DS Interview Study Guide Part II: Software Engineering

This post continues my series on data science interviews. One of the major difficulty of doing data science interviews is that you must show expertise in a wide variety of skills. In particular, I see four key subject areas that you might be asked about during an interview:

1. Statistics
2. Software Engineering/Coding
3. Machine Learning
4. “Soft” Questions

This post focuses on software engineering & coding. It will be primarily a resource for aggregating content that I think you should be familiar with. I will mostly point to outside sources for technical exposition and practice questions.

I’ll link to these as appropriate throughout the post, but I thought it would be helpful to put up front a list of the primary resources that I’ve used when studying for interviews. Some of my favorites are:

• Data Structures and Algorithms in Python, for a good introduction to data structures such as linked lists, arrays, hashmaps, and so on. It also can give you good sense of how to write idiomatic Python code, for building fundamental classes.
• SQLZoo for studying SQL and doing practice questions. I particularly like the “assessments”.
• Cracking the Coding Interview for lots of practice questions organized by subject, and good general advice for the technical interviewing process.

I also use coding websites like LeetCode to practice various problems. I also look on Glassdoor to see what kinds of problems people have been asked.

As always, I’m working to improve this post, so please do leave comments with feedback.

What Languages Should I Know?

In this section of data science interviews, your are generally asked to implement things in code. So, which language should you do it in? Generally, the best answer is (unsurprisingly) that you should work in Python. The next most popular choice is R; I’m not very familiar with R, so I can’t really speak to it’s capabilities.

There are a few reasons you should work in Python:

1. It’s widely adopted within industry.
2. It has high-quality, popular packages for working with data (see pandas, numpy, scipy, statsmodels, scikit-learn, matplotlib, etc).
3. It bridges the gap between academic work (e.g. using NumPy to build a fast solver for differential equations) and industrial work (e.g. using Django to build webservices).

This is far from an exhaustive list. Anyways, I mostly work in Python. I think it’s a nice language because it is clear and simple to write.

If you want to use another language, you should make sure that you can do everything you need to - this includes reading & writing data, cleaning/munging data, plotting, implementing statistical and machine learning models, and leveraging basic data types like hashmaps and arrays (more on those later).

I think if you wanted to do your interviews in R it would be fine, so long as you can do the above. I would strongly recommend against languages like MATLAB, which are proprietary and not open-source.

Languages like Java can be tricky since they might not have the data-oriented libraries that Python has. For example, I’ve worked profesionally in Scala, and am very comfortable manipulating data via the Spark API within it, but still wouldn’t want to have to use it in an interview; it just isn’t as friendly for general-purpose hacking as Python.

So is Python all you need? Well, not quite. You should also be familiar with SQL for querying databases; we’ll get into that later. I don’t think the dialect you use particularly matters. SQLZoo works with MySQL, which is fine. Familiarity with bash and shell-scripting is useful for a data scientist in their day-to-day work, but generally isn’t asked about in interviews. For the interviews, I’d say if you know one general-purpose language (preferably Python, or R if need be) and SQL, then you’ll be fine.

General Tips for Coding Interviews

Coding interviews are notorious for being high-stress, so it’s important that you practice in a way that will maximize your comfort during the interview itself - you don’t want to add any unnecessary additional stress into an already difficult situation. There are a wide variety of philosophies and approaches to preparing yourself for and executing a successful interview. I’m going to talk about some points that resonate with me, but I’d also recommend reading Cracking the Coding Interview for a good discussion. Of course, this isn’t the final word on the topic - there are endless resources available online that address this.

How to Prepare

When preparing for the interview, make sure to practice in an environment similar to the interview environment. There are a few aspects of this to keep in mind.

• Make sure that you replicate the writing environment of the interview. So, if you’ll be coding on a whiteboard, try to get access to a whiteboard to practice. At least practice on a pad of paper, so that you’re comfortable with handwriting code - it’s really quite different than using a text editor. If you’ll be coding in a Google Doc, practice doing that (protip: used a monospaced font). Most places I’ve interviewed at don’t let you evaluate your code to test it, so you have to be prepared for that.
• Time yourself! It’s important to make sure you can do these things in a reasonable amount of time. Generally, these things last 45 minutes per “round” (with multiple rounds for on-site interviews). Focus on being efficient at implementing simple ideas, so that you don’t waste a bunch of time with your syntax and things like that.
• Practice talking. If you practice by coding silently by yourself, then it might feel strange when you’re in the interview and have to talk through your process. The best is if you can have a friend who is familiar with interviewing play the interviewer, so that you can talk to them, get asked questions, etc. You can also record yourself and just talk to the recorder, so that you get practice externalizing your thoughts.

