# Teaching Recursion with the N Queens Problem

Posted in Computer Science

## A Gentle Introduction to Recursion

Recursion, particularly recursive backtracking, is far and away the most challenging topic I cover when I teach the CSE 143 (Java Programming II) course at South Seattle College. Teaching the concept of recursion, on its own, is challenging: the concept is a hard one to encounter in everyday life, making it unfamiliar, and that creates a lot of friction when students try to understand how to apply recursion.

The key, as I tell students from day one of the recursion unit, is to always think in terms of the base case and the recursive case. The base case gives your brain a "trapdoor" to exit out of an otherwise brain-bending infinite conceptual loop. It helps recursion feel more manageable. But most importantly: it enables thinking about recursion in terms of its inputs and outputs.

More specifically, to understand recursion requires (no, not recursion) thinking about two things: where you enter the function and when you stop calling the function. These are the two least complicated cases, and they also happen to be the two most important cases.

College courses move at an artificially inflated pace, ill-suited for most community college students, and the material prescribed must be presented at the given pace mostly independent of any real difficulties the students face (there is only minimal room for adjustment, at most 2-3 lectures).

This means that, before the students have had an opportunity to get comfortable with the concept of recursion, and really nail it down, they're introduced to yet another mind-bending topic: recursive backtracking algorithms.

These bring a whole new set of complications to the table. Practice is crucial to students' understanding, and all too often, the only way to get students to practice (particularly with difficult subject matter like recursion) is to spend substantial amounts of time in class. My recursion lectures routinely throw my schedule off by nearly a week, because even the simplest recursion or backtracking exercise can eat up an hour or more.

## Recursive Backtracking

Backtracking is an approach for exploring problems that involve making choices from a set of possible choices. A classic example of backtracking is the 8 Queens problem, which asks: "How many ways are there of placing 8 queens on a chessboard, such that no queen attacks any other queen?"

The problem is deceptively simple; solving it requires some mental gymnastics. (By the way, most people who have actually heard of the problem are computer scientists who were exposed to it in the process of learning how to solve it, leading to the hipster effect - it's often dismissed by computer scientists as an "easy" problem. The curse of knowledge at work.)

The recursive backtracking algorithm requires thinking about the squares on which to place the 8 queens in question as the set of choices to be made.

The naive approach ignores the constraints, and makes all 8 choices of where to place the 8 queens before ever checking if the queen placements are valid. Thus, we could start by placing all 8 queens in one single row on the top, or along one single column on the left. Using this approach, we have 64 possibilities (64 open squares) for the first queen, then 63 possibilities for the second queen, then 62 possibilities for the third queen, and so on. This gives a total number of possible combinations of:

$$\dfrac{64!}{(64-8)!} = 178,462,987,637,760$$

(By the way, for those of you following along at home, you can do this calculation with Python:)

>>> from scipy import *
>>> math.factorial(64)/math.factorial(64-8)
178462987637760L


Even for someone without a sense of big numbers, like someone in Congress, that's still a pretty big number. Too many for a human being to actually try in a single lifetime.

## Paring Down the Decision Tree

But we can do better - we can utilize the fact that the queen, in chess, attacks horizontally and vertically, by doing two things:

• Limit the placement of queens so that there is one queen per column;

• Limit the placement of queens so that there is one queen per row.

(Note that this is ignoring diagonal attacks; we'll get there in a minute.)

This limits the number of solutions as follows: the first queen placed on the board must go in the first column, and has 8 possible squares in which it can go. The second queen must go in the second column, and has 7 possible squares in which it can go - ignoring the square corresponding to the row that would be attacked by the first queen. The third queen goes into the third column, which has 6 open squares (ignoring the two rows attacked by the two queens already placed).

That leads to far fewer solutions:

$$8! = 40,320$$

and for those following along at home in Python:

>>> from scipy import *
>>> math.factorial(8)
40320


To visualize how this utilization of information helps reduce the problem space, I often make use of a decision tree, to get the students to think about recursive backtracking as a depth-first tree traversal.

(By the way, this is a strategy whose usefulness extends beyond the 8 queens problem, or even recursive backtracking problems. For example, the problem of finding cycles in a directed graph can be re-cast in terms of trees.)

So far, we have used two of the three directions of attack for queens. This is also enough information to begin an implementation of an algorithm - a backtracking algorithm can use the fact that we place one queen per column, and one queen per row, to loop over each row, and steadily march through each column sequentially (or vice-versa).

