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Old 05-05-2012, 06:34 PM
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Traveling on all lines of the Paris Métro


It’s no secret (but I suppose it’s not well known either) that I’ve always loved the Paris Métro. One of the things that I’ve always wanted to do is to ride on every single line. Since I’ve lived in Paris before, I’ve covered most of them, but not in a systematic way, and I’m pretty sure I’ve never been on line 9.

So, I’m thinking about going to Paris in the fall, and I would very much like to correct this situation by traveling on every single line all at once, just to get it over with. Being the kind of person that I am, I eventually got curious as to what would be the most efficient way (defined as being the way with the fewest stations) to cover all the lines, including the 3 and 7 bis, but not the tram lines or the RER. And now that’s what I want to do. Also, for elegance, I’d like the route to start and end at the same station.

The problem is that I have no idea how to figure this out or even if it is feasible for a computer non specialist to do for myself. So, I would greatly appreciate any help you may offer to help me through figuring this out. In return, when I actually do this, I will provide you with photographs of me traveling, perhaps in various comical poses.


Here’s a map of the Paris Métro in case you need one: http://mapsof.net/map/paris-metro

(I apologize if this is the wrong place for this thread. I could be soliciting facts or opinions I guess.)
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Old 05-05-2012, 07:29 PM
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Do you just want to be on each of them at some time, or do you want to travel the full length of each?
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Old 05-05-2012, 07:35 PM
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Do you just want to be on each of them at some time, or do you want to travel the full length of each?
What I want is, at least, to travel from one stop to another stop on each of the lines.
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Old 05-05-2012, 07:40 PM
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So you're not trying to hit every station, but just to plot a course across the city that lets you ride every line briefly?
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Old 05-05-2012, 09:13 PM
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So you're not trying to hit every station, but just to plot a course across the city that lets you ride every line briefly?
Yes.
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Old 05-06-2012, 08:13 AM
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Interesting problem.

On an unrelated note, I've a friend who's gig is to go to a city and bike to all the metro stops. He says it's a great way to see the whole city, tourist and non-, ugly and beautiful. I think he's done it in Paris, I know he's done Berlin.

Are you determined to get to all the stops only via the metro, or can we get off the map?

I'm thinking it might be more efficient to travel to the end of one line, then take surface transport to the terminus of the next adjacent line, travel its entire length, then take alternative transport to the terminus of the next line, etc. That way each line could be transited from end to end only once, with limited need for retracing your path.

Last edited by Attack from the 3rd dimension; 05-06-2012 at 08:18 AM.
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Old 05-06-2012, 08:47 AM
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Here's a map of the metro with streets, so a non-London-Underground projection. Assuming we can leave the metro, you could start off with missing Line 9 at Pont de Sèvres, travel the whole line to Mairie de Montreuil, then walk 20 minutes or get a cab to Parc de Bagnolet / Gallieni and transit the #3 line to Pont de Levallois — Bécon. You'd still have some bits to sort out.
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Old 05-06-2012, 10:31 AM
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Are you determined to get to all the stops only via the metro, or can we get off the map?
To my mind, leaving the metro violates the spirit of the problem.
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Old 05-06-2012, 11:43 AM
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To my mind, leaving the metro violates the spirit of the problem.
That's why I want the constraints defined. I don't want to spend 20 minutes with a map of the metro if what we really need is some outside-the-metro thinking.
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Old 05-06-2012, 02:37 PM
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What I want is, at least, to travel from one stop to another stop on each of the lines.
Another question, then: Does the Paris metro include any lines that overlap, and if so, does a leg on the overlap portion count for both?
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Old 05-06-2012, 03:26 PM
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Another question, then: Does the Paris metro include any lines that overlap, and if so, does a leg on the overlap portion count for both?
There are certainly overlaps - for example lines 8 and 9 run parallel and hit the same stations from Richelieu Drouot all the way to République. Judging by the spirit of the OP, I would say it only counts for the one you're currently riding (although I suppose you could go half of the way in one, then change and finish on the other).