There are some services online that will do “practice” interviews for you. When I was practicing for a software engineer interview with Google, I used Gainlo for this - they were kind of expensive, but you interview with real Google software engineers, which I found helpful.

However, the interviews for a software engineering position at Google are very standardized in format. I haven’t used any of the services that do this for data science, and the interviews you’ll face are so varied. Therefore, I imagine it is harder to do helpful “mock interviews”. If you’ve used any of these services, I’d be very curious to hear about your experience.

Tips for Interviewing

There are some things it’s important to keep in mind as you do the interview itself.

• Talk about your thought process. Don’t just sit sliently thinking, then go and write something on the board. Let the interviewer into your mind so that they can see how you are thinking about the problem. This is good advice at any point in a technical interview.
• Start with a simple solution you have confidence in. If you know that you can quickly write up a suboptimal solution (in this case, maybe insertion sort), then do that! You can discuss why that solution is sub-optimal, and they will often brainstorm with you about how to improve it. That said, if you are just as confident in writing up something more optimal (say, quicksort) then feel free to jump right to that.
• Sketch out your solution before doing real code. This is not necessary, but sometimes for complicated stuff it’s nice to write out your approach in pseudocode before jumping into real code. This can also help with exposing your thought process to the interviewer, and making sure they’re on board with how you’re thinking about it.
• Think about edge cases. Suppose they ask you to write a function that sorts a list. What if you’re given an empty list? What if you’re given a list of non-comparable things? (In Python, this might be a list of lists.) What does your function do in this case? Is that what you want it to do? There’s no right answer here, but you should definitely be thinking about this and asking the interview how they want the function to behave on these cases.
• Be sure to do a time complexity analysis on your solution. They want to know that you can think about efficiency, so unless they explicitly ask you not to do this, I’d recommend it. We’ll discuss more about what this means below.

For a more thorough discussion of preparation and day-of techniques, I’d recommend Cracking the Coding Interview.

Tips for Coding

There are few things specifically in how the interviewee writes code that I think are worth mentioning. This kind of stuff usually isn’t a huge deal, but if you write good code, it can show professionalism and help leave a good impression.

• Name your variables well. If the variable is the average number of users per region, use num_users_per_region, or users_per_region, not avg_usr or num_usr. Unlike in mathematics, it’s good to have long, descriptive variables.
• Use built-ins when you can! Python already has functions for sorting, for building cartesian products of lists, for implementing various models (in statsmodels and scikit-learn), and endless other things. It also has some cool data structures already implemented, like the heap and queue. Get to know the itertools module; it has lots of usefull stuff. if you can use these built-ins effectively, it demonstrates skill and knowledge without adding much effort on your part.
• Break things into functions. If one step of your code is sorting a list, and you can’t use the built-in sorted() function, then write a separate function def sort() before you write your main function. This increases both readability and testability of code, and is essential for real-world software.
• Write idiomatic Python. This is a bit less important, but make sure to iterate directly over iterables, don’t do for i in range(len(my_iterable)). Also, familiarize yourself with enumerate and zip and know how to use them. Know how to use list compreshensions, and be aware that you can do a similar thing for dictionaries, sets, and even arguments of functions - for example, you can do max(item for item in l if item % 2 == 0) to find the maximum even number in l. Know how to do string formatting using either .format() for f-strings in Python 3.¹

I’m only scratching the surface of how to write good code. It helps to read code that others have written to see what you don’t know. You can also look at code in large open-source libraries.

With all that said, let’s move on to some of the content that might be asked about in these interviews.

Working with Data

One of the fundamental tasks of a data scientist is to load, manipulate, clean, and visualize data in various formats. I’ll go through some of the basic tasks that I think you should be able to do, and either include or link to Python implementations. If you work in R, or any other language, you should make sure that you can still do these things in your preferred language.