## The Pseudocode

There is still a bit more to do to cut down on the problem space that needs to be explored, but before we do any of that, we should first decide on an approach and sketch out the psuedocode.

The structure of the explore method pseudocode thus looks like:

explore(column):
if last column:
# base case
else:
# recursive case
for each row:
if this is a safe row:
place queen on this row
explore(column+1)
remove queen from this row


## The Actual Code

Over at git.charlesreid1.com/charlesreid1/n-queens I have several implementations of the N Queens problem:

## Row, Column, and Diagonal Attacks

We have already utilized knowledge that there will only be one queen per column, and one queen per row. But one last bit of information we can utilize is the fact that queens attack diagonally. This allows us to eliminate any squares that are along the diagonals of queens that have already been placed on the board.

How to eliminate the diagonals? It basically boils down to two approaches:

1. Use a Board class to abstract away details (and the Board class will implement "magic" like an isValid() method).

2. Hack the index - implement some index-based math to eliminate any rows that are on the diagonals of queens already on the board.

The first approach lets you abstract away the details, possibly even using a Board class written by a textbook, which is lazy fine, if you are working on a practical problem and need some elbow grease, but not so much if you are a computer science student learning the basic principles of software design.

The second approach requires some deep thinking about how the locations of the N (or 8) queens are being represented in the program.

## Accounting for Diagonal Attacks

At some point, when you use the above pseudocode, you are going to want to know the answer to the following question: for a given column k, what rows are invalid because they are on diagonals of already-placed queens?

To answer this, think about where the diagonal indices of chess board squares are located, and how to find the diagonals on column X attacked by a queen placed in column Y.

The following diagram shows a queen on row 3 of column 2, and the diagonal attack vectors of that queen. Each of the squares along those diagonal vectors can be ruled out as possible squares to place a queen. When selecting a square for the third queen, which goes in the third column, the second and fourth rows can both be ruled out due to the diagonals. (The third row, of course, can also be ruled out, due to the one-queen-per-row rule.)

However, the effect of the already-placed queen propagates forward, and affects the choice of possible squares for each queen after it. If we jump ahead in the recursive algorithm, to say, queen number 6, being placed on column number 6 (highlighted in blue), the queen in column 2 (row 3) still affects the choice of squares for that column (as do all queens previously placed on the board). In the case pictured in the figure, the seventh row (as well as an off-the-board row) of column 6 can be ruled out as possible squares for the placement of the 6th queen.

Accounting for these diagonal attacks can lead to substantial speed-ups: each queen that is placed can eliminate up to two additional squares per column, which means the overall decision tree for the N queens problem becomes a lot less dense, and faster to explore.

## Why the N Queens Problem?

Invariably, some students will deal with this difficult problem by questioning the premise of the question - a reasonable thing to wonder.

This leads to a broader, more important question: why do computer scientists focus so much on games?

Games, like computers, are self-contained universes, they are abstract systems, they remove messy details and complications. They allow you to start, from scratch, by setting up a board, a few rules, a few pieces - things that are easy to implement in a computer.

Mazes, crossword puzzles, card games, checkers, chess, are all systems with a finite, small number of elements that interact in finite, small numbers of ways. The beauty of games is that those small rule sets can result in immensely complex systems, so that there are more branches in the chess decision tree (the Shannon number, $$10^{120}$$) than there are protons in the universe (the Eddington number, $$10^{80}$$).

That simplicity is important in computer science. Any real-world problem is going to have to be broken down, eventually, into pieces, into rules, into a finite representation, so that anything we try to model with a computer, any problem we attempt to solve computationally, no matter how complex, will always have a game-like representation.

(Side note: much of the literature in systems operations research, which studies the application of mathematical optimization to determine the best way to manage resources, came out of work on war games - which were themselves game-ified, simplified representations of real, complex systems. Econometrics, or "computational economics," is another field where game theory has gained much traction and finds many practical applications.)

Recursion, too, is a useful concept in and of itself, one that shows up in sorting and searching algorithms, computational procedures, and even in nature.

But it isn't just knowing where to look - it's knowing what you're looking for in the first place.

