Tuesday, November 6, 2007

The Most Efficient Walking Path Between Two Points on the Manhattan Grid

By: Abhas Gupta
    Note: Yes, this isn't an insight into Health care economics. I'm so shell shocked from the OB/Gyn shelf that I'm desperately looking for creative outlets. Stay tuned for two articles that I hope to post shortly--the first is on medical malpractice and the second is on the health care payment system.

    Like many New Yorkers, I think about the same question every time I get out of the subway: What is the fastest way to get to where I'm going? And every time, I simply walk in the general direction of my destination, letting traffic lights dictate my specific path. I asked my better half about her method to getting places quickly and she replied, "It's easy! Just time the lights." Although this may be good advice for the seasoned New Yorker traveling at the speed of traffic, I suggest the following, more generalizable approach.

    First, let's break the problem down into three simple cases:

    Case 1 - the two possible paths here are of the same distance and have no additional variables (traffic lights, stop signs, etc.); therefore, both paths are equivalent.

    Case 2 - the two possible paths here involve crossing one street and one avenue. The order is determined by which direction has the green light first.

    Case 3 - here, all of the possible paths have the same distance and the same number of "obstacles" (two avenues and two streets to cross). This example can be easily extended to more complicated cases. In fact, all paths on an ideal grid--one where all streets are parallel, all avenues are parallel, and all streets are perpendicular to all avenues--have the same distance. Coincidentally, this distance is called the Manhattan distance. So how does one determine the most efficient path?

    First, we need to identify the inherent differences between crossing a typical street and a typical avenue in Manhattan. Streets frequently have less traffic and fewer lanes. Streets can also be easily crossed when a light is red. Avenues, on the other hand, typically have more traffic and more lanes, making it difficult to cross unless the light is green. Finally, lights for avenues remain green for a longer period of time, so multiple streets can be traversed without stopping at a light, whereas the same is not the case for multiple avenues.

    The fastest path between any two points obviously avoids getting stuck at a light. The solution then boils down to which path has a lower probability of getting stuck at a light. Since streets are easier to cross than avenues, the fastest path would lend priority to crossing avenues (since they are more difficult to cross). Also, depending on the distance that needs to be traveled, waiting a few seconds at an intersection in order to cross an avenue could be advantageous over proceeding to the next intersection. To summarize, the most efficient walking path can be achieved by following these simple rules:
    1. When given an equal choice of direction (like Case 1), travel towards the next avenue (thereby maximizing opportunities to cross avenues)
    2. When the path requires crossing multiple avenues, consider waiting at an intersection to cross the avenue instead of proceeding to the next intersection
    3. Treat crosstown streets like avenues (since they have the same risk of getting stuck at a light)

    1 comment:

    Anonymous said...

    This has to be the nerdiest post I've ever read....to think this guy actually used to be cool in college...=)

    What has the world come to ?