Anyway, for rive gauche lines I figure the most efficient way to use all lines would be to start at Denfert, 6 to Raspail, 4 to Montparnasse, 13 to Duroc, 10 to Sevre-Babylone, 12 to Concorde, 8 to Madeleine, 14 to St. Lazare. That's 7 lines in 13 stations, not too bad. That leaves only 7 untouched, and you can easily hook up with it on rive droite.

The problem is that you then have to go all the way back east to get a spot on the short 3bis and 7bis lines (I didn't even know those existed till now ! ) ; and using the 1 at all will be tricky efficiency-wise.
Maybe a loop NW could take care of that ? The Most Useless Line* to Villiers, 2 to Charles de Gaulle, 1 to Palais Royal, 7 to Chaussée d'Antin, 9 to République, 5 to Gare de l'Est, 7 once more to Louis Blanc, 7bis to Place des Fêtes, 11 to Porte des Lilas, 3bis to Gambetta.

And then you're fucking nowhere , but you did use all 16 lines in a mere 47 stops ! Can anyone top that ?

* That's 3. Goes from nowhere to nowhere, and stops at only one major hub leading to somewhere. Yes, it's the one I live on, why do you ask ?

Last edited by Kobal2; 05-06-2012 at 03:27 PM.
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Old 05-06-2012, 03:31 PM
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Wait, I missed the "end at the start" condition. Hmm. Then add a ride on the 3 to Gambetta, then 4 to Denfert, adding 18 stops (fewer if you switch to the RER B at Châtelet).

With that condition in play, it might be more worthwile to start in Châtelet, since more or less everything goes there, meaning the trip back from anywhere you end up once you've travelled on all lines would be shorter. And then you declare Mornington Crescent, easy peasy.

Last edited by Kobal2; 05-06-2012 at 03:32 PM.
  #13  
Old 05-06-2012, 03:36 PM
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Quote:
Originally Posted by Attack from the 3rd dimension View Post
That’s why I want the constraints defined. I don’t want to spend 20 minutes with a map of the metro if what we really need is some outside-the-metro thinking.
I would prefer metro-only transportation. The basic idea is that I get on the metro, and later get off, and in between, I’ve traveled on every line for at least one stop.


So, just to clarify things, the problem is more of an intellectual one than a practical one, except that I do intend to travel the route. So the absolute geography of my itinerary is of no particular importance. I’m pretty sure this is an application of graph theory, but I don’t really know how. Anyway, here are all the details of my idea that I can think of.

- Travel on each of the 16 lines (that is lines 1–14 and 3 and 7 bis) of the Paris Métro from one stop until at the least the next stop.

- Not leave the métro system until the very end, which means staying within the parts of the métro system where having a valid ticket is required. No other form of transportation, except foot, and only then to get from one platform to another, can be used.

- Transfer from one line to another at stations that have transfers marked on the map. This means, in terms of planning my itinerary, that stations without transfers might as well not exist, except for the first and last stops.

- The itinerary can begin and end at any stop of the system (and if anyone cares, I will most likely travel to the beginning stop by bus, but the bus travel doesn’t count as part of the official itinerary).

- I can only be on one line at any given moment. (So, even though lines 8 and 9 follow the same trajectory for a bit, that doesn't cover both lines. I would have to transfer.)


My optimization concerns have changes a bit since I’ve had time to sleep on it. But here they are.

- Minimize the number of transfers to complete the route. The absolute minimum is 15, and it would be nice to hit every line exactly once, but if it’s not possible, it’s not possible.

- It’s no longer necessary to end up where I start, but it would still be nice. This would mean that I would have to start the trip at a transfer station, leaving on one line and coming back on another.

- Finally, I wonder what the shortest itinerary (by number of stops, including stops that aren’t transfer stations) that fulfills these constraints would be.



An example of an acceptable route, albeit one that doesn’t start and stop in the same place, is as follows:

Start at the Gare de Lyon (which is about a third of the way from the right edge and a third of the way from the bottom of the map).

