Empty Pipes

A few months ago somebody asked the following question on AskReddit:

“If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around?”

As usual, the comments were highly informative with various users running simulations to show that two people would generally find each other faster when both moved around. The question was lost in the detritus of my mind until a recent late-night conversation after a bioinformatics conference. The context in this conversation was different but the underlying question was the same.

“Are two proteins more likely to encounter each other when both are mobile or when one is attached to a cell membrance while the other is free to float around the interior of the cell?”

After bringing up the reddit post, I tried my best to explain why it would make sense that fixing the position of one person (or protein) would make it easier for the other to find it.

Alas, my memory had failed me and this conclusion was contrary to that suggested by the reddit comment simulations. What struck me, however, is that I couldn’t make a coherent case for why standing still is better, and wasn’t convinced by the other’s argument that moving around was better. Frustrated, I decided to make a little tool to perform a simulation of my own and to illustrate the phenomenon. So, without further ado, here we go.

Scenario 1: One person waits

One person (green circle) waits, while the other (red circle) searches. Both start at random positions at the start of each simulation.

Scenario 2: Both people move

Both people (green and red circles) move around. Both start at random positions at the start of each simulation.

Verdict

To save you some time, I took the liberty of repeating these simulations for a variety of grid sizes and recorded the results.

Number of Moves Required to Meet

It’s clear that the better strategy is for both people to move at random than to have one person wait while the other moves randomly.

But wait…

Is this really a realistic scenario? How often do people just move around randomly? In most cases, when people are looking for something, they search in some methodic manner. The rest of this post will explore two fundamentally similar strategies and how they affect the search time.

Scanning Strategy

The first strategy involves simply scanning the grid up and down, left to right.

Avoiding Strategy

The avoiding strategy involves keeping track of where the person has been, and at every point, visiting the least visited neighbor. If there is more than one least visited neighbor, pick one at random and continue.

Centering Strategy

A reddit user recently pointed out that a reasonable strategy might be for one person to just go to the center because there they would have a larger chance of being found by randomly moving person. An illustration of this appears below where the red person moves to the center and green person wanders around randomly.

Pairwise Comparison of Strategies

To determine what the best course of action is when two people are separated somewhere, we need to do a pairwise comparison of the strategies on an 8x8 grid:

Median Moves Requires to Meet

Generally, standing still is the best strategy if the other person is not moving around randomly. Even if they are, the best strategy appears to be to first go the center and stay put there. Moving around randomly is only optimal if the other person is doing an avoiding walk. Scanning and avoiding work well in general with one major caveat: if both people are scanning, it can lead to a situation where the two peole never meet, no matter how long they walk.

The avoiding strategy avoids (heh), this outcome by introducing a little bit of stochasticity into the searching process by picking a random position to go to when there are equally many least visited options available.

Conclusion

The best strategy is for people to choose a point at which they meet if they get lost. When both people go to the center, it takes a median of 3 moves and will never take more than four moves (on an 8x8 grid).

Often times, however, this doesn’t enter the conversation and an ad-hoc strategy must be selected. The second best option is for one person to stop moving and for the other to scan the search area looking for them. This also requires a certain amount of prior coordination to determine who will stop. If both people decide to stop, then it stands to reason that they’ll never find each other. Thus, when there is no agreed upon strategy, it appears that the best choice is to walk around avoiding places that you’ve already been and taking a random turn here or there.

Finally, if you’re really lazy it pays off to walk to the center and wait there. This will actually lead to a faster rendezvous when the other person is walking around randomly than walking around randomly yourself.

Good luck!

Read on for a look at the distributions of finding times and the application that made all of this possible.

Appendix

The Distribution of the Number of Moves Required to Meet

	Standing	Random	Scanning	Avoiding	Centering
Standing
Random
Scanning
Avoiding
Centering

Key Points from the Histograms:

• Having one person sit still severly restricts the maximum number of moves that need to be taken if the other person is using the avoiding or scanning strategy.
• One person standing and the other moving randomly can lead to an exorbitantly long search time.
• Both people standing or both people scanning can lead to infinite search times.
• Most of the time, the search will be faster rather than slower, but the times can vary a lot. Much of the bias toward faster search times is actually due to the fact that the starting points are chosen randomly, and closer starting points are more likely than further starting points.

In Three Dimensions

A few people wondered how different the results would be on a 3D grid. It’s large-ish step to add 3D rendering to the current animations, but changing the code to use 3D dimensions wasn’t a huge deal. The results are in the table below.

Median Moves Requires to Meet

The results a roughly analagous to those on the 2D grid. Naturally, it takes more moves for people to find each other in 3D, but the best strategy remains standing - avoiding, and moving in a random fashion is better than standing when the other person is also moving randomly. For those that are interested, the histograms below show the distribution of search lengths on a 3D grid.

The Distribution of the Number of Moves Required to Meet on a 3D Grid

	Standing	Random	Avoiding
Standing
Random
Avoiding

Make Your Own Simulation

Here’s an application that you can use to run your own simulation.

Source

The source code for the application to run these simulations can be found on github.

    var breaks = [0, 4.5, 9, 13.5, 18];
    var data = [[18, 13, 10, 9, 10, 13, 18],
        [13, 8, 5, 4, 5, 8, 13],
        [10, 5, 2, 1, 2, 5, 10],
        [9, 4, 1, 12, 1, 4, 9],
        [10, 5, 2, 1, 2, 5, 10],
        [13, 8, 5, 4, 5, 8, 13],
        [18, 13, 10, 9, 10, 13, 18],
        [18, 13, 10, 9, 10, 13, 18]];