Road Coloring Problem Solved

ArieKremen writes "Israeli Avraham Trakhtman, a Russian immigrant mathematician who had been employed as a night watchman, has solved the Road Coloring problem. First posed in 1970 by Benjamin Weiss and Roy Adler, the problem posits that given a finite number of roads, one should be able to draw a map, coded in various colors, that leads to a certain destination regardless of the point of origin. The 63-year-old Trakhtman jotted down the solution in pencil in 8 pages. The problem has real-world implementation in message and traffic routing."

The writeup is misleading

[karl marx is my hero (1222496)]

The guy is actually a mathematician who had to work as a security guard right after he immigrated to Israel, which is common for most immigrants. This guy had lots of formal training solving equations like this.

The parent is misleading

[PDAllen (709106)]

The road colouring problem is not 'an equation'.

'deterministic automaton' basically means: computer program which doesn't have any form of randomness involved. If you're in this particular state in the program and you do this particular thing then always the same thing happens.

The road colouring problem amounts to: For any program, I claim that there is one single sequence of actions so that if you do this sequence of actions, from whatever start point, when you finish you will be in a specific state (`target', say) of the program.

If you take any real normal program, it's `obvious' what to do: for example if you're looking at 'edit' and you want a typed copy of Shakespeare in Times 12pt, you hit backspace as many times as edit can have characters (lots, but it's a finite number) then you go through the menus by keypress and set each style to the correct thing, then you start typing in the works of Shakespeare which eventually gives you the target state. And it doesn't matter if the monitor was off, you know that this method definitely works, whatever state edit was in when you got there, whether it was in the blank just loaded state, or whether it was editing font size in War and Peace, whatever.

It is not so easy to prove that in fact for any program you can find such a sequence.

In fact, it isn't even true. A really simple example is the program which every time you press a key changes the screen between red and blue. Now you're supposed to give a sequence of actions - keystrokes - which are guaranteed to end up with the screen blue (target state). But really what the keystrokes are is irrelevant. This program doesn't care if you press 'a' or 'ESC', it just changes screen colour every stroke. So really your sequence of actions comes down to 'press keys however many times'. How many times? Well, if the screen starts red (and remember the monitor is off, so you don't know if that's true) then you'd need to press keys an odd number of times to get it to end up blue. So you have to do that. But then what happens if the screen started blue? It ends up red. This sequence of actions doesn't always reach the target state - and there cannot be any sequence of actions which does.

So you have to impose a condition to make the result true. In the example I gave, the program has two states and it loops between them. It's a program with exactly one loop, of length two, and the greatest common divisor (GCD) of these loops is of course 2. Basically, having the ability to loop between states like this (or in a loop of length 3, or 4, or whatever) is the only 'obvious' barrier to having a sequence of actions which goes to a target state. So you impose a condition: the GCD of all the loops is 1 (this doesn't mean there are no loops, but the guy proves it means you can find a way out of all the loops). If you don't have that, then you can find automatons (programs) which work like the example and you cannot possibly have the sequence of actions you need. If you do have it, there is no 'obvious' barrier to the sequence of actions existing. And what this paper shows is that there isn't any hidden catch: once you get rid of the 'obvious' barrier then you can find a sequence of actions which definitely puts the program you're looking at into the target state.

Re:

[PDAllen (709106)]

I didn't even mention NP-completeness. Mainly because it has nothing to do with the problem. Are you just slinging buzzwords on a subject you don't understand?

Applications?

[DigiShaman (671371)]

How soon can we see this implemented in my Garman GPS unit or Google Maps?

Re:Applications?

[MightyYar (622222)]

Egads, please no! This only guarantees that you will arrive at a destination from an arbitrary start point. Even if you could somehow use it for Google directions, your route would be crazy. You could start out next door to your destination and it may take you a day-and-a-half of travel to finally wind up there as you tour all over the city. I can see this being interesting to networking, though. A guaranteed route somewhere from any start point sounds pretty cool, even if a bit inefficient.

It's a theorem

[pikine (771084)]

It's a theorem which states that if a graph is strongly connected and aperiodic, then there exists a road coloring. It was conjectured by Weiss and proven by Trahtman 30 years later, which is what the article is about. A graph that is periodic may still have a road coloring, but that's not the scope of the theorem.

Most roads are strongly connected but periodic (not aperiodic), in the sense that you can drive around a block in 4 segments, and drive around 2 blocks in 6. The period in this case is 2. You can't guarantee a computer network to be aperiodic either. These graphs may still have a road coloring, but the theorem doesn't apply. Therefore, this theorem has little application in practice.

