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Ultra-Fast BroadbandBy Elliot Paton-Simpson
Ultra-Fast Broadband
Owairaka and the surrounding region has decided to install an ultra-fast broadband. The network spans areas and has certain broadband towers located near certain spots.
It will cost a lot of money and the council wants to keep costs to a bare minimum... To do so, the minimum time must be spent on installing the optical fibres.
The graph in the following slide shows the towers' location.
Unitec
MOTAT
Avondale College Pak'nsave
Gladstone Primary Owairaka
Domain
Mt Roskill Grammar School
Owairaka Park
Mt Albert Grammar School
Rocket ParkSt Luke's
Plant Barn
1.2km2.5km0.8km
1km2.3km
3.3km2.3km
0.7km
1.7km
2.2km
4.7km
3.4km
1.4km
5km
1.8km
2.8km
3.6km
4km
2.3km
2.1km2.4km
3.9km
1.8km
Minimum Spanning Tree
Because of costs, the council would like to start off by having the bare minimum amount of optic fibre to run the broadband, then would install the other paths later. To do this they must use a minimum spanning tree to figure out the shortest possible connection that joins all the broadband towers.
The method used is repeatedly by finding the shortest distance and if it does not loop into the new graph add it to the new graph. Eventually, you should join all the vertices.
Unitec
MOTAT
Avondale College Pak'nsave
Gladstone Primary Owairaka
Domain
Mt Roskill Grammar School
Owairaka Park
Mt Albert Grammar School
Rocket ParkSt Luke's
Plant Barn
1.2km2.5km0.8km
1km2.3km
3.3km2.3km
0.7km
1.7km
2.2km
4.7km
3.4km
1.4km
5km
1.8km
2.8km
3.6km
4km
2.3km
2.1km2.4km
3.9km
1.8km
All vertices have been
joined.
The total distance is
20.3km
The starting route
Now the council has the opportunity to gain enough profit before continuing with the full installation. This will save a lot of money but will still be satisfactory in including all of the required vertices.
Traversable or Not?
Unfortunately, certain streets will undergo road works and a paper boy has to travel on every route, travelling the shortest distance possible that goes on every path.The boy is paid depending on the distance he travels and the council would prefer to pay the smallest amount possible.
If the graph is a Eulerian trail or a semi-Eulerian trail, it is traversable. Otherwise it is not traversable and must find a way to cover all routes with the shortest possible excess distance.
Unitec
MOTAT
Avondale College Pak'nsave
Gladstone Primary Owairaka
Domain
Mt Roskill Grammar School
Owairaka Park
Mt Albert Grammar School
Rocket ParkSt Luke's
Plant Barn
1.2km2.5km0.8km
1km2.3km
3.3km2.3km
0.7km
1.7km
2.2km
4.7km
3.4km
1.4km
5km
1.8km
2.8km
3.6km
4km
2.3km
2.1km2.4km
3.9km
1.8km
Key
• - Odd Vertice
• - Even Vertice
Not Traversable!!!
The graph is not traversable. To be traversable it either must have no odd vertices or 2 odd vertices. This graph has 4 odd vertices. This means that we will have to add in an extra path. The shortest way to travel across every path is to add in another path that is the same as the other, that turns two odds to an even…
In this case we should add a path between Plant Barn and St Luke’s, only giving 1.2km extra distance to travel. Now Mt Roskill Grammar School and Owairaka Park are the only odd points. Therefore they will be the starting and finishing points.
Unitec
MOTAT
Avondale College Pak'nsave
Gladstone Primary Owairaka
Domain
Mt Roskill Grammar School
Owairaka Park
Mt Albert Grammar School
Rocket ParkSt Luke's
Plant Barn
1.2km2.5km0.8km
1km2.3km
3.3km2.3km
0.7km
1.7km
2.2km
4.7km
3.4km
1.4km
5km
1.8km
2.8km
3.6km
4km
2.3km
2.1km2.4km
3.9km
1.8km
Key
• - Odd Vertice
• - Even Vertice
1.2km
The Paper Boy’s Path
The paper boy follows this route:Owairaka park → Pak’nsave → Avondale College → MOTAT → Pak’nsave →
Owairaka Domain → Owairaka Park → Mt Roskill Grammar School → Owairaka Domain → Gladstone Primary → MOTAT → Owairaka Domain → Mt Albert Grammar School → MOTAT → Unitec → Plant Barn → St Luke’s → Plant Barn → Rocket Park → Unitec → Mt Albert Grammar School → Rocket Park → St Luke’s → Mt Albert Grammar School → Mt Roskill Grammar School
The path is 58.4km long, 1.2km longer than all the routes added together.
This isn’t bad as 1.2km probably would only cost a minor sum of money.
Shortest Path
An repairman from Unitec must travel to Owairaka park to fix one of the Broadband towers. Due to time being money, the council wants him to spend the least amount of time to travel to Owairaka park.
To do so, he must find the shortest way to reach Owairaka park.
Unitec
3.3km
2.3km
2.3km
2.5km
Plant Barn
Rocket Park
MAGS
MOTAT
Rocket Park
St Luke’s
3.7km
3.3km
Plant Barn3.1km
St Luke’s3.3km
MAGS3km
3km
Rocket Park
4km
St Luke's
Mt Roskill Grammar
School7km
Owairaka Domain
3.7km
MOTAT4.5km
MAGS5.5km
Owairaka Domain
6.7kmGladstone Primary
6.1km
Pak'nsave6.9km
Avondale College
7.3km
Plant Barn4.5kmMAGS
5km
Owairaka Domain
12km
Owairaka Park10.9km
Mt Roskill Grammar
School8.7km
Owairaka Park
6.1km
Pak'nSave5.8km
Gladstone Primary
5.5km
Owairaka Park7.6km
Avondale College
8.1km
MOTAT9.4km
Pak'nSave9.6km
MOTAT8.3km
The Repairman's Destinations.
The repairman takes the path:
Unitec → Mt Albert Grammar School → Owairaka Domain → Owairaka Park
The one thing is that he doesn't have to travel along the Broadband cable routes and could travel by road with half the distance.
Conclusion
The broadband cable installation went ahead as planned and was highly profitable. There were no complaints from any of the affected households and it was money efficient.
The main problem was that the repairman could quite have easily taken a direct route to Owairaka Park instead of stopping off at other places on the way.