the travelling salesman problem involves visiting every town (node) on a graph and returning to the...
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The travelling Salesman problem involves visiting every town (node) on a graph and returning to the start in the shortest distance.The slides will show how to find 2 values that the actual journey lies between called the Upper and Lower Bound.The French towns below have to be visited
Travelling Salesman Problem
The following slides show how to find an Upper Bound to the travelling Salesman Problem using the Nearest Neighbour Algorithm and a Lower Bound by removing a town and using the Minimum Connector Algorithm
Travelling Salesman Problem
Link to Nearest Neighbour Algorithm to find an Upper Bound
Link to finding a Lower Bound using Kruskals Algorithm
Nearest Neighbour Algorithm
This is will give an Upper Bound to the Travelling Salesman problem.
It may be possible to complete the journey in a smaller distance but the Upper Bound is an upper limit to the required distance and is definitely possible.
The smaller the Upper Bound the better
Nearest Neighbour from a Table
The chart shows the distances between French towns
1. Choose a starting town2. Find the nearest neighbour3. Repeat step 2 ensuring that you have not already been there.4. When all the towns have been visited return to the starting
town using the shortest route.5. Repeat using a different starting place and the answer giving
the lowest total distance is the Upper Bound
Start at LyonsReading across nearest neighbour is Dijon – 192 milesReading down nearest neighbour is St Etienne – 59 milesChoose St Etienne
Lyons St Etienne
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2
Carry on from St EtienneReading across nearest neighbour is Dijon – 244 milesReading down nearest neighbour is Toulouse – 528 milesChoose Dijon
Lyons St Etienne
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2
3
Dijon
Carry on from DijonCannot choose Lyons as this has already been usedReading across nearest neighbour is Calais – 543 milesReading down nearest neighbour is Orleans – 297 milesChoose OrleansLyons St Etienne Dijon
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2
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4
Orleans
Carry on from OrleansReading across nearest neighbour is Calais – 421 milesReading down nearest neighbour is Poitiers – 212 milesChoose Poitiers
Lyons St Etienne Dijon Orleans
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2
3
45
Poitiers
Carry on from PoitiersReading across nearest neighbour is Bordeaux – 247 milesReading down nearest neighbour is Toulouse – 435 milesCannot choose St Etienne as already visitedChoose BordeauxLyons St Etienne Dijon Orleans Poitiers
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2
3
45
Bordeaux
6
Carry on from BordeauxReading down nearest neighbour is Toulouse – 244 milesCannot read acrossChoose Toulouse
Lyons St Etienne Dijon Orleans Poitiers
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2
3
45
BordeauxToulouse
6
7
Carry on from ToulouseReading across nearest neighbour is Marseille – 405 milesCannot read downChoose Marseille
Lyons St Etienne Dijon Orleans Poitiers
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2
3
45
BordeauxToulouse
6
7
Marseille
8
Carry on from Marseille Reading across nearest neighbour is Calais – 1067 miles (Rest have all been visited)Reading down nearest neighbour have all been visitedChoose Calais
Lyons St Etienne Dijon Orleans Poitiers
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2
3
45
BordeauxToulouse
6
7
Marseille
8
8
Calais
Carry on from CalaisAll towns have been visited so return to Lyons – 751 miles
Lyons St Etienne Dijon Orleans Poitiers
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2
3
45
BordeauxToulouse
6
7
Marseille
8
8
Calais
Total distance = total of circled values = 3526 miles
Upper Bound = 3526
Now repeat using a different starting valueThe lowest value obtained is the Upper BoundThis means that the shortest route is equal or less than the Upper Bound
Lower BoundThis is the smallest journey that it may be possible to do.
It may not in fact be possible to actually achieve this distance but it is a value to aim for.
The actual journey length lies between the upper and lower bound.
Lower Bound from a Table
The chart shows the distances between French towns
1. Remove one of the towns2. Find the minimum connector of the remaining towns using
Prim or Kruskal3. Add the town removed back in using the shortest 2 arcs.4. Find the total length of all the arcs used.
Remove MarseilleNow do Kruskals AlgorithmFind the shortest distance : Draw it
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2
St EtienneLyons
St Etienne to Lyon 59 miles
Find the next shortest distance Draw it
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2
St EtienneLyonsDijon
3
Dijon to Lyons 192 miles
Find the next shortest distance Draw it
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2
St EtienneLyonsDijon
3
PoitiersOrleans
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5
Orleans to Poitiers 212 miles
Find the next shortest distance Ignore Dijon to St Etienne as this makes a cycleDraw it
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
6
5
7
Bordeaux to Toulouse 244 miles
Find the next shortest distance Draw it
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
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7
6
Bordeaux to Poitiers 247 miles
Find the next shortest distance Draw it
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
6
5
7
Orleans to Dijon 297 miles
Find the next shortest distance Ignore Orleans to St Etienne and Poitiers to St Etienne as these make a cycleDraw it
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
6
5
7
8
Calais
Calais to Orleans 421 miles
Now all the towns are connected with the minimum connectorFinally join Marseille back in with the shortest 2 arcs
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
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5
7
8
Calais
Now all the towns are connected with the minimum connectorFinally join Marseille back in with the shortest 2 arcs
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
6
5
7
8
Calais
Marseille
Adding up all the circled distances gives a lower bound This can be repeated to find another lower bound by removing a different town
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2
St EtienneLyonsDijon
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PoitiersOrleans
4
ToulouseBordeaux
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5
7
8
Calais
Marseille
Lower Bound = 2297
The highest lower bound is the best as the actual answer lies between the Lower and Upper bound. If the Lower and Upper bound are the same then that is the answer
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2
St EtienneLyonsDijon
3
PoitiersOrleans
4
ToulouseBordeaux
6
5
7
8
Calais
Marseille
Lower Bound = 2297
Lyons St Etienne Dijon Orleans Poitiers
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2
3
45
BordeauxToulouse
6
7
Marseille
8
8
Calais
Total distance = 3526 miles
Upper Bound = 3526
Upper Bound = 3526Lower Bound = 2297
The actual answer lies between the Lower and Upper bound and can only be found by trial and error.The Upper and Lower Bound give an idea as to what the actual answer lies betweenThe smaller the gap the better.Remember need High Lower Bound and Low Upper Bound.