Networks & Minimum Spanning Trees. Cambridge is installing fibre optic cabling between the...
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Transcript of Networks & Minimum Spanning Trees. Cambridge is installing fibre optic cabling between the...
Networks & Minimum Spanning Trees
• Cambridge is installing fibre optic cabling
between the surrounding villages.
• All the villages must be connected.
• We need to minimise the amount of cable that
is used.
Networks and Minimum Spanning Trees
Cambridge
Milton
Bottisham
Fulbourn
Great Shelford
Grantchester
Coton
Girton
Impington
6 miles
6 miles
5 miles4 miles
3 miles
6 miles
3 miles 3 miles
4 miles
7 miles
6 miles
5 miles
3 miles
4 miles
5 miles
5 miles
Cambridge
Milton
Bottisham
Fulbourn
Great Shelford
Grantchester
Coton
Girton
Impington
6 miles
6 miles
5 miles4 miles
3 miles
6 miles
3 miles 3 miles
4 miles
7 miles
6 miles
5 miles
3 miles
4 miles
5 miles
5 miles
C
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6
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3
6
3 3
4
7
5
3
4
55
6
This network can now be used to model our situation.
Networks and Minimum Spanning Trees
• A network is a type of graph used in
Decision Maths.
• Points (called nodes or vertices) are
connected by lines (called arcs or edges).
• The lines (edges) have values attached to
them, these may represent a cost or a
distance.
Networks and Minimum Spanning Trees
• A minimum spanning tree is a section of the
network.
• All points (nodes) must be included but you
should not have any cycles.
• A cycle links points together in a closed ring.
• The minimum spanning tree links the points
(nodes) using the lines (edges) that have
the smallest values.
Cycle ✖ ✔
Networks and Minimum Spanning Trees
• Algorithms have been created to help
us decide which edges should be
used to create the minimum
spanning tree.
• We are going to use Kruskal’s
Algorithm.
Kruskal's’s Algorithm
1. Write down all the edges in size order
2. Select the shortest edge in a network
3. Select the next shortest edge which does not create a cycle
4. Repeat step 3 until all vertices have been connected
C
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6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3
IM – 3
GC – 3
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
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I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3
GC – 3
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
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B
F
S
G
O
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I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3
GC – 3
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
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B
F
S
G
O
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6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
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B
F
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O
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6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
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6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
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6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
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B
F
S
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O
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6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
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6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
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I
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6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6 ✖ (cycle)
BF – 6
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6 ✖ (cycle)
BF – 6 ✔
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6 ✖ (cycle)
BF – 6 ✔
BM – 6
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6 ✖ (cycle)
BF – 6 ✔
BM – 6 ✖ (cycle)
CB – 7
C
M
B
F
S
G
O
T
I
6
6
54
3
6
3 3
4
7
5
3
4
55
6
TI – 3 ✔
IM – 3 ✔
GC – 3 ✔
GO – 3 ✔
MC – 4 ✔
SG – 4 ✔
OC – 4 ✖ (cycle)
IC – 5 ✖ (cycle)
TC – 5 ✖ (cycle)
SC – 5 ✖ (cycle)
SF – 5 ✔
TO – 6 ✖ (cycle)
CF – 6 ✖ (cycle)
BF – 6 ✔
BM – 6 ✖ (cycle)
CB – 7 ✖ (cycle)
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3
3 3
4
3 The minimum spanning tree
contains only the edges we have
selected.
The total weight of the tree comes
from the summation of our
distances.
3 + 3 + 4 + 3 + 3 + 4 + 5 + 6 = 31
So, the city needs a total of 31 km
in fibre optic cabling!
Tunneling Students
• In an attempt to get out of the rain and away
from the tourists, the students of the
University of Cambridge are setting up some
underground tunnels between the university
buildings!
• Suggest a network of tunnels they could use,
minimising the amount of tunneling required.
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1KQ – 0.2KT – 0.2TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2KT – 0.2TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2KT – 0.2TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5 ✔TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5 ✔TCl – 0.5SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5 ✔TCl – 0.5 ✖ (cycle)SjJ – 0.5SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5 ✔TCl – 0.5 ✖ (cycle)SjJ – 0.5 ✖ (cycle)SjCl – 0.7
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.7
0.5
0.5
0.5
0.20.1
0.3
0.5
0.40.2
KPb – 0.1 ✔KQ – 0.2 ✔KT – 0.2 ✔TSj – 0.2 ✔PbE – 0.3 ✔TE – 0.4 ✖ (cycle)ClS – 0.5 ✔SQ – 0.5 ✔EJ – 0.5 ✔TCl – 0.5 ✖ (cycle)SjJ – 0.5 ✖ (cycle)SjCl – 0.7 ✖ (cycle)
Cl
S
Q
Pb
K
T
SJ
J
E
0.2
0.5
0.5
0.20.1
0.3
0.5
0.2
The students should tunnel between the
university buildings as shown.
The total length of tunneling needed is
2.5 miles.