I haven't read the paper, but there are generally two ways to prove the theorem: (1) show a coloring algorithm that makes only the assumptions of strong connectivity and aperiodicity, or (2) show the contraposition, that if there is no road coloring, then it implies either the graph is not strongly connected or there is period. Only (1) is useful in practice because you have a method to generate road coloring and instructions to reach all vertices, but it's harder to verify that (1) is correct. (2) is not very useful because although you have a proof, but you don't end up with an algorithm that generates road coloring.

In practice, algorithms get things done. Proofs are only certificates.

Bad timing.

[Rob T Firefly (844560)]

Unfortunately for him, a cheap, safe, mass-market flying car was announced an hour later.

Link to the solution

[wiredog (43288)]

Since the one in the write-up is borken. Here [arxiv.org] it is.

wikipedia

[ZiggyM (238243)]

from Wikipedia: In the real world, this phenomenon would be as if you called a friend to ask for directions to his house, and he gave you a set of directions that worked no matter where you started from. http://en.wikipedia.org/wiki/Road_Coloring_Conjecture/ [wikipedia.org]

Re:

[nherc (530930)]

Oddly Wikipedia directory/article URL listings seem case sensitive and the given one pointing at 'Road_Coloring_Conjecture' does not work, but 'Road_coloring_conjecture' does. The correct Wikipedia link to the Road Coloring Conjecture page is http://en.wikipedia.org/wiki/Road_coloring_conjecture [wikipedia.org].

Re:wikipedia

[Stephan202 (1003355)]

Drop the trailing slash: http://en.wikipedia.org/wiki/Road_Coloring_Conjecture [wikipedia.org]

Re:wikipedia

[Jasin Natael (14968)]

The problem is more constrained than that. Think about it more like this: You're in a town that's been planned very carefully. At any intersection, the possible roads away from that intersection are labeled with colors. I'll assume three colors, and that each intersection has exactly three roads leading away from it (the number of roads that lead into the intersection doesn't matter).

Your friend tells you that his address is "Red-Blue-Blue". This means that, no matter which intersection you start from, by repeatedly following these directions, you WILL end up at his house.

A Subquadratic Algorithm for Road Coloring

[Frans Faase (648933)]

Apart from the referenced paper being some months old, the author has an extended paper with an efficient algorithm. See A Subquadratic Algorithm for Road Coloring [arxiv.org].

Old news?

[amplt1337 (707922)]

1. Here's wtf the problem even is [wikipedia.org], for those of us who aren't all up in the "mathematical curiosities" business. Basically the question is, for a specific kind of graph (where you can go from point A to a finite number of points B, C, or D, etc) can you label the possible paths from each point so that, starting from anywhere, you can follow an invariable rule that will get you to a specific destination point. (Check the link, the picture makes much more sense).

2. Apparently his proof was published last September. It's "news" because it's just now hitting the semi-mainstream press. You people fail at nerddom.

Re:

[huckamania (533052)]

Apparently his proof was published last September. It's "news" because it's just now hitting the semi-mainstream press. You people fail at nerddom.

No, they're just applying the normal slashdot skepticism. The guy was a security guard for crying out loud. And he's old. Really old. No way he could have used his brain to solve something involving math. Besides, he's probably only trying to get seed money so he can start a pyramid scheme selling perpertual motion.

Re:

[Reason58 (775044)]

But when you get to that point, it loses all meaning. R on Lincoln L on 1st St L on R Ave R on Prescott How is that any harder than saying: Orange Burnt Amber Sanguine Desire Ghost Egg White Not to mention the more complicated the street system the more colors you would need and that would cause extreme confusion as people try and tell shades apart.

Actually

[The MAZZTer (911996)]

He did it in 6.5 pages, the rest is references, whitespace at the very end, and the abstract and title.

Re:

[The MAZZTer (911996)]

Err, I mean 5.5 pages.

Re:

[qw(name) (718245)]

The proof takes up 5.5 pages all typed up and pretty. His original proof was hand-written and consumed 8 pages.

Re:Actually

[spookymonster (238226)]

Actually, if he were to put it on a SD card, it'd take up less than 1 square inch...

Where's my PEDANTIC:-1 option?

Sign of a true Genius.

[Lumpy (12016)]

"Some people think they need to be complicated. I think they need to be nice and simple."