Take line 14 southeast 1 stop to Bercy.
Take line 6 southwest 11 stops to Montparnasse–Bienvenüe.
Take line 13 northwest 7 stops to Saint-Lazare.
Take line 12 south 6 stops to Sèvres–Babylone.
Take line 10 east 2 stops to Odéon.
Take line 4 north 7 stops to Strasbourg–Saint-Denis.
Take line 8 east 8 stops to Reuilly–Diderot.
Take line 1 northeast 1 stop to Nation.
Take line 2 north 4 stops to Père Lachaise.
Take line 3 east 1 stop to Gambetta.
Take line 3bis north 3 stops to Porte des Lilas.
Take line 11 west 2 stops to Place des Fêtes.
Take line 7bis around the curve 7 stops to Louis Blanc.
Take line 7 southwest 2 stops to Gare de l’Est.
Take line 5 south 2 stops to République.
Take line 9 southeast 1 stop to Oberkampf.

So, that’s my first attempt, driven by trial-and error. 15 transfers (which is good), 65 stops (which I don’t know if it’s good or not), and it doesn’t start where it ends (which is undesirable). I admit there isn’t likely to be too much that is going to be better to deal with the 3bis and 7bis, but the rest, especially the line 6 bit, seems open for modification.

I made a drawing, that maybe you can see here: http://www.keepandshare.com/photo/41...-map?fv=y&ifr=

Last edited by RadicalPi; 05-06-2012 at 03:38 PM.
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Old 05-06-2012, 03:41 PM
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The metro is a graph. Stations are vertices, and the edges are the trains you can take. Each edge has a type associated with it, and you're looking for the shortest path that contains at least one edge of each type.

One possibility is to look at a related graph, where the edges in the metro system are the vertices, and the edges of the new graph correspond to adjacent edges in the original. Then you're looking at a shortest path problem on vertices, which may be easier to solve.
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Old 05-06-2012, 06:21 PM
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Here’s the best I’ve managed to do so far by trial and error. Many thanks to Kobal2 for bringing up the idea of using Châtelet. You, and anyone one else who wants to, are more than welcome to come with when I do this later this year.

Start at Porte des Lilas, one of the terminuses of Line 3bis.

Take line 3bis south 3 stops to Gambetta.
Take line 3 west 1 stop to Père Lachaise.
Take line 2 south 4 stops to Nation.
Take line 6 southeast 5 stops to Bercy.
Take line 14 northwest 2 stops to Châtelet.
Take line 4 south 4 stops to Odéon.
Take line 10 west 4 stops to Duroc.
Take line 13 north 4 stops to Champs-Élysées–Clemenceau
Take line 1 southeast 1 stop to Concorde.
Take line 12 north 1 stop to Madeleine.
Take line 8 east 2 stops to Richelieu Drouot.
Take line 9 east 4 stops to République.
Take line 5 north 2 stops to Gare de l’Est.
Take line 7 northeast 2 stops to Louis Blanc.
Take line 7bis east 5 stops to Place des Fêtes.
Take line 11 east 2 stops to Porte des Lilas, which is the beginning spot.

15 transfers (best possible), 46 stops (which is 1 worse than Kobal2’s), but the itinerary does start where it ends.

http://www.keepandshare.com/photo/41...-try?fv=y&ifr=

Quote:
Originally Posted by Chronos View Post
Another question, then: Does the Paris metro include any lines that overlap, and if so, does a leg on the overlap portion count for both?
No. If there are two lines that take the same path, each line is on separate tracks. So, even though the 8 and the 9 follow each other in the center of the city, line 8 is actually on top of line 9. A leg on one line only counts for that one line.

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The metro is a graph. Stations are vertices, and the edges are the trains you can take. Each edge has a type associated with it, and you're looking for the shortest path that contains at least one edge of each type.
This is correct, as far as I’ve been able to tell. The problem is that I've never taken a class about graph theory ever in my life, so, even with the transformation you suggest, I don’t know what to do with it. Hopefully, you could point me to somewhere that explains all this more clearly. I have a few months, after all, to work on this.


