objective:Understand what an algorithm is and be able to use them to solve a simple problem

Monday’s child is fair of faceTuesday’s child is full of grace

Wednesday’s child is full of woeThursday’s child has far to go

Friday’s child is loving and givingSaturday’s child works hard for its living

And a child that is born on the Sabbath dayIs fair and wise and good and gay

If you don’t know what day of the week you were born on you can work it out using Zeller’s algorithm

Zeller’s algorithmLet day number = DMonth number = MYear number = Y

If M is 1 or 2 add 12 to M and subtract 1 from Y

Let C be the first two digits of YAnd Z be the last two digits of Y

Add together the integer parts of (2.6M – 5.39), (Z÷4) and (C÷4), thenadd on D and Z and subtract 2C

Find the remainder when this value is divided by 7.

If the remainder is 0 the day was SundayIf the remainder is 1 the day was Monday etc.

Eg 15th May 1991D =15 M = 5 Y = 1991

C = 19Z = 91

7 + 22 + 4 + 15 + 91 – 38 = 101

3

So Wednesday

Russian peasant’s algorithm for long multiplication

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 of that number ignoring any remainder.

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. Add up the remaining numbers in the left column. This is the answer you need.

24 16348 8196 40192 20384 10768 51536 23072 13912

Euclid’s algorithm for finding the HCF of two integers

Step 1: Pick two numbers

Step 2: Find the difference between the two numbers to get a third number.

Step 3: Keep the two smallest numbers of the threeand cross out the third number

Step 4: If the two numbers you are left with are the same then this number is the HCF of the original two numbers. If not then go back to Step 2.

Ways to communicate algorithms:

1) As a series of instructions expressed in words

2) As a Flow Chart

3) As a computer program – using ‘Pseudo Code’

Euclid’s algorithm for finding the HCF of two integers in words

Step 1: Pick two numbers

Step 2: Find the difference between the two numbers to get a third number.

Step 3: Keep the two smallest numbers of the three and cross out the third number

Step 4: If the two remaining numbers are equal then this number is the HCF of the original two numbers. If not go back to Step 2.

Euclid’s algorithm for finding the HCF of two integers in a flow chart

Read two numbers x,y

x>y?Subtract y from x to get a new

value for x

x<y?

x=y?

Subtract x fromy to get a new

value for y

Output x

Euclid’s algorithm for finding the HCF of two integers in ‘Pseudo Code’

Input xInput yRepeatIf x>y then x = x-yIf y>x then y = y-xUntil x=yPrint x

yes

yes

no

no

yes

no

Page 26 q.3, Page 27 q.5, Page 29 q 11, Page 41 q.19

Algorithmic Complexity

There are many algorithms that can be used to solve some problems.

The effectiveness of an algorithm can be judged in two ways: Its success rate and its efficiency

Ways to judge success

Some algorithms always find the best solution (these are the best kind of algorithms)

Some algorithms find a solution, but not always the best (these are called Heuristic algorithms)

Ways to judge efficiency

The efficiency of an algorithm is a measure of its speed.

This usually depends on the number of calculations required to complete the task

A more efficient algorithm is one that completes the task using fewer calculations

Measuring the efficiency of an algorithm

Consider calculating the value of a polynomial for some value of x

For a linear:

You will be required to do a*x + b 1 multiplications and 1 addition

For a quadratic:

You will be required to do a*x*x + b*x + c 3 multiplications and 2 additions

For a cubic:

You will be required to do a*x*x*x + b*x*x + c*x + d 6 multiplications and 3 additions

For a quartic:

You will be required to do a*x*x*x*x + b*x*x*x + c*x*x + d*x + e

10 multiplications and 4 additions

In general you need to do 1/2n(n+1) mults and n additions. In total: 1/2n2 + 3/2n calculations.

We say the problem has quadratic complexity. The time to solve it is proportional to the size2.

If you double the ‘size’ of the problem it will take four times longer to solve.

If you triple the ‘size’ it will take nine times longer. etc

Measuring the efficiency of an algorithm

If you rewrite each polynomial in ‘nested’ form like this to begin with: (((ax + b)x + c)x + d)x + e

Then for a linear a*x + b 1 multiplication and 1 addition

For a quadratic: (a*x + b)*x + c 2 multiplications and 2 additions

For a cubic: ((a*x + b)*x + c)*x + d 3 multiplications and 3 additions

For a quartic: (((a*x + b)*x + c)*x + d)*x + e 4 multiplications and 4 additions

In general you need n multiplications and n additions, a total of 2n calculations.