The man is a True Genius. Insight to all of out out there, that is the proper way of thinking.

Re:Sign of a true Genius.

[Jamil Karim (931849)]

The man is a True Genius.

Real Men of Genius!
Today, we salute you, Mr. former-nightwatchman-turned-mathmetician!

HIRED!

[TheGreatOrangePeel (618581)]

Well, SOMEONE is about to be hired by google... ...you suppose it's too late to learn Russian, and become his friend?

Slashdottings in real life?

[tepples (727027)]

The conjecture essentially assumed it's possible to create a "universal map" that can direct people to arrive at a certain destination, at the same time, regardless of starting point. Experts say the proposition could have real-life applications in mapping and computer science.

If a thousand people arrive at a shopping mall, at the same time, isn't that called a flash mob [wikipedia.org]? Then I guess this conjecture has already been solved online, and it is called Slashdot.

What about the cyclical traps?

[Muad'Dave (255648)]

Given this graph [wikipedia.org], what about the 'traps' that exist for directions 'rrr' and 'bbb'? If you number the nodes top to bottom, left to right as 1-8, nodes 1, 2, and 4 all are destinations for 'rrr' depending on your starting point. Similarly, nodes 6, 7, and 8 are all destinations for 'bbb' depending on where you start. Does the problem definition say you must follow the whole direction, i.e. you must always traverse 3 arcs, or is it 'legal' to stop when you've reached your destination (assuming you know where

Washington - RNC ???

[zappepcs (820751)]

"Say you've lost an e-mail and you want to get it back -- it would be guaranteed," he said. "Let's say you are lost in a town you have never been in before and you have to get to a friend's house and there are no street signs -- the directions will work no matter what."

I wonder if this can be retroactively applied to certain email servers in Washington?

What is road-coloring (and why we should care)

[kevinatilusa (620125)]

The question behind road coloring is this: Given a directed network with out-degree 2 (from any place you can get to exactly two other places), we want to color the edges leading out of each node red or blue so the following "universal directions" condition exists: For any final destination A, there is a set of directions (e.g. take red three times, then blue twice, then red again) that gets you to A no matter where you started from. It may not be the shortest path, but you'll get there. There is one obvious obstruction and one slightly less obvious obstruction: If your network is disconnected so you can't get to A from your starting point no matter what path you follow, you're clearly stuck. On the other hand, there's also a "periodicity" obstruction if all of the cycles of the graph are multiples of the same number. For example, suppose that you were trying to give universal directions for a square, where the roads in question connected every vertex to its two neighbors. If I want to go from my starting point to an adjacent vertex, I have to take an odd number of steps. If I want to go from my starting point to the opposite vertex, I have to take an even number of steps. This means I can't even know how many steps I have to take (let along which steps) unless I knew where I started. It was conjectured, and Trahtman showed, that these two are the only possible obstructions. In particular, he even gives an algorithm for figuring out how to label the roads quickly. I guess the applications I'd see in this are for algorithms and the design of autonomous systems. The idea here is that, if the robot gets stuck somewhere in a multi-step procedure, you may want it to restart from the beginning. However, this can be difficult if the robot does not know where it is in the procedure, or even where it is physically. Trahtman's algorithm could allow you to exchange some computation at the beginning of the procedure (which can be done before the robot goes out, for example), for this "reset" functionality. I don't know whether this is feasible or not though.

I knew it!

[Teilo (91279)]

Proof at last, that that guy who kept saying, "You can't get there from here" is a bloody liar.

and the evil professor says ...

[B3ryllium (571199)]

Now adjust the algorithm for colour-blindness! :)

Show your work.

In related news ....

[PPH (736903)]

Based upon his ground-breaking research on the road coloring problem, night watchman Trakhtman has been promoted to the Department of Transportation pavement striping crew.

Re:XKCD

[by Anonymous Coward]

No, today's XKCD is about the traveling salesman problem. They are very different.

Re:

[Jawbreaker4Fs (974108)]

Yeah... this would only apply to the XKCD comic if the road coloring problem was NP-complete.

Re:

[darthflo (1095225)]

I wouldn't consider myself mathematically literate, but applying the findings of Trakhtman to navigation instructions in the real world would, if I understand the theorem [wikipedia.org] correctly, make finding any point anywhere as easy as "at intersections, follow the pattern red-blue-blue".
What I don't quite get, is the efficiency of this. The WP example looks like, transferred to the real world, a trip from Ohio to Cleveland may very well go through Indiana and Pittsburgh. Not what I'd consider efficient/fast routing.