# Undergraduate Research Project: Wireless Sensor Networks for Internet of Things Applications (Part 2: The Technologies)

Posted in Wireless

# Undergraduate Research Project (UGR): The Technologies

In this post we'll cover some of the technologies that were used in our South Seatte College undergraduate research project. The project involved an ensemble of different technologies to complete each component of the data analysis pipeline. Some components were planned for, but other components were implemented due to "surprise" challenges that cropped up during the course of the project, while yet more technologies were integrated into the pipeline to avoid extra costs.

## Overview of the UGR Project

Before we go further, let's recap what the project was all about. As the research project mentor, I was leading a group of five undergraduate students in a project entitled "Wireless Sensor Networks for Internet of Things Applications." This involved guiding students through the construction of a data analysis pipeline that would utilize a set of sensors, each collecting data about wireless networks in the vicinity, and collect the data into a central database. We then impemented data analysis and visualization tools to analyze the sensor data that was collected and extract meaningful information from it.

There were three major sets of tools used - those used onboard the Raspberry Pi sensors (to extract and transfer wireless data), those used to store and organize the wireless sensor data (NoSQL database tools), and those used to process, analyze, and visualize the data colleted (Python data analysis tools).

The technologies used can be classified two ways:

• Student-Led Components - the software components of the pipeline that students learned about, and whose implementation was student-led.

• Backend Components - the software components of the pipeline that were too complicated, too hairy, and/or too extraneous to the project objectives to have students try and handle. These were the components of the project that "just worked" for the students.

## Student-Led Components

### Raspberry Pi

The Raspberry Pi component presented some unique challenges, with the chief being, enabling the students to actually remotely connect via SSH to a headless Raspberry Pi.

This deceptively simple task requires an intermediate knowledge of computer networking, and coupled with the obstreperous Raspberry Pi, a restrictive college network, the additional complications of students running Linux via virtual machines on Windows (all of the students were using Windows)... It ended up taking more than a month to be able to consistently boot up the Pi, remotely SSH to the Pi, and get a command line using either a crossover cable or a wireless network.

Part of this was induced by hardware, but part was due to unfamiliarity with SSH and Linux, the problems that constantly cropped up ("X is not working in the virtual machine") that were trivial for me to solve, but enigmas for the students, who often did not possess Google-fu.

### Question Skills

This last point is subtle but important: the simple skill of knowing what questions to ask, and how to ask them, be they questions asked of a machine or a person or a data set, was one of the most important skills the students gained during this process. These skills go beyond the usual computer science curriculum, which consists of learning structured information in terms of languages and functionality, and require students to solve unstructured problems that are complex - so complex, they simply do not care about languages or functionality.

The flexibility to use many tools was a key element of this project, and a principal reason to use a scripting language (Python) that was flexible enough to handle the many tasks we would be asking of it.

• Issues were due to a combination of hardware and networking problems

• Many issues required multi-step workarounds

• Workarounds introduced new concepts (DHCP, subnets, IP configuration schemes, IPv6)

• Each new concept introduced led students to feel overwhelmed

• Students had a difficult time telling what steps were "normal" and which were esoteric

• There is a lot of documentation to read - especially difficult for non-English speakers

Each of the multitude of problems students experienced arose from different aspects of the machines. Each problem (networking, hardware, physical power, cables, networking, packet dropping, interfaces, incorrect configuration, firewalls) led to more concepts, more software, more commands.

It can be difficult to troubleshoot networking and hardware issues. It is even more difficult to explain the problem while you are troubleshooting it, and also explain things are important and that students should learn more about, versus some concept that is of questionable usefulness. (Case in point: regular expressions.) On top of that, it is difficult to constantly make judgment calls about what things are important, how important they are, and also helping students not to feel overwhelmed by all the things they don't know yet.

All the while, you are also teaching Google-fu. Did I mention that many of the students do not speak English as their first language?

### Aircrack/Airodump

Once the students had reached the Raspberry Pi command line, we moved on to our next major tool - the aircrack-ng suite. This was a relatively easy tool to get working, as it was already available through a package manager (yet another new concept for the students), so we did not waste much time getting aircrack operational and gathering our first sensor data. However, to interpret the output of the tool required spending substantial time covering many aspects of networking - not just wireless networks, but general concepts like packets, MAC addresses, IP addresses, DHCP, ARP, encryption, and the 802.11 protocol specification.