The problem is said to have linear complexity and the time to solve is proportional to the size.

Doubling the ‘size’ of the problem doubles the time to solve.

Tripling the ‘size’ of the problem triples the time to solve etc.

Efficiency Problems

1) A sorting algorithm has quadratic efficiency. It takes 3 seconds to sort a list of 4000 items.

How long will it take to sort a list of a) 8000 items?

b) 13000 items?

2) An algorithm has cubic efficiency. It takes 10 seconds to solve a problem of ‘size’ 12. How long will it take to solve a problem of ‘size’ 30?

3) An algorithm takes 5 minutes to solve a problem of ‘size’ 30.

How much longer would it take to solve a problem of size 40 if the algorithm has cubic efficiency rather than quadratic efficiency?

Bin Packing algorithms

Suppose we have 11 vehicles to load onto a ferry with 5 lanes, each 15m long.

The vehicles are labelled A to K and have different lengths as denoted.

A 8m

B 7m

C 4m

D 9m

E 6m

F 9m

G 5m

H 5m

I 6m

J 7m

K 8m

Can we fit all the vehicles onto the ferry?

Algorithms for solving this problemFirst Fit algorithm: Take the first item and place it in the first available ‘bin’ working from

one end. Repeat until all the bins are full or until all items are packed.

First Fit Decreasing algorithm: As above but sort the items into decreasing order

of size first

Full bins algorithm: Look for combinations of items that will make ‘full bins’ first and fill

these bins, then use the First Fit algorithm to pack any remaining items

1 23

6

2 35

3

A B C D E F

4

Each block will be fitted into the first bin that has room for it.Each block will be fitted into the first bin that has room for it.

Bin packing – First fit algorithm

6 units

A B C D E F

4

The first block fits into the first binThe first block fits into the first bin

1 23

6

2 35

3


6 units

A B C D E F

4

The second block fits in the first binThe second block fits in the first bin

1

23

6

2 35

3


6 units

A B C D E F

4

The third block will not fit in the first binThe third block will not fit in the first bin

1

23

6

2

35

3


6 units

A B C D E F

4

1

23

6

35

3

2

But there is room in the second binBut there is room in the second bin


6 units


The third bin is the first one the 5 will fit inThe third bin is the first one the 5 will fit in

A B C D E F

4

1

2

6

3

5

3

2

3

5

56 units

The second bin has room for the 3The second bin has room for the 3

A B C D E F

4

1

2

6

33

2

35

3


6 units

The fourth bin is the first one with room for the next one

The fourth bin is the first one with room for the next one

A B C D E F

4

1

2

6

33

2

35

2 2

2


6 units

The next one also fits in the fourth binThe next one also fits in the fourth bin

A B C D E F

4

1

6

3

2

35

2

3 3 3

3


6 units


No room until the fifth bin for the 6No room until the fifth bin for the 6

A B C D E F

4

1

6

3

2

35

2

3

6 6 6

66 units

The 3 has to start a new binThe 3 has to start a new bin

A B C D E F

4

1

2

35

2

3 6

3 3 3 33

3

Total usage is 6 bins.Total usage is 6 bins.


6 units

1 23

6

2 35

3

A B C D E F

4

With the first fit decreasing algorithm we sort the blocks into descending order first.


Bin packing – First fit decreasing algorithm

6 units



233345

1

6

2

A B C D E F


6 units

23312

A B C D E F

Now we use the first fit algorithmNow we use the first fit algorithm

5 4 3

6


6 units


2331

6

2

A B C D E F


5

4 3

5

6 units

2331

6

2

A B C D E F


5

4

4

3

4


6 units

2331

6

2

A B C D E F


5 4

33

3

3


6 units

231

6

2

A B C D E F


5 4 3

33

3

3


6 units

2 1

6

2

A B C D E F


5 4 3

3

33

3

3

3


6 units

1

6

2

A B C D E F


5 4 3

3

3

22

2


6 units

1

6

A B C D E F


5 4 3

3

3

2

22

2 2

2


6 units

6

A B C D E F


5 4 3

3

3

22

1

We have packed them into 5 bins.We have packed them into 5 bins.

1


6 units

Sorting Algorithms

Algorithms for Sorting Items

1) Interchange sort algorithm

2) Bubble sort algorithm

3) Shuttle sort algorithm

4) Quick sort algorithm

The Interchange Sort Algorithm

• Step 1: Search down the list to find the smallest number and interchange it with the first number on the list

• Step 2: Starting with the 2nd number on the list, search down the list to find the smallest number left and interchange it with the 2nd number on the list.