In Python, the key technologies are the packages pandas (for loading, cleaning, and manipulating data), numpy (for efficiently working with unlabeled numeric data), and matplotlib (for plotting and visualizing data).

Loading & Cleaning Data

This tutorial on DataCamp nicely deals with the basics of using pd.read_csv() to load data into Pandas. It is also possible to load from other formats, but in my experience writing to and from comma- or tab-separated plaintext is by far the most common approach for datasets that fit in memory.²

For example, suppose you had the following data in a csv file:

name,age,country,favorite color
steve,7,US,green
jennifer,14,UK,blue
franklin,,UK,black
calvin,22,US,

You can copy and paste this, into Notepad or whatever text editor you like³, and save it as data.csv.

You should be able to

• load in data from text, whether it is separated by commas, tabs, or some other arbitrary character (sometimes things are separated by the “pipe” character |). In this case, you can just do df = pd.read_csv('data.csv') to load it.
• Filter for missing data. If you wanted to find the row(s) where the age is missing, for example, you could do df[df['age'].isnull()]
• Filter for data values. For example, to find people from the US, do df[df['country'] == 'US']
• Replace missing data; use df.fillna(0) to replace missing data with zeros. Think for yourself about how you would want to handle missing data in this case - does it make sense to replace everything with zeros? What would make sense?

Dealing with missing data is, in particular, an important problem, and not one that has an easy answer. Towards Data Science has a decent post on this subject, but if you’re curious, there’s a lot to read about and learn here.

More advanced topics in pandas-fu include using groupby, joining dataframes (this is called a “merge” in pandas, but works the same as a SQL join), and reshaping data.

As I said before, loading and manipulating data is one of the fundamental tasks of a data scientist. You should probably be comfortable doing most or all of these tasks if asked. Pandas can be a bit unintuitive, so I’d recommend practicing if you aren’t already comfortable with it. Doing slicing and reshaping tasks in numpy is also an important skill, so make sure you are comfortable with that as well.

Visualization

Another essential aspect of data work is visualization. Of course, this is an entire field unto itself; here, I’ll mostly be focusing on the practical aspects of making simple plots. If you want to start to learn more about the overarching principles of the visual representation of data, Tufte’s book is the classic in the field.

In Python, the fundamental tool used for data visualization is the library matplotlib. There exist many other libraries for more complicated visualization tasks, such as seaborn, bokeh, and plotly, but the only one that you really need to be comfortable with (in my opinion) is matplotlib.

You should be comfortable with:

• plotting two lists against one another
• changing the labels on the x- and y-axis of your plot, and adding a title
• changing the x- and y-limits of your plot
• plotting a bar graph
• plotting a histogram
• plotting two curves together, labelling them, and adding a legend

I won’t go through the details here - I’m sure you can find many good guides to each of these online. The matplotlib pyplot tutorial is a good place to start.⁴

It’s worth noting that you can plot directly from pandas, by doing df.plot(). This just calls out to matplotlib and plots your dataframe; I will often find myself both plotting from the pandas DataFrame.plot() method as well as directly using pyplot.plot(). They work on the same objects, and so you can use them together to make more complicated plots with multiple values plotted.

Data Structures & Algorithms

Designing and building effective software is predicated on a solid understanding of the basic data structures that are available, and familiarity with the ways that they are employed in common algorithms. For me, learning this material opened up the world of software engineering - it illuminated the inner workings of computer languages. It also helped me understand the pros and cons of various approaches to problems, in ways that I wouldn’t have been able to before.

This subject is fundamental to software engineering interviews, but for data scientists, its importance can vary drastically from role to role. For engineering-heavy roles, this material can make up half or more of the interview, while for more statistician-oriented roles, it might only be very lightly touched upon. You will have to use your judgement to determine to what extent this material is important to you.

I learned this material when I was interviewing by reading the book Data Structures and Algorithms in Python.⁵ It’s really a great book - it has good, clear explanations of all the important topics, including complexity analysis and some of the basics of the Python language. I can’t recommend it highly enough if you want to get more familiar with this material.⁶ You can buy it, or look around online for the PDF - it shouldn’t be too hard to find.