Re:

[notgm (1069012)]

not fast, but reliable. if you don't care what time you get in to cleveland, this guarantees you'll never *end* up in pittsuburgh.

wait, ohio to cleveland?

Re:

[alcmaeon (684971)]

True, you will never end up in Pittsburgh unless you finally tire of traveling, but you may pass through Pittsburgh many times on your way to Cleveland. Hell, you might pass though Lima, Peru on your way to Cleveland, Ohio. For that matter, you might pass though Cleveland, Ohio itself several times before you complete the trip according to the instructions.

Re:Yeah, yeah, First Post, but...

[gomiam (587421)]

I am not a mathematician either, but I think being able to provide a pattern that allows you to reach any given point in the graph would allow for faster switching at the nodes (once you reach certain speeds switching becomes the bottleneck). The problem isn't concerned with efficiency but reachability, anyway.

Re:

[ColdWetDog (752185)]

Yah..I thought the airlines solved this long ago with their baggage systems.

What airlines have you been using lately? Not the ones I fly, certainly.

Re:Night Watchman?

[Yvanhoe (564877)]

In Soviet Russia they say that because Americans were so poor mathematicians, they had to invent the computer...

Re:Night Watchman?

[Applekid (993327)]

It's way better to be good at math, though, if you have to pick one. Sure, computers are quite practical for solving difficult problems, but you don't get nearly as much insight from simulation as you do from deriving an analytical formula for something (assuming such a formula is possible).

True, but the math needed to draw one frame of Doom alone took me a week, and I still haven't finished filling the boxes on my sheet of graph paper.

HE'S NOT A WATCHMAN

[mcmonkey (96054)]

If you RTFA (yes, I must be new here) he worked as a night watchman when he first moved to Isreal. He's been working in mathematics for over a decade.

Originally from Yekaterinburg, Russia, Trahtman was an accomplished mathematician when he came to Israel in 1992, at age 48. But like many immigrants in the wave that followed the breakup of the Soviet Union, he struggled to find work in the Jewish state and was forced into stints working maintenance and security before landing a teaching position at Bar Ilan in 1995.

You might as well say the 2002 Nobel prize in economics went to a lieutenant in the army. It's just a minor detail that he was a lieutenant 50 years before winning the prize.

Re:

[BattleApple (956701)]

It's just like the "psychologist" that entered the Netflix contest
http://developers.slashdot.org/article.pl?sid=08/03/04/2348257 [slashdot.org]

Re:Night Watchman?

[reset_button (903303)]

According to Wikipedia [wikipedia.org], he is a mathematician at Bar-Ilan University in Israel. Here is his homepage [biu.ac.il] hosted by the university. Maybe he was a night watchman, but it looks to me like he's a professor now...

Re:Night Watchman?

[DirePickle (796986)]

He's in Israel, not the US, dude. Save your diatribe for another time.

Re:

[qw(name) (718245)]

Reread what the OP said.

The person to whom you replied never said he was Israeli. He said "He's in Israel."

Re:

[DavidShor (928926)]

This happened in Israel, not the America.

And in my experience, you mainly see that America-only stuff in the public sector(and even then, not too often). The business world tends to know better.

Re:

[by Anonymous Coward]

Come on AC, your premise is patently false. I'm at one of the finer educational institutes in the US, and we have people from all over the world who earned their Ph.D.s from all over the world. If you came from one of the highly respected universities throughout the world, and are brilliant, you will get a job. If you have your Ph.D. from the University of Lower Slobobia, of course its going to be difficult. The quality of education in places in Asia, India, and even Russia varies wildly from their best Uni

Re:

[Sparks23 (412116)]

Trakhtman is a Russian Jew who immigrated to Israel in 1992 after the breakup of the Soviet Union, and the article says that he had trouble finding work during the influx of refugees. Not because of his background, but because of a sudden unemployment glut. This is why he previously ended up serving as a night watchman, a laborer, a maintenance worker or whatever else he did.

However, the article ALSO says he's been teaching mathematics at Bar Ilan University since 1995; the university is where he solved t

Re:

[pembo13 (770295)]

I'd be curious to see a digitized copy of his notes

Re:OK, but...

[KDR_11k (778916)]

Forty-two.

Re:

[ScrewMaster (602015)]

How many paths must a man walk down before you can call him a man?

Well, unfortunately for all us Free Worlders out there, that path is apparently a Red one.