Initially I had thought to use a Python library called Scapy, which provides functionality for interacting with wireless cards and wireless packets directly from Python. My bright idea was to use aircrack to show students what kind of information about wireless networks can be extracted, and to write a custom Python script that would extract only the information we were interested in.

Unfortunately, the complexity of Scapy, and the advanced level of knowledge required of users (even to follow the documentation), meant the tool overwhelmed the students. We wound up practicing putting wireless USB devices into monitor mode from the command line, and starting the wireless network signal profiling tool.

The approach we adopted was to collect wireless network data using aircrack-ng's airodump-ng tool, and to dump the network data at short intervals (15 seconds) to CSV files. These CSV files were then post-processed with Python to extract information and populate the database.

By the end of the first quarter of the project, we were able to utilize airodump-ng to collect wireless network data into CSV files, and parse the data with a Python script.

### Pi CSV Files

Further complicating the process of collecting wireless network data from Raspberry Pis was the fact that we were gathering data from the Pis in a variety of different environments - most of which were unfamiliar, and would not reliably have open wireless networks or networks that the Pi was authorized to connect to. Even on the South Seattle campus, the network was locked down, with only HTTP, HTTPS, and DNS traffic allowed on ports 80, 443, and 53, respectively.

This meant we couldn't rely on the Pis making a direct connection to the remote server holding the central database.

Instead, we utilized rsync to synchronize the CSV files gathered by the Pi with the remote server, and we offloaded the process of extracting and analyzing data from the CSV files to a script on the remote server.

That way, the Pis gather the raw data and shuttle the raw data to the remote server (whenever it is available), and the data extraction and analysis process can be performed on the raw data in the CSV files as many times as necessary. If the analysis required different data, or needed to be re-run, the process could simply be updated and re-run on the databae server, with the Raspberry Pi removed from the loop.

### NoSQL Database

We needed a warehouse to store the data that the Raspberry Pis were gathering. The aircrack script was dumping CSV files to disk every 15 seconds. Rather than process the data on-board the Raspberry Pi, the script to extract and process data from the CSV files was run on the computer running the database.

This is a best practice I learned form experience:

• Extract and process the sensor data on-premises (i.e., near or where the data is stored)

• Keep the original, raw data whenever possible, transport it to the data storage

• Assume the components of your pipeline will be unreliable or asychronously available

• Build the pipeline to be robust and handle failures.

