Do you park in the first place you come across when you arrive at a parking lot? Do you spend a while looking for the “perfect parking lot” Today mathematics moves away from the movements of the stars and the rhythms of nature to shed light on a mundane decision: they offer us the answer to the best way to park.
This classic optimization problem has received a new update thanks to the physicists Paul Krapivsky of Boston University (USA) and Sidney Redner of the Santa Fe Institute of New Mexico (USA) that has been published in the journal Journal of Statistical Mechanics.
The problem has to do with something that many of us can relate to when we are overwhelmed or desperate to find a parking space. And this is the key: the best parking space is the one that minimizes the time we spend in the parking lot . So the gap next to the front door is ideal, unless you have to go around three times to get it. In order to reduce the time spent driving through the parking lot and the consequent walk through it, an efficient driver must decide whether to go through the confined space, park further away, or settle for something in between.
“Mathematics allows you to make smart decisions,” says Redner. “It allows us to approach a complex world with some ideas.”
Three parking strategies
In their article, Krapivsky and Redner outline three simple strategies for parking in an idealized single-row parking lot. They are summarized in: submissive, prudent and optimistic.
Drivers who take the first available space follow what the authors call a strategy“submissive”. “They don’t waste time looking for a place to park,” leaving spaces near the driveway unfilled. Those who bet to find a space right next to the entrance are“optimistic”. They drive all the way to the entrance, then back up to the nearest gap. The drivers“prudent”they take the middle road. They drive beyond the first available space, betting on the availability of at least one other space further away. When they find the closest space, they take it. If there are no gaps between the farthest parked car and the driveway, prudent drivers back into space that a submissive driver would have immediately claimed.
Despite the simplicity of the three strategies, the authors had to use multiple techniques to calculate their relative merits.Oddly enough, the submissive strategy reflected a dynamic seen in microtubules that provide scaffolding within living cells. A car that parks immediately after the farthest car corresponds to a monomer projecting at one end of the microtubule. The equation that describes the length of a microtubule, and sometimes a dramatic shortening, also describes the chain of “tame” conductors that accumulate at the other end of the lot.
“Sometimes there are connections between things that seem to have no connection,” says Redner. “In this case, the connection to microtubule dynamics solved the problem.”
To model the optimistic strategy, the authors wrote a differential equation. Once they started to mathematically express the setting, they spotted a logical shortcut that greatly simplified the number of spaces to consider.
The prudent strategy, according to Redner, was “inherently complicated” given the many spaces at stake. The authors approached this by creating a simulation that allowed them to calculate, on average, the average density of points and the amount of recoil required.
Which is the best option?
According to experts, as the name suggests, it would be the prudent strategy . In general, this way of parking costs drivers the least amount of time, closely followed by the optimistic strategy. The submissive strategy was “laughably inefficient,” as the sheer number of empty spaces it left led to a long walk to the entrance.
Redner acknowledges that the optimization problem sacrifices a lot of real-world applicability in exchange for a mathematical insight. Leaving out the competition between cars, for example, or assuming that the vehicles follow a uniform strategy in each scenario, are unrealistic assumptions that the authors could address in a future model.
“If you really want to be an engineer, you have to consider how fast people drive, the actual designs of the parking lot and the spaces,” he says. “Once you start to be completely realistic, every parking situation is different and you lose the ability to explain anything,” concludes the expert.
Referencia: P L Krapivsky, S Redner. Simple parking strategies. Journal of Statistical Mechanics: Theory and Experiment, 2019; 2019 (9): 093404 DOI: 10.1088/1742-5468/ab3a2a
