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During the summer holidays last year, I went on a cycling holiday. I arrived in Roscoff, by ferry from Plymouth, and stayed at a friend’s house for a few days and then set off around Brittany on my bicycle. I found the cycling hard work at the beginning, but after a few days my legs were no longer sore, even after five hours of continuos cycling. At the end of three weeks I had visited many of the towns and villages in Brittany and had nicely tanned legs.

 

This year I am planning to go cycling with a friend. To avoid cycling along a route which is repetitive and contains long stretches of road to cycle in a day. I will draw a network to plan my route. I have linked up the main towns in west Brittany and distances between them along roads shown in the Michelin road map. I intend to cycle from one town to another during the day, and then stay the night in the destination town. I will start and finish the trip in the small town of Roscoff, where my friend and I will stay at my French friend’s house in the town centre.

 

 

I will try to cycle along the shortest possible route between town. I can only look at a few of the routes between the 10 nodes out a possible (10-1)! ÷ 2 = 181440. This would be far too many to individually analyse. Some of the possible arcs will be left as the routes between the towns either do not exist or the route passes directly through another node. I feel my network is adequately sophisticated as there are 10 nodes and many of the nodes are joined to each other. Distances between nodes are calculated by adding up the distances of sections of road on the map. All the distances on my network are in kilometres, avoiding main roads with large visitors and fast cars. I will use Prim’s algorithm to join all the nodes on the network together, leaving out some routes which are not on the map. Afterwards, certain nodes will be methodically omitted to obtain a list of minimum connectors, out of which I find he the highest minimum connector.

 

 

	Roscoff	Morlaix	Lannion	St. Brc	Brest	Carhaix	Pontivy	Quimp.	Conc.	Lorient
Roscoff	-	25	¥	¥	64	¥	¥	103	¥	¥
Morlaix	25	-	45	105	65	50	119	82	102	¥
Lannion	¥	45	-	70	¥	64	92	114	¥	134
St. Brc	¥	105	70	-	147	84	59	¥	112	100
Brest	64	65	¥	147	-	97	¥	121	141	162
Carhaix	¥	50	64	84	97	-	54	61	62	75
Pontivy	¥	119	92	59	¥	54	-	94	87	50
Quimp.	103	82	114	¥	121	61	94	-	24	67
Conc.	¥	102	¥	112	141	62	87	24	-	64
Lorient	¥	¥	134	100	162	75	50	67	64	-

 

I proceeded to omit each node in turn to find the minimum connector, the minimum connector rule. Each node that was left out was later joined by an arc to and from the node. If a tour is formed than it is the best solution. Normally the solution is more than this number and the highest minimum connector is taken to be the lower bound for where the solution may lie.

Omitted Node	Min Conn	Minus	Total
Roscoff	473	70	543
Morlaix	387	109	496
Lannion	373	129	507
St. Brieuc	368	129	497
Brest	401	104	505
Carhaix	403	104	507
Pontivy	408	85	494
Quimper	405	86	494
Concarneau	382	114	496

 

Here is a diagram of the lowest band value, the minimum spanning-tree, therefore the solution must be less than 2 × 432 = 864km. It is important to find the boundaries where the solution will lie, as this is the only way to judge if the solution can be improved or if it is adequate in the given situation. The solution must be more than the highest minimum connector, 494km. The range of the solution:

highest minimum connector < solution £ minimum connector × 2     =     494 < solution £ 864

I will proceed to make shortcuts along the minimum connector × 2, to avoid travelling along the same route twice. The solution would be better if it is closer to 494km, rather than 864km.

Minimum connector – travelling twice along the same routes. 864km	Minimum connector with shortcuts, avoiding travelling along the same routes to reduce distance and boredom. 559km

 

Roscoff ® Morlaix ® Lannion ® Pontivy ® Lorient ® Concarneau ® Quimper ® Brest ® St. Brieuc ® Morlaix ® Rocoff

Here is the tour, starting from Roscoff, and travelling to closest unvisited town. The distances between nodes get larger, with this example, as nodes left unvisited near the end of the tour are far apart. This results in a very large distance travelled.

