Friday, April 22, 2005

There and back again

Anyone who is serious about walking in Manhattan knows that if you're going diagonally across and either up or down, there's one rule you must obey: always go with the light.

If you don't know, the streets of Manhattan are laid out in a grid. So getting going up or down town is easy--a straight line. Same with going cross town. But if you need to get across and either up or down, you've got to zig-zag. And this zig-zagging gives you a whole bunch of route choices. Do I head east first? Do I go down a block and then over? The rule makes the choices for you. You go where the green lights lead you.

This is the situation I face about twice a day as I make my way from Penn Station at 33rd and 7th to my office at 42nd and 2nd and then back again. So everyday I put the rule to the test. And everyday, I get a sneaking feeling that the rule is wrong. Sometimes the rule forces me into one side of the box that circumscribes my diagonal. I may end up on 42nd and Park with no choice but to continue straight down 42nd. More often I get to 2nd ave. at around 38th, so it's up 2nd to 42nd.

When I get into these forced choices and then get stuck at a long light, the question starts to nag. If I had just waited at that light back there could I be still making progress here? The rule works on maximizing the local choices. But it can't look ahead at the whole range of choices. And therein lies the rub.

There must be a mathematical proof whether the rule actually is the best way to get around Manhattan or not. During my walk I sometimes imagine creating a computer simulation of my walk and running different versions of the rule a bunch of times to compare their outcomes. Then I remember I'm not a computer scientist and I've got too much to do anyway.

Maybe someone has already done this. Any mathematicians out there want to comment?


boredoom said...

The practice is known as "checkerboarding," and allows skillful cabbies to zip through the city without stopping, as long as it's diagonally. It's kind of like we're all rooks.

I was about to say that this may mean that walking diagonally is faster than walking orthogonally the same distance, but that's probably not so - we still pay a penalty for zig-zagging with the blocks when going diagonally.

boredoom said...

Thinking more about it, I realize I used a modified form of checkerboarding. I use the rule "Always cross an avenue when you can." For those of you who don't know, the north-south running avenues are broader and further apart than than the east-west running streets.

I can often jaywalk across streets, but never across avenues. So checkerboarding is usually a matter of walking along an avenue till I get a green light to cross it. Then I walk over to the next avenue and walk along it till I can cross it.

I don't think you need to worry about whether waiting out some lights is more efficient than checkerboarding. As you note, sometimes you end up at one edge of the square that circumscribes the route. Walking along the sides of the square is equivalent time-wise to following any preset route within that square, waiting out the lights at each crossing. But most of the time that doesn't happen when you checkerboard, which should mean that on average you save time.

liberal elite said...

Interesting that it's "checkerboarding" when we're restricted to moving like rooks!

I like the Avenue codicil. But I'm not sure if it's necissarily true that you will save time on average by checkerboarding. That's why I was thinking of the computer simulation as a way to test the hypothesis. I also thought about keeping a log of my walking times everyday and adhering strictly to the checkerboarding principle--I admit that I don't always. If I were nerdier I would have done this long ago...

boredoom said...

The only way I see to beat checkerboarding is to have advance knowledge of when the signals switch. Given that the avenue signals are synchronized, that may be possible. There might be a certain path that will beat the average checkerboarding time based on the relationship between your walking speed and the signal switch times.

liberal elite said...

It may be that using some simple pattern, like a straight, L will consistently beat checkerboarding if you have luck with the lights. Checkerboarding definitely beats the worst case scenario: you hit every light. But it's not obvious that waiting early couldn't pay off in the end.

Anecdotally both last night and this morning, I did checkerboarding. I didn't hit the edges of the board until I was 2 or 3 blocks from my destination. So I definitely did ok by it.

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