A courier
with one
urgent
delivery
to make
A satellite Sainsburys
navigation Home
Delivery
system
A gritting lorry
gritting all
the roads
in a town
Cable TV
Ambulance company
travelling linking towns
using as little
to an
emergency cable as
possible
Council workers Road builder
joining a few
re − painting
the lines in the
villages as
middle of the
economically
roads
as possible
A milk
tanker
from a dairy
collecting
milk
from farms
Courier
with
several
deliveries
to make
A parking
A person
official
planning
patrolling
the route
to a holiday all the
resort in UK streets in
an area
Highways
Authority
inspecting
roads for
fallen trees
after a storm
Pedestrian
precinct
being created
to connect
places of
interest in a
town centre
Water pipelines
being laid to
connect
pumping
stations as
economically
as possible
A family on
a shopping
trip visit
several shops
then return to
their car
Deciding on
the best site
for a doctor's
surgery serving
four villages
A network
Finding the
connecting stitching a
minimum cost
several
logo on a
of a tour of
several towns computers track suit
in a building
Sheet A A
B
C
D
E
F
G
H
I
J
K
L
M
N
Cook Sausages
Cook Bacon
Chop Mushrooms
Cook Mushrooms
Cook Tomatoes
Fry eggs
Make Toast
Butter Toast
Clear Table
Boil Kettle
Make Tea
Set Table
Sheet B Activity
A Cook Sausages
B Cook Bacon
Chop
C
Mushrooms
Cook
D
Mushrooms
E Cook Tomatoes
F Fry eggs
G Make Toast
H Butter Toast
I
Clear Table
J
Boil Kettle
K Make Tea
L Set Table
M
N
Duration
(Mins)
Immediate
Predecessor
Sheet C Start Preparation
Finish
Preparation
o
hn
wit ges
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A w eat ed
rep
h
p
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repeated vertices
h
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Simple Graph
alu
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A subset of the vertices
and edges of a larger graph
nG
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No
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Line from vertex to vertex
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Point where edges meet
ree
T
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n
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edges crossing
Bipartite Graph
Mo
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t he t han
sam one
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of betw
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tic een
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...
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Complete Graph
Gr
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mo t jo t s
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Path
Eulerian Cycle
x
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Trail
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te
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Odd node
Arc
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hw
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Gr
ho
it h
dd
no
deg
ree
... connects all vertices with
the least total edge length
wit
Cy
cle
e
e
r
T
Weight
Complete graph
with five vertices
No
de
N
e
od
Co
on nnec
e p ted
air
G
of rap
od
d v h wit
ert h
ice
s
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cyc
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rap ect io
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a
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Dif sam
t he
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Eu
aph
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nce ertex
Isomorphic Graphs
Gr
of aph c
arc on
s a sist
nd
i
nod ng
es
V
a
le
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c
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Planar Graph
h
pat
sed
clo
A sequence of
consecutive edges
h
p
a
gr
b
Su
... a
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Ed
De
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Point where edges meet
Va
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Connected Graph
f
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4
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... where every vertex is
joined to every other vertex
Hamiltonian Cycle
Ed
ge
A furniture manufacturer makes square
dining tables, round dining tables and chairs.
Each square table needs £40 of raw materials
and takes 12 hours to make.Each round table
needs £50 of raw materials and takes 14 hours to
make. Each chair needs £40 of raw materials
and takes 16 hours to make. He must make at
least 4 times as many chairs as tables.
There are 500 hours available and £1500 of
raw materials.Each chair makes £40 profit each
square table makes £30 profit and each round
table makes £35 profit
Jo is making raspberry and chocolate cakes
for a charity fair
A raspberry cakes needs 200g of flour and
200g of sugar and 2 eggs
A chocolate cake needs 225g of flour and
150g of sugar and 2 eggs
There are 3Kg of flour, 2.5Kg of sugar and 28
eggs available.
A raspberry cake sells for £2.50 and a chocolate
cake sells for £3.00
A company produces three soft toys, an antelope,
a bear and a cat. For one day's production run it
has available 11 m 2 of fur fabric, 24 m 2 of wool
fabric and 30 glass eyes.
The antelope requires 0.5 m 2 of fur fabric, 2 m 2 of
wool fabric and two eyes. Each sells at a profit of £3.
The bear requires 1 m 2 of fur fabric, 1.5 m 2 of
wool fabric and two eyes. Each sells at a profit of £5.
The cat requires 1 m 2 of fur fabric, 1 m 2 of wool
fabric and two eyes. Each sells at a profit of £2.
Brad makes and sells three types of birdfood
A, B and C.
A contains 4 kg of bird seed, 2 suet blocks and
1 kg of nuts, B contains 5 kg of bird seed, 1 suet
block and 2 kg of nuts and C contains 10 kg of
bird seed, 4 suet blocks and 3 kg of nuts.
Each week Brad has 140 kg of bird seed, 60 suet
blocks and 60 kg of nuts available for the packs.
The profit made on each pack of A, B and C is
£3.50, £3.50 and £6.50 respectively.
A small factory makes 2 types of inflatable boats,
a 2 person and 4 person. Each 2 person boat
requires 0.9 hours in the cutting department and
0.8 hours in the assembly department. Each four
person boat requires 1.8 hours in the cutting
department and 0.5 hours in the assembly department.