We used a cheap, $5/month virtual private server from Linode to run the database. The database technology we chose was MongoDB, mainly because it is a ubiquitous, open-source, network-capable NoSQL database. The NoSQL option was chosen to give students flexibility in structuring the database, and avoid the extra pain of making a weakly-typed language like Python talk to a strongly-typed database system like SQLite or PostgreSQL (which would raise so many questions from students about what is "normal" or "not normal" that I would start to feel like the parent of a bunch of teenagers). Think of the long-term influence that research mentors can have: simply by showing students how to use vim, and not emacs, I have set them on the path to enlightenment. We ran the database on the server, but conceptualizing the database was difficult for the students. To this end, I set up an instance of Mongo Express, which provided a password-protected, web-based interface for administering MongoDB that enabled the students to deal with and visualize information more easily. MongoDB also provided Python bindings via PyMongo, and it was all available for students to install on their local virtual machines and experiment with basic database operations. The MongoDB documentation provides some good examples. The main struggle that students had was transferring what they had learned about wireless signals and aircrack to the database. Knowing what questions to ask the database proved to take most of their time. ## Backend components During the process of getting each component working, the project occasionally encountered difficulties. The chiefest among these was the fact that the wireless network at our college allowed traffic only on ports 80, 443, and 53, meaning SSH, Rsync, and MongoDB traffic would not make it past the school's firewall. ### Stunnel I have written about Stunnel before on this blog, and have some notes on Stunnel on the charlesreid1.com wiki. This tool proved invaluable for overcoming some of the difficulties on the back-end for the Raspberry Pis. To allow the Raspberry Pis to securely send data to the database server, I wrote a script that would run on boot and would look for a list of trusted wireless networks, connect to them, and establish an stunnel connection with the remote database server. The script then used rsync over stunnel to synchronize any raw data collected by the Raspberry Pi with the remote database server. This also satisfied the criteria that the data pipeline be robust and capable of handling failure - this system used stunnel to punch out of a restrictive firewall, and rsync handled comparisons of raw data on the remote and local ends to ensure that only the minimum possible amount of data was transferred between the two. The raw data was plain text and consisted of text files of modest size, making the job easy for rsync. This was implemented in a boot script, so one simply connected one of the Raspberry Pis to a portable power source (battery pack), and the Pi would look for networks that it trusted, join those networks, and make an stunnel connection over the network to transfer its data (CSV files) to the database server. ### Virtual Private Server Another bit of infrastructure that was provided on the back end was the virtual private server from Linode, so that the students did not have to find a workaround to SSH out of the school's restrictive firewall. A domain for the server was also purchased/provided. ### Docker The virtual private server ran each service in a Docker container - stunnel, MongoDB, MongoExpress, and the long list of Python tools needed to run the Jupyter notebooks for data analysis. Each Docker container exposed a particular port, making it accessible at an appropriate scope, and by connecting containers to other containers, each component could also seamlessly communicate. Thus one Docker container ran the MongoDB, while another container ran MongoExpress, which established a connection to the MongoDB container. Using Docker was not strictly necessary, but it was a good opportunity to learn about Docker and get it set up to help solve real-world infrastructure and service problems. ## Technologies Flowchart The following flowchart shows the technology stack that was used to coordinate the various moving parts between the Raspberry Pi clients and the remote database server. # Undergraduate Research Project: Wireless Sensor Networks for Internet of Things Applications (Part 1: The Project) Posted in Wireless Table of Contents: # Overview of the Undergraduate Research (UGR) Project ## South Seattle UGR Project For the past year, in addition to my duties as a computer science and math instructor at South Seattle College, I have served as a research mentor for an NSF-funded undergraduate research project involving (off-and-on) five different South Seattle students - all of whom have expressed interest in transferring to the University of Washington's computer science program after they finish at South Seattle College. The students have various levels of preparation - some have taken calculus and finished programming, while others are just starting out and have no programming experience outside of "programming lite" languages like HTML and CSS. But it's also been an extremely rewarding opportunity. I have gotten the chance to kindle students' interests in the vast world of wireless security, introduced them to essential technologies like Linux, helped them get hands-on experience with NoSQL databases, and guided them through the process of analyzing a large data set to extract meaningful information - baby data scientists taking their first steps. These are all skills that will help equip students who are bound for university-level computer science programs, giving them both basic research skills (knowing the process to get started answering difficult, complex questions) and essential tools in their toolbelt. Two students who I mentored as part of a prior UGR project last year (also focused on wireless networks and the use of Raspberry Pi microcomputers) both successfully transferred to the University of Washington's computer science program (one in the spring quarter of 2016, the other in the fall of 2016). Both students told me that one of the first courses they took at the University of Washington was a 2-credit Linux laboratory class, where they learned the basics of Linux. Having already installed Linux virtual machines onto their personal computers, and having used technologies like SSH to remotely connect to other Linux machines, they both happily reported that it was smooth sailing in the course, and it was one less thing to worry about in the process of transferring and adjusting to the much faster pace of university courses. ## Engineering Design Project The project was entitled "Wireless Sensor Networks for Internet of Things Applications," and was intended to get students introduced to the basic workflow of any internet of things system: a sensor to collect data, a wireless network to connect sensors together, a warehouse to store data collected from sensors, and a workflow for analyzing the data to extract meaningful information. The focus was to implement a general workflow using tools that could extend to many internet of things applications, be they commercial, residential, or industrial. However, the NSF grant provided only a modest amount of funding, intended to go toward stipends to pay students and mentors a modest amount during the quarter, with only modest amounts of money for basic equipment. (We were basically running a research project on a$100 budget.)

That meant the project had to be flexible, scrappy, and run on a shoestring budget. This meant we were limited to cheap, off-the-shelf technologies for the sensors, the sensor platform, and the back-end infrastructure. Two technologies in particular lent themselves nicely to these constraints:

• Wireless USB antennas - USB wifi dongles are cheap ($10), and the ubiquity of wireless networks and wifi signals meant this would provide us with a rich data set on the cheap. • Raspberry Pi - the Raspberry Pi is a credit-card sized microcomputer that runs a full stack Linux operating system. With the low price point ($30) and the many free and open-source tools available for Linux, this was a natural choice for the sensor platform.

The result was a set of wireless sensors - Raspberry Pis with two wireless antennas - one antenna for listening to and collect wireless signal data in monitor mode, and one antenna to connect to nearby wireless networks to establish a connection to a centralized data warehouse server.