• Step 3: Repeat Step 2 starting with the 3rd, then 4th number on the list until the bottom of the list is reached.

OriginalList

5126943

After 1Pass

1526943

After 2Passes

1256943

After 3Passes

1236945

After 4Passes

1234965

After 5Passes

1234569

After 6Passes

5126943

The Bubble Sort Algorithm

• 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 them as necessary.

• Step 3: If no swaps have occurred then stop. Otherwise, ignore the last element of the list and return to Step 1.

Example

OriginalList5126943

Example

OriginalList5126943

“Compare numbers in pairs”

Example

OriginalList5 11 526943

“swapping them as necessary”

Example

OriginalList5 11 526943

Making a pass down the list

Example

OriginalList5 11 22 56943

Example

OriginalList5 11 22 56943

Example

OriginalList5 11 22 56 6943

Example

OriginalList5 11 22 56 6943

Example

OriginalList5 11 22 56 69 943

Example

OriginalList5 11 22 56 69 943

Example

OriginalList5 11 22 56 69 44 93

Example

OriginalList5 11 22 56 69 44 93

Example

OriginalList5 11 22 56 69 44 33 9

Example

OriginalList

1st pass

5 11 22 56 69 44 33 9

“ignore the last element”

Example

OriginalList

1st pass

5 11 22 56 69 44 33 9

“return to Step 2”

Example

OriginalList

1st pass

5 11 22 56 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 56 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 56 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 6 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 6 69 44 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 6 49 4 64 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 6 49 4 64 33 9 9

Example

OriginalList

1st pass

5 1 11 2 22 5 56 6 49 4 34 3 63 9 9

Example

OriginalList

1st pass 2nd pass

5 1 11 2 22 5 56 6 49 4 34 3 63 9 9

Example

OriginalList

1st pass 2nd pass

5 1 11 2 22 5 56 6 49 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 56 6 49 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 56 6 49 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 5 56 6 49 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 5 56 6 49 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 5 46 6 4 59 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 5 46 6 4 59 4 34 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass

5 1 1 11 2 2 22 5 5 46 6 4 39 4 3 54 3 6 63 9 9 9

Example

OriginalList

1st pass 2nd pass 3rd pass

5 1 1 11 2 2 22 5 5 46 6 4 39 4 3 54 3 6 63 9 9 9

Example

OriginalList


5 1 1 11 2 2 22 5 5 46 6 4 39 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 46 6 4 39 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 46 6 4 39 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 4 46 6 4 39 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 4 46 6 4 39 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 4 36 6 4 3 49 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList

1st pass 2nd pass 3rd pass 4th pass

5 1 1 1 11 2 2 2 22 5 5 4 36 6 4 3 49 4 3 5 54 3 6 6 63 9 9 9 9

Example

OriginalList


5 1 1 1 11 2 2 2 22 5 5 4 36 6 4 3 4 49 4 3 5 5 54 3 6 6 6 63 9 9 9 9 9

Example

OriginalList


5 1 1 1 1 11 2 2 2 2 22 5 5 4 36 6 4 3 4 49 4 3 5 5 54 3 6 6 6 63 9 9 9 9 9

Example

OriginalList


5 1 1 1 1 11 2 2 2 2 22 5 5 4 36 6 4 3 4 49 4 3 5 5 54 3 6 6 6 63 9 9 9 9 9

Example

OriginalList


5 1 1 1 1 11 2 2 2 2 22 5 5 4 3 36 6 4 3 4 49 4 3 5 5 54 3 6 6 6 63 9 9 9 9 9

Example

OriginalList

1st pass 2nd pass 3rd pass 4th pass 5th pass

5 1 1 1 1 11 2 2 2 2 22 5 5 4 3 36 6 4 3 4 49 4 3 5 5 54 3 6 6 6 63 9 9 9 9 9

“If no swaps have occurred then stop.”

The Shuttle Sort

Algorithm

The Shuttle Sort Algorithm• 1st 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.

• And so on, through the entire list.