The company makes profit of £25 on each 2 person
boat and £40 on each 4 person boat.The maximum
hours available each month in the cutting department
is 864 and the maximum hours available each month
in the assembly department is 672.
Anna is making birthday cards in 2 designs,
flowers and boats, to sell.She has enough card
to make 16 birthday cards but she needs to get
them all made in the next 6 hours. Flower cards
take 30 mins to make and sell for £1.25. Boat
cards take 20 mins to make and sell for £1.40
A manufacturer makes three products X, Y and Z.
They all require resources A, B, C and D which are
in short supply. The table shows the amount of each
resource needed and the profit on each product
A B C D profit
X
20 0 20 40
6
Y
50 20 40 30
4
Z
40 10 20 20 10
available 600 100 700 1800
Fine Foods Ltd buys vegetable oil from two
sources A and B.The oil from A contains
50% olive oil 10% sunflower oil and 40% corn
oil while the oil from B contains 20% olive oil,
60% sunflower oil and 20% corn oil.Fine foods
must make a blend with at least 30% olive oil
and at least 40% sunflower oil.They know that
they can sell up to 35,000 litres of the blended
oil at a profit of 25p per litre.
maximise P = 35x + 30y + 40z
subject to
4x + 10y + 4z ≤ 150
6x + 7y + 8z ≤ 250
x + y − 4z ≤ 0
maximise P = 2.5x + 3y
subject to
8x + 9y ≤ 120
4x + 3y ≤ 50
3x + 2y ≤ 28
maximise P = 3x + 5y + 2z
subject to
x + 2y + 2z ≤ 22
4x + 3y + 2z ≤ 48
x + y + z ≤ 15
maximise P = 3.5x + 3.5y + 6.5z
subject to
4x + 5y + 10z ≤ 140
2x + y + 4z ≤ 60
x + 2y + 3z ≤ 60
maximise P = 25x + 40y
subject to
3x + 6y ≤ 2880
8x + 5y ≤ 6720
maximise P = 1.25x + 1.4y
subject to
x + y ≤ 16
3x + 2y ≤ 36
maximise P = 6x + 4y + 10z
subject to
2x + 5y + 4z ≤ 60
2y + z ≤ 100
2x + 4y + 2z ≤ 70
4x + 3y + 2z ≤ 180
maximise P = 25x + 25y
subject to
2x − y ≤ 0
−x+y≤ 0
x + y ≤ 35
P x
y
1 − 2.5 − 3
1 − 35 − 30 − 40 0 0 0 0
0 4
10
4 1 0 0 150 0
8
9
0 6
7
8 0 1 0 250 0
4
3
P
x
y
0
1
1
z
s t u rhs
−4 0 0 1
0
0
3
2
s
0
1
0
s t u rhs P x
y
z
0 0 0 0 1 − 3.5 − 3.5 − 6.5
5
10
1 0 0 22 0 4
1
4
0 1 0 48 0 2
0
0 0 1 15
1
1
0
1
2
3
x
1 −6
0 2
0 0
0 2
0 4
y
z
− 4 − 10
5
4
2
1
4
2
3
2
s t u v rhs
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
60
100
70
180
P x
y s
1 − 25 − 25 0
0 2
−1 1
0 −1
1 0
0
1
1
s t u rhs
0 0 0 0
1 0 0 140
0 1 0 60
0 0 1 60
P x
y s t rhs
P
x
y
1 − 25 − 40 0 0 0
1 − 1.25 − 1.4
0 3
6 1 0 2880
0
1
1
0 8
5 0 1 6720
0
3
2
P
u rhs
0 0
0 120
0 50
0 0 1 28
P x
y
z
1 −3 −5 −2
0 1 2
2
0 4 3
2
1
t
0
0
1
s t rhs
0 0 0
1 0 16
0 1 36
t
0
0
1
u rhs
0 0
0 0
0 0
0 0 1 35
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Babylonians develop first algorithms
1600 BC
Euclid's algorithm
C. 300 BC
Sieve of Eratosthenes
C. 200 BC
Gaussian elimination
described by Liu Hui
263
Al−Khawarizmi described
an algorithm for solving linear
equations and quadratic equations
John Napier develops
method for performing
calculations using logarithms
Between 813 and 833
1614
Newton−Raphson method developed by
Isaac Newton and independently
by Joseph Raphson
1671 − 1690
Simplex algorithm developed
by George Dantzig
1947
Kruskal's algorithm developed
by Joseph Kruskal
1956
Prim's algorithm developed
by Robert Prim
1957
Dijkstra's algorithm developed
by Edsger Dijkstra
1959
Quicksort developed by
C. A. R. Hoare
1962
RSA encryption algorithm
discovered by Clifford Cocks
1973
RSA encryption algorithm
rediscovered by Ron Rivest,
Adi Shamir, and Len Adleman
1977
Merge sort developed
by John von Neumann
1945
The Ford−Fulkerson algorithm for
finding the maximum flow in a
network. developed by L. R. Ford, Jr.
and D. R. Fulkerson
1962
rsync algorithm (for synchronising files
and directories from one location to
another while minimising data transfer)
developed by Andrew Tridgell
1998
The Lempel−Ziv −Markov chain
Algorithm (LZMA) for data
compression
2001