Now, after spending most of the afternoon doing planning routes, I have to get back to work.
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Old 05-06-2012, 06:51 PM
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No. If there are two lines that take the same path, each line is on separate tracks. So, even though the 8 and the 9 follow each other in the center of the city, line 8 is actually on top of line 9. A leg on one line only counts for that one line.
Oh, OK, I was thinking of a situation in my home town where two of the lines were identical until they diverged: Going from downtown to Shaker Square (or anywhere in between), the Blue Line and Green Line were interchangeable (in fact, the Red Line also overlapped for three stations, but that one used different cars and different platforms).
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Old 05-06-2012, 08:26 PM
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Assuming we can leave the metro
That's only allowed if the Marseilles Gambit is in effect, unless of course if you're playing by the Modified Rochambeau edition.
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Old 05-06-2012, 10:35 PM
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Are we trying to hit all lines in the least amount of time?
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Old 05-06-2012, 11:59 PM
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So, I’m thinking about going to Paris in the fall, and I would very much like to correct this situation by traveling on every single line all at once, just to get it over with.
I think this is only possible under the "many worlds" interpretation of quantum mechanics.

And, on that interpretation, you will surely have done it already.
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Old 05-17-2012, 12:34 PM
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Quote:
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Are we trying to hit all lines in the least amount of time?
Not necessarily the least amount of time, but the fewest number of stops. Obviously, the two criteria are somewhat related.

Quote:
Originally Posted by njtt View Post
I think this is only possible under the "many worlds" interpretation of quantum mechanics.

And, on that interpretation, you will surely have done it already.
If I'd known it would require quantum physics to do this, maybe I shouldn't have even started.
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Old 05-21-2012, 05:29 PM
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The metro is a graph. Stations are vertices, and the edges are the trains you can take. Each edge has a type associated with it, and you're looking for the shortest path that contains at least one edge of each type.
I asked around, and it turns out that this problem is NP-complete by reduction from the Hamiltonian path problem. So using one of the solutions upthread is likely your best option.
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Old 05-21-2012, 07:52 PM
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That's only allowed if the Marseilles Gambit is in effect, unless of course if you're playing by the Modified Rochambeau edition.
But according to Directive EC 67/202.10 originally promulgated by Lavallière and Bouton, it's also allowed for any non-French citizen with a valid visa. This essentially holds for anyone with a Schengen visa now.
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Old 05-22-2012, 08:59 AM
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Only marginally related here, but if you're a fan of the Paris Metro (or even if you're not), you might enjoy this little artistic video montage: Paris Metro as You've Never Seen It. I saw it on The Daily Beast today and thought of this thread.

Loves me some Paris Metro!
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Old 06-21-2019, 04:15 AM
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I know that it’s been a while, but I’ve finally made my way back to Paris, and using the route below, I completed the circuit of all the lines of the métro.

Quote:
Originally Posted by RadicalPi View Post
Here’s the best I’ve managed to do so far by trial and error. Many thanks to Kobal2 for bringing up the idea of using Châtelet. You, and anyone one else who wants to, are more than welcome to come with when I do this later this year.

Start at Porte des Lilas, one of the terminuses of Line 3bis.

Take line 3bis south 3 stops to Gambetta.
Take line 3 west 1 stop to Père Lachaise.
Take line 2 south 4 stops to Nation.
Take line 6 southeast 5 stops to Bercy.
Take line 14 northwest 2 stops to Châtelet.
Take line 4 south 4 stops to Odéon.
Take line 10 west 4 stops to Duroc.
Take line 13 north 4 stops to Champs-Élysées–Clemenceau
Take line 1 southeast 1 stop to Concorde.
Take line 12 north 1 stop to Madeleine.
Take line 8 east 2 stops to Richelieu Drouot.
Take line 9 east 4 stops to République.
Take line 5 north 2 stops to Gare de l’Est.
Take line 7 northeast 2 stops to Louis Blanc.
Take line 7bis east 5 stops to Place des Fêtes.
Take line 11 east 2 stops to Porte des Lilas, which is the beginning spot.

15 transfers (best possible), 46 stops

The only differences are that I started and ended at Gare de l’Est, since that was most convenient for me. Also, there are only 3 stops on line 4 between Châtelet and Odéon.

The trip took 2 hours and 27 minutes from starting platform to ending platform, only 3 minutes longer than I had planned. As you might imagine, most of the time was spent waiting for the trains to arrive, at about a 60/40 ratio.

A montage picture of the information panels in the stations I took is here.

