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.