## Project Components: Extract, Store, and Analyze

The wireless sensor network project had three major components:

• Extract - using a wireless USB antenna, the Raspberry Pi would listen to wireless signals in the area, creating a profile of local network names, MAC addresses, signal strengths, encryption types, and a list of both clients and routers. Students used the aircrack-ng suite to extract wireless signal information.

• IMPORTANT SIDE NOTE - students also learned about wiretapping laws and various legal aspects of wireless networks - the difference between monitoring ("sniffing") wireless traffic versus simply building a profile of wireless traffic.

• Store - students learned about NoSQL databases (we used MongoDB) and set up a NoSQL database to store information about wireless signals. This also required some basic Python programming, as the wireless signal information was exported to a large number of CSV files and had to be programmatically collated and extracted.

• Analyze - the pinnacle of the project was in the analysis of the wireless signal data that was captured. Students ran several "experiments," collecting wireless signals for 2 hours using a portable battery and a Raspberry Pi with wifi dongles. By running experiments under different conditions (at the college library, at a coffee shop, on a bus), a diverse set of data was gathered, allowing students to extract meaningful information about each experiment from each data set.

## The Internet of Things: Not Just a Buzzword

One of the biggest challenges starting out was in getting the students into the right "mindset" about the Internet of Things. This was a challenge that I did not forsee when I came up with the project title. As a chemical engineer working on natural gas processing at a startup company, I knew the value of creating wireless infrastructure to extract data from sensors, throw it into a giant bucket, and utilize computational tools to analyze the data and extract information from it.

But the students involved in the project had no exposure to this kind of workflow. To them, the Internet of Things meant toasters and TVs that were connected to the internet, so they were expecting a design project in which we would make a prototype consumer device intended to connect to the internet.

Further complicating things was the fact that we were focusing on building a data acquisition system - a data analysis pipeline - a workflow for extracting, storing, and analyzing sensor data. We were not focused on the specific types of questions that our specific type of data could answer. This was a bit puzzling for the students (who could not see the intrinsic value of building a data analysis pipeline). Much of their time was spent struggling with what, exactly, we were supposed to be doing with the data, and getting past a contaminated, consumer-centric view of the term "Internet of Things."

It was, therefore, a major breakthrough when one of the students, as we were diving deeper into the data analysis portion, utilizing Python to plot the data, quantitatively analyze it, and better understand it, told me, "Looking back, I realize that I was thinking really narrowly about the whole project. I thought we were going to build a 'smart' device, like a business project. But now I realize our project has a bigger scope, because of the analysis part."

That, in a nutshell, was precisely the intention of the project.

## University of Washington Undergraduate Research Symposium

Next week the students present the culmination of their research project at the University of Washington's Undergraduate Research Symposium, where they will have a poster that summarizes their research effort, the results, and the tools that were used.

It is clear to anyone attending the Undergrad Research Symposium that community college students are among the minority of students who are involved in, and benefiting from, research projects. The intention of most of the projects showcased at the symposium is to launch undergraduate students into a graduate level research career and prepare them to hit the ground running, and have a stronger resume and application, when they have finished their undergraduate education and are applying to graduate schools. Many of the research posters at the symposium showcase research using expensive equipment, specialized materials and methods, and complex mathematical methods. Many of the students are mentored by world-class research professors with deep expertise and small armies of graduate and postgraduate researchers.

Despite our research efforts being completely outmatched by many of the undergraduate researchers from the University of Washington (out-funded, out-manned, and out-gunned), our group managed to pull together a very interesting and very ambitious design project that collected a very rich data set. The students were introduced to some useful tools and fields of computer science (wireless networks, privacy and security, embedded devices, databases, Linux), and exposed students to a totally new way of thinking about the "internet of things" that allows them to move beyond the shallow hype of internet-connected toothbrushes. The students have developed the ability to build a data pipeline that could be used by a company to address real, significant problems and needs around data.

All in all, this was an extremely worthwhile, high-impact project that's equipping the next generation of computer scientists with the cognitive tools to anticipate and solve data problems, which (as hardware becomes cheaper and embedded devices become more ubiquitous) are problems that will only become more common in more industries.

## The Poster

Here's a rough draft of the poster we will be showing at the UGR symposium: