## The Problem

I currently consult a local startup company that uses IoT sensors to monitor the status of an organizations trashcans and dumpsters. More important than trashcans is likely dumpsters, however, since more money is wasted by collecting an empty dumpster as opposed to an empty trash can 50 feet away from the next.

IoT systems create lots of data, which can be really fun to analyze. I have been doing a lot of data science lately, and I realized there are a lot of features we can add that may help the trashcan situations. Let’s say that I have a nice campus, like Disneyland, and I want to make sure that if someone has trash, they can throw it away. How can I ensure that I have my trashcans in the right place?

## Method

We are going to look at several similar existing problems to see if there is already a good way of solving our problem. Likely its been solved, but there’s a chance that it hasn’t been formalized and we will have to adapt an existing problem or algorithm to our situation. The goal here is to fully understand our problem, it’s complexity and it’s simplicity. If we can avoid a complicated algorithm, its best to do that.

## Set Cover Problem (NP-Complete)

Immediately, this seems to be an NP-Complete problem. I need to cover another set by adding a trashcan. This scared me at first, however, since Set-cover is a known NP-Complete problem. So with the 100’s of trashcans that Disneyland has, we’re exceeding practical run times. But then I realized that’s not my input-set cardinality, that’s my solution set size. How can we choose a way to represent the sets? Maybe we can triangulate between restaurants and rides to call each significant point a point in our sets and specify a bound of 50 feet to allow a trashcan cover a ride, but that doesn’t take into account the amount of trash or popularity of the area. So I decided that this wouldn’t be the best-fit problem to think of.

## The Art Gallery Problem (NP-Hard)

Then I found the art gallery problem, which is simply this: Where can I put the minimum number of security guards to monitor every area in a museum. This works well for our problem, since we are an optimization problem, dealing with location and placement, but again, this doesn’t deal directly with popularity, i.e. the Mona Lisa wouldn’t be any more heavily guarded than a child’s hand-turkey in these problems. This problem is also NP-Hard, which means that if we could simplify this problem we might be able to find a good algorithm for it, but thinking of our problem like this will prove to make it inefficient to solve.

## K-means Clustering

During my Big Data class, we talked about K-means clustering, which could be a great method of visualizing the data. If we plot the quantity of trash produced as a data-point, with one point for each unit, and the x and y coordinates are simply location, we can create some good clusters to see areas where we are not currently collecting on a map, and given a high density of data points(every trashcan’s data is in the same spot), if we force our algorithm to add a cluster (\(k = k + 1\)) then one of the clusters should divide, right? We will revisit clustering.

## Bubble Charts

I bring this up, as this is how I started to visualize the problem. The size of the bubble represents the amount of trash generated in an area, the location of the center of the bubble is the location (lat/lng) of the trashcan. So if I have two distant bubbles that overlap, I need to insert another trashcan in the middle. If I have a bubble that gets too big, I need to add another trashcan nearby to alleviate the load on the other trashcan. The problem here is this isn’t an algorithm or problem, just a visualization. But now we have something we can use set-cover and k-means on. Using a bubble chart, we can plot points proportionally using a gaussian normal distribution and create a dataset that we can then cover with a density based set-cover algorithm or some density-based clustering algorithm.

## DBSCAN

This algorithm is great for density-based clustering. DBSCAN will, given an epsilon and min-points, cluster the points accounting for noise, and add clusters as needed. This can be used to analyze how the trash collection is currently, answering the question, “Do I need to add another trashcan?” If the number of clusters produced by a DBSCAN don’t match the actual number of trashcans, we have an abnormal distribution of density in some area. This assumes, however, that you don’t have any trashcans in the same place, or side-by-side.

This algorithm may help us determine if we need to add another trashcan, based on density. But ultimately, that’s over complicating the issue. Why not just look at how fast a trashcan is filling up? That’s not our only question here. We need to know where to place the next trashcan.

## Conclusion

Our algorithm will be as follows:

- We look at the frequency of collection for all trashcans.
- We compare that to pre-established thresholds.
- We determine who the “neighbors” are for the trashcan that exceeds the threshold.
- We generate random points(Gaussian distribution) using a proportional number of points per trashcan to trash generated per trashcan.
- Run K-means on the existing dataset with the three centroids being the current trashcan locations
- Add a new centroid and rerun K-means to determine the optimal placement for the containers.
- Rerun 3-6 for each trashcan exceeding the threshold.

Analyzing this algorithm we just created, our most complex part is K-means which has a complexity of \(O(k*N*T)\) where T is the number of iteration, k is the number of clusters, and N is the number of samples. This will be repeated for every trashcan, so in worst case, our algorithm will likely be \(O(k^2*N*T)\) Not bad for a set-cover type problem.

This solution ignores infrastructure concerns, such as, “that’s a building, I can’t put a trashcan there” but gives a good solution if we are looking at a campus type location.