 

Starting Node	Route	Distance
Roscoff	R® M® L® C® P® Lo® Co® Q® B® S® M® R	724
Morlaix	M® B® C® P® Lo® Co® Q® L® S® M® R® M	667
Lannion	L® M® R® B® C® P® Lo® Co® Q® C® S® L	629
St. Brieuc	S® P® Lo® Co® Q® C® M® R® B® M® L® S	577
Brest	B® R® M® L® C® P® Lo® Co® Q® C® S® B	673
Carhaix	C® M® R® B® M® L® S® P® Lo® Co® Q® C	577
Pontivy	P® Lo® Co® Q® C® M® R® B® M® L® S® P	577
Quimper	Q® C® M® R® B® M® L® S® P® Lo® Co® Q	580
Concarneau	Co® Q® C® M® R® B® M® L® S® P® Lo® Co	577
Lorient	Lo® P® C® M® B® R® M® L® S® Co® Q® Lo	626

 

The solution lies between 864km and 494km at 559km. It is closer to the highest minimum connector which means the result is satisfactory considering the large distances involved. The minimum connecting route has many dead ends and therefore resulted in a rather large difference between the highest minimum connector and the shortest route. Quite a few shortcuts were made, added to the distance of the tour.

If we were to cycle the route in 10 days, staying in each town overnight, we would cycle an mean-average of 55.9km a day. This is the normal distance covered by a person, cycling for a day. The route does not travel along the same arcs twice, this will make the cycling holiday more enjoyable. As we will be cycling during the summer holidays, the weather will be very hot.

The route is good due to the fact that much of it lies near the coast, where the temperature is slightly cooler. In the planning of the cycling tour of Brittany, I did not include other variables such as gradient and quality of roads, the sites of interest along the routes and whether we will stay the night at each town we visit.

Carhaix is located on a hill in the centre of Finistere. This would mean that the journey towards the town would be more difficult to cycle and the average speed would lower, meaning a longer time spent per kilometre than on other arcs. In departing from the town, the opposite scenario would occur, more frequent downhill cycling, less time needed per kilometre and a higher average speed.

Roads near the coast are often over many small hills and valleys, but these are not so difficult, as uphill riding is often brief. The longer the route the greater the chance of loosing our way, so with longer routes away from towns or recognisable natural features, estimated times should be increased to allow for any delays occurring. This said, all alterations will be minimal and will not effect the final tour, so I will not redo the tour building.

Our average speed would be 10km/h, this includes short breaks along the route. Here is a timetable of our tour of Brittany.

 

Time and Day	Distance	Planned Activities
16.30 Monday		Arrive off passenger ferry in Roscoff
Monday Night		Stay at my friend’s house in Roscoff
Tuesday		Go sailing during the day around the harbour ion Roscoff
Tuesday Night		Stay a second night at my friend’s house
11.00 - 13.00 Wednesday	25km	Cycle to Morlaix
Wednesday Night		Wonder around the Wednesday market, book a night at Youth Hostel
10.30 - 13.00 Thursday	45km	Cycle to Lannion
19.30 - 21.30 Thursday		Watch a movie at the cinema, stay for the night at a Youth Hostel
10.00 - 17.00 Friday	70km	Cycle to St. Brieuc and then along the East coastal road
18.00 - 19.00 Friday		Visit the sites of the town, including historic town centre and castle
Friday Night		Stay at a Youth Hostel by the river
10.30 - 16.30 Saturday	59km	Cycle to Pontivy and stay over night in a B&B
11.00 - 16.00 Sunday	50km	Cycle to Lorient
18.00 - 23.00 Sunday		Leave baggage at Youth Hostel and have a night in the city
11.00 - 17.30 Monday	64km	Cycle to Concarneau
18.30 - 20.30 Monday		Walk along seafront and go to a restaurant or order a takeaway
Monday Night		Stay the night at the Youth Hostel by the sea
10.00 - 12.30 Tuesday	24km	Travel to Quimper
13.30 - 15.00 Wednesday		Leave baggage in Youth Hostel, have lunch, then explore the town.
15.30 - 18.30 Wednesday		Visit the leisure centre and go for a swim, ice-skating, or bowling
Wednesday Night		Stay at Youth Hostel in Quimper
10.00 - 17.30 Thursday	61km	Cycle to Carhaix in the highland of Brittany, stay at a B&B.
9.30 - 18.30 Friday	97km	Longest cycle, downhill the city of Brest
Friday Night		Stay in Youth Hostel near "Sea World"
10.00 - 16.30 Saturday	64km	Return to small town of Roscoff
8.00 - 18.00 Sunday		Travel by ferry back to Plymouth
19.30 Sunday		Catch train back to Penzance (if the train is on time)

 

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