What is the most efficient way to walk across Manhattan? 
Specifically, what is the heuristic you should use when coming to an
intersection? (For example, do you always cross when given a green
light or do you wait until you are forced to cross?)  Also, what is
the  potential time savings by using the best method? (or are all
methods the same?)

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Request for Question Clarification by pafalafa-ga on 25 Apr 2006 12:36 PDT

pbandj1599-ga,

I'm not sure just what type of information/evidence you're looking for.

As an experienced New York pedestrian, however, I can tell you that
the biggest difference in walking times boils down to whether you walk
along -- or avoid -- the most crowded streets.

There's a big difference between trying to cut across town on 34th
Street (crowded) or 32nd St. (not so bad).  So too with other really
crowded streets -- 42nd St, 23rd St, parts of 5th Ave, Times Sq, etc,
etc.

Stick to the roads less traveled (or walked) and you'll have the
fastest route available.

What do you  think?

paf

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Clarification of Question by pbandj1599-ga on 25 Apr 2006 12:48 PDT

I was asking a more hypothetical / programming question.  Let's assume
that all streets are equally busy and all avenues are equally busy
(but waiting for an avenue light is longer than waiting for a street
light).  Given that simplification, what is the best way to go about
walking.  The answer I'm looking for should have some quantitative
reasoning behind it.

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Clarification of Question by pbandj1599-ga on 25 Apr 2006 14:42 PDT

I think by posing the question as "across Manhattan" I may have sent
the question in a direction I did not intend.  I assume the heuristic
(or some slight deviation from it) should work in any city with
north/south and east/west blocks.  I used Manhattan as an example of
city laid out as such, but I never meant, specifically in Manhattan.

As an example, I assume that one of the slowest ways would be to walk
east/west until you get to the right longitude and then walk
north/south until you get to the right latitude.  Walking this way
would create an "L" shape but would likely lead to one of the highest
numbers of times of getting stuck at a crosswalk.  I assume some sort
of criss-cross pattern (where you alternate in some thought out way
when to travel north/south and when to travel east/west and when to
wait if necessary for a light to change).  Even if there is only a
slight difference between the criss-cross and the "L", that is the
relatively computational interesting question I am looking to solve. 
There must be an optimum heuristic to follow.  I am looking for that
heuristic and if possible, a bonus would be what the quantifiable time
savings would be between the best heuristic and the worst.

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Clarification of Question by pbandj1599-ga on 25 Apr 2006 19:42 PDT

To clarify points from below:
Questions:
(1) how many streets versus avenues are you trying to travel
(2) how do street blocks and avenue blocks compare in length
(3) what are the speeds you can travel on streets versus avenues (I
assume you can travel faster on streets because avenues are busier
because the lights on avenues are longer, but how much faster)
(4) what are the lengths of lights on streets and avenues (we know
lights on avenues are longer, but by how much)

Response: 
I'm looking for a solution which is general enough to work for issue
#1.  In other words, there should be variables in solution for how
many streets and/or blocks.

For simplicity purposes, let's assume that 1 avenue block is 3.5 times
a street block.  And let's assume that besides the length differences,
time spent traveling an avenue or a block is equal (it takes equal
time to walk 1 avenue or 3.5 streets). And the length of a walk light
is 50 seconds at a street and 30 seconds on an avenue.  Also as a
point of clarification assume there is no "timed" lights so the point
of the light cycle when you get to a corner is random.  (Of course if
someone can come up with a more general solution that accounts for all
those variables that would be amazing!)

Walk it and find out.  Take a stop watch or two.

I would take a cab.

Pbandj1599--
Well, as any good Manhattan-ite and taxi driver knows, stop lights are
sequenced the best on AMSTERDAM AVENUE, FIRST AVENUE, 11TH AVENUE --
among others -- so you can drive / walk for longer before hitting a
light.

"Best kept secret" STREETS are 56th Street going East and 31st Street Going West. 

A tip to Google Researchers, look up "Sam Schwartz" aka "Gridlock
Sam," who writes a column in the New York Daily News. For years he has
devised and tested all kinds of traffic shortcuts for New Yorkers.
It's possible he may have even discsussed how to efficiently walk
Manhattan.

P.S. PBandJ -- I read your question to a few of my office colleagues,
who said, "Don't forget to factor in getting mugged and being asked
for directions by tourists!"

Don't take a cab, it's faster walking. ;)

Paf may correct me, but my recollection is that it takes 30 seconds to
walk an up-/downtown block  - street to street -  and 180 seconds for
a crosstown block  - avenue to avenue, plus time for any lights.
If you are moving diagonally, crossing several streets and avenues,
jaywalk in the middle of a block when crossing a street, avoiding
waiting for the light, and fairly safe, since they are all oneway.  
Wait for the light when crossing most avenues, unless it is a Sunday
in July or August  - or unless you have trained in Paris traffic.

The time saving is avoiding the lights when crossing streets, but
maybe some of them will be green anyway.  Jaywalking diagonally across
streets won't reduce the time much (walking the hypotenuse instead of
crossing the street and walking the length of the block) because you
will have to slow down to dodge traffic.

If you can incorporate Paf's suggestions of heavily walked streets to
avoid, you might be able to calculate approximate times for any route.

You have clarified that this:

(1) is an academic exercise and doesn't specifically refer to Manhattan,
(2) all streets are equally busy and all avenues are equally busy (but
I assume, avenues and streets are not equally busy)
(3) traffic lights are longer on avenues than streets

The answer still depends on:

(1) how many streets versus avenues are you trying to travel
(2) how do street blocks and avenue blocks compare in length
(3) what are the speeds you can travel on streets versus avenues (I
assume you can travel faster on streets because avenues are busier
because the lights on avenues are longer, but how much faster)
(4) what are the lengths of lights on streets and avenues (we know
lights on avenues are longer, but by how much)

The length of a walk light is a bit more complicated.  You may start
crossing as long as it is green, but you will have to wait while it is
red, which on avenues will probably be longer than the green phase. 
So on average, you have to wait half of the red phase,
No, correction!  On average you will arrive in the middle of the
complete red-green cycle.  Let's say, 30 seconds green, 90 seconds
red, total 120 seconds.  You arrive at a random time in the cycle, so
you will on average have a 25% chance of catching the green phase and
75%, the red, so I expect that you would wait on average 1/2 of 3/4 of
the 90 second red phase = 34 seconds.
(This is very much open to correct! :)

In your rectilinear city plan, moving diagonally is still a series of
right angle steps, in principle the same distance as walking up the
avenue till you get to the desired crosstown street.  But by
jaywalking and taking slightly diagonal paths from the curb to the
inside of a corner where you won't be cross traffic, you may be able
to shorten the distance somewhat.  In real life, this is probably only
theoretical  (maybe not in Indianapolis :)  because with any volume of
ambling pedestrians, you will find it faster to pass people,
disregarding the geometric advantage of walking the optimum diagonal.

So, assuming as before that you accept jaywalking streets, I reckon
that you can avoid all street lights but on average must wait for
avenue ligts by whatever the correct formula is for the actual
red-green phases.  I have discounted any advantage of diagonals along
the streets, which with the longer blocks between avenues would be
fairly minimal, anyway.  Thus the time would be your mph tempo applied
to the distance straight uptown and crosstown, plus the average time
for avenue lights.

Of course, with some experience with the lights, you could probably
avoid waiting for a couple by continuing up the avenue to the next
street, anticipating correctly that when you got there the light would
be green for you.

How's that?   Now some one correct my waiting time calculation.