Example

OriginalList

5126943

Example

OriginalList

1st pass

512 26 69 94 43 3

“1st pass”

Example

OriginalList

1st pass

512 26 69 94 43 3

“compare the 1st and 2nd numbers”

Example

OriginalList

1st pass

5 11 52 26 69 94 43 3

“swap if necessary”

Example

OriginalList

1st pass 2nd pass

5 11 52 26 6 69 9 94 4 43 3 3

“2nd pass”

Example

OriginalList

1st pass 2nd pass

5 11 52 26 6 69 9 94 4 43 3 3

“compare the 2nd and 3rd numbers”

Example

OriginalList

1st pass 2nd pass

5 11 5 22 2 56 6 69 9 94 4 43 3 3


Example

OriginalList

1st pass 2nd pass

5 11 5 22 2 56 6 69 9 94 4 43 3 3

“if a swap has occurred, compare the 1st and 2nd numbers”

Example

OriginalList

1st pass 2nd pass

5 1 11 5 22 2 56 6 69 9 94 4 43 3 3


Example

OriginalList


5 1 11 5 22 2 56 6 69 9 9 94 4 4 43 3 3 3

“3rd pass”

Example

OriginalList


5 1 11 5 22 2 56 6 69 9 9 94 4 4 43 3 3 3

“compare 3rd and 4th numbers”

Example

OriginalList


5 1 11 5 22 2 5 56 6 6 69 9 9 94 4 4 43 3 3 3


Example

OriginalList


5 1 1 11 5 2 22 2 5 56 6 6 69 9 9 94 4 4 43 3 3 3

“if a swap has occurred….”

Example

OriginalList


5 1 1 11 5 2 22 2 5 56 6 6 69 9 9 94 4 4 4 43 3 3 3 3

Example

OriginalList


5 1 1 11 5 2 22 2 5 56 6 6 69 9 9 94 4 4 4 43 3 3 3 3

Example

OriginalList


5 1 1 11 5 2 22 2 5 56 6 6 6 69 9 9 9 94 4 4 4 43 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 69 9 9 9 94 4 4 4 43 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 69 9 9 9 94 4 4 4 43 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 69 9 9 9 94 4 4 4 43 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 69 9 9 9 9 44 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 69 9 9 9 9 44 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 6 49 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 56 6 6 6 6 49 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList

1st pass 2nd pass 3rd pass 4th pass 5th pass 6th pass

5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 93 3 3 3 3 3

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 9 33 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 64 4 4 4 4 9 33 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 6 34 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 59 9 9 9 9 6 34 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 5 39 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 46 6 6 6 6 5 39 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 4 36 6 6 6 6 5 49 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 22 2 5 5 5 4 36 6 6 6 6 5 49 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 11 5 2 2 2 2 22 2 5 5 5 4 36 6 6 6 6 5 49 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 1 11 5 2 2 2 2 22 2 5 5 5 4 36 6 6 6 6 5 49 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

OriginalList


5 1 1 1 1 1 11 5 2 2 2 2 22 2 5 5 5 4 36 6 6 6 6 5 49 9 9 9 9 6 54 4 4 4 4 9 63 3 3 3 3 3 9

Example

• The example required 14 comparisons and 9 swaps to take place.

• The Bubble Sort algorithm requires 20 comparisons and 9 swaps for the same list.

• This makes the Shuttle Sort the more efficient algorithm.

The Quick Sort Algorithm• Choose the first number in the list as a pivot and

ring it.• Pass through the list writing those numbers lower

than or equal to the pivot before it in the list and those numbers larger than the pivot after it. This will split the list into two sub-lists with the pivot as a divide.

• Repeat the algorithm on each sub-list, choosing the first number in each sub-list as the pivot for that sub-list.

• Keep splitting the sub-lists in this way until no sub-list has more than one number in it

Example

OriginalList

5129643

Example

Ring the pivot

OriginalList

5129643

ExamplePass down the list, placing

the numbers on the correct side of the pivot

OriginalList

5129643

First Pass

1243596

Example

Ring the pivots in the sub-lists

OriginalList

5129643

FirstPass

1243596

Example

OriginalList

5129643

FirstPass

1243596

Second Pass

1243569

Pass down the list, placing the numbers on the

correct side of each pivot

Example

OriginalList

5129643

FirstPass

1243596

SecondPass

1243569

Ring the pivots in the new sub-

lists

Example

OriginalList

5129643

FirstPass

1243596

SecondPass

1243569



ThirdPass

1243569

Example

OriginalList

5129643

FirstPass

1243596

SecondPass

1243569

ThirdPass

1243569

Ring the pivots in the sub-lists

Example

OriginalList

5129643

FirstPass

1243596

SecondPass

1243569

ThirdPass

1243569

FourthPass

1234569



Example

OriginalList

5129643

FirstPass

1243596

SecondPass

1243569

ThirdPass

1243569

FourthPass

1234569

No sub-lists of more than one so stop

Algorithms for ‘searching’ for an item on a list

The Binary Search Algorithm Use the Binary Search pps

Documents

Lesson Objective: Understand what an algorithm is and be able to use them to solve a simple problem