I am, on the whole, prouder than I probably should be. Thanks to everyone in 2012 who helped me figure this out.
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Old 06-21-2019, 05:39 AM
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Now do New York City. The record is just under 24 hours.
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Old 06-21-2019, 06:43 AM
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It's been seven years and nobody's bothered to track down or construct a machine-readable node list for the Paris Métro graph? Bah!!
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Old 06-21-2019, 07:47 AM
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I am, on the whole, prouder than I probably should be.
Nonsense. You had a dream, you took steps to make it happen, you executed it to perfection. Congrats!
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Old 06-21-2019, 09:11 AM
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It's a good job you did it now. They're building four more metro lines: 15, 16, 17 and 18.

See: https://www.societedugrandparis.fr/i...ct-europe-1061

I imagine that may complicate the matter somewhat.
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Old 06-21-2019, 09:47 AM
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If you fancy taking the London Tube Challenge here is a handy map to put you on the right track (groan).

The basic rules are:
1. You don’t have to cover every stretch of track or step on every platform, but you must arrive at or depart from every Underground station by train. (Overground, National Rail and Docklands Light Railway stations not included).

2. You may take buses or run between stations (but not taxis, private cars or skateboards).

If you are going for the record (currently 16 hours, 14 minutes and 10 seconds) you also need a witness with a master stopwatch.

https://secure.i.telegraph.co.uk/mul...e_3078591a.jpg
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Old 06-28-2019, 04:30 AM
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Having participated in several Tube challenges and being acquainted with most of the recent record holders, I can tell you one or two things about doing this sort of thing.

It's a variant on the travelling salesman problem and there is no easy solution. The last time we held a Tube challenge it brought competitors from around the world, many of them mathematicians. Some people used neural networks, some used various efficiency routines, some did it by dead reckoning, others relied on luck or experience of the system. There's all kinds of extra variables such as which carriage to get on for the most direct exit or transfer, minor branches (eg Kensington Olympia) that don't get many trains, etc. It's quite a problem to solve. Even with the right plan, you can arrive in a station and encounter an unexpected train, meaning you have to be very adaptive. And there's the problem of food, drink and toilets. And sometimes you need the experience of actually knowing the system as there's some points where you're better off travelling on foot overground or taking a car or bike between awkward spurs. Even a seemingly perfect plan can get screwed up when certain stations (eg Victoria) get closed for overcrowding, a section gets suspended for a signal failure, etc. There's a lot to taken into account. Even a seemingly perfect route can get screwed up when a minor delay throws everything out of whack.

Well done to RadicalPi for realising his dream!
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Old 06-28-2019, 01:51 PM
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Isn't this simply a version of the Travelling Salesman problem, a common & well-studied problem in operations research?
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Old 06-29-2019, 07:01 AM
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Isn't this simply a version of the Travelling Salesman problem, a common & well-studied problem in operations research?
The basic problem, just finding a route regardless of time, is like that. And the Metro is small enough that the time complexity isn't a real problem.

But if you throw in requirements regarding reducing time then it gets complicated. Add in the dynamic, everchanging aspects of trains being delayed, etc. and it gets nasty.

Even trying to be "approximately optimal" only partially helps then.
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Old 07-06-2019, 10:54 PM
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Nonsense. You had a dream, you took steps to make it happen, you executed it to perfection. Congrats!
Hear, hear - and thanks for giving us an update!
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Old 07-06-2019, 11:12 PM
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I know that it’s been a while, but I’ve finally made my way back to Paris, and using the route below, I completed the circuit of all the lines of the métro.




The only differences are that I started and ended at Gare de l’Est, since that was most convenient for me. Also, there are only 3 stops on line 4 between Châtelet and Odéon.

The trip took 2 hours and 27 minutes from starting platform to ending platform, only 3 minutes longer than I had planned. As you might imagine, most of the time was spent waiting for the trains to arrive, at about a 60/40 ratio.

A montage picture of the information panels in the stations I took is here.

I am, on the whole, prouder than I probably should be. Thanks to everyone in 2012 who helped me figure this out.
Thanks for the followup 7 years later! But I just have one question. From everything I've heard about the Paris Métro (and also the airports and other public places) when you finally arrived back at Gare de l'Est, did you still have your wallet?
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