Time and Space Complexity Analysis

Before you begin writing algorithms, you need to know how to analyze their complexity. The “complexity” of an algorithm tells you how the amount of time (or space) that the algorithm takes depends on the size of the input data.

It is formalized using the so-called “big-O” notation. The precise mathematical definition of $\mathcal{O}(n)$ is somewhat confusing, so you can just think of it roughly as meaning that an algorithm that is $\mathcal{O}(n)$ “scales like $n$”; so, if you double the input size, you double the amount of time it takes. If an algorithm is $\mathcal{O}(n^3)$, then, doubling the input size means that you multiply the time it takes by $2^3 = 8$.⁷ You can see how even a $\mathcal{O}(n^2)$ algorithm wouldn’t work for large data; even if it runs in a reasonable amount of time (say, 5 seconds)for 10,000 points, it would take about 15,000 years to run on 1 billion data points. Obviously, this is no good.

So complexity analysis is critical. You don’t want to settle for a $\mathcal{O}(n^2)$ solution when a $\mathcal{O}(n)$ or $\mathcal{O}(n \log n)$ solution is available. I won’t get into how to do the analysis here, besides saying that I often like to annotate my loops with their complexity when I’m writing things. For example, here’s a (slow) approach to finding the largest k (unique) numbers in a list:

def get_top_k(k, input_list):
    top_k = []
    for _ in range(k):  # happens k times
        remaining = [num for num in input_list if num not in top_k]  # O(n)
        if remaining:
            top_remaining = max(remaining)  # O(n)
            top_k.append(top_remaining)  # O(1)
    return top_k

I know that the outer loop happend k times, and since finding the maximum of a list is $\mathcal{O}(n)$, the total task is $\mathcal{O}(nk)$.⁸ To learn more about how to do complexity analysis, I’d look at DS&A, Cracking the Coding Interview, or just look around online - I’m sure there are plenty of good resources out there.

You can also consider not just the time of computation, but the amount of memory (space) that your algorithm uses. This is not quite as common as time-complexity analysis, but is still important to be able to do.

A very useful resource for anyone studying for a coding interview is the big-O cheat sheet, which shows the complexity of access, search, insertion, and deletion for various data types, as well as the complexity of searching algorithms, and a lot more. I often use it as a reference, but of course it’s important that you understand why (for example) an array has $\mathcal{O}(n)$ insertion. Just memorizing complexities won’t help you much.

Arrays & Hashmaps

In my opinion, the two essential data structures for a data scientist to know are the array and the hashmap. In Python, the list type is an array, while the dict type is a hashmap. Since both are used so commonly, you have to know their properties if you want to be able to design efficient algorithms and do your complexity analysis correctly.

Arrays are a data type where a piece of data (like a string) is linked to an index (in Python, this is an integer, starting with 0). I won’t go too deep into the details here, but for arrays, the important thing to know is that getting any element of an array is easy (i.e. doing mylist[5] is $\mathcal{O}(1)$, so it doesn’t depend on the size of the array) but adding elements (particularly in the beginning or middle of the array) is difficult; doing mylist.insert(k, 'foo') is $\mathcal{O}(n-k)$, where $k$ is the position you wish to insert at.⁹

Arrays are what we usually use when we’re building unordered, unlabelled collections of objects in Python. This is fine, since insertion at the end of an array is fast, and we’re often accessing slices of arrays in a complicated fashion (particularly in numpy). I generally use arrays by default, without thinking too much about it, and it generally works out alright.

Hashmaps also link values to keys, but in this case the key can be anything you want, rather than having to be an ordered set of integers. In Python, you build them by specifying the key and the value, like {'key': 'value'}. Hashmaps are magical in that accessing elements and adding elements are both $\mathcal{O}(1)$.¹⁰ Why is this cool? Well, say you wanted to store a bunch of people’s names and ages. You might think to do a list of tuples:

names_ages = [('Peter', 12), ('Kat', 25), ('Jeff', 41)]

Then, if you wanted to find out Jeff’s age, you would have to iterate through the list and find the correct tuple:

for name, age in name_ages:  # happens n times
    if name == 'Jeff':
        print(f"Jeff's age is {age}")

This is $\mathcal{O}(n)$ - not very efficient. With hashmaps, you can just do

name_ages = {'Peter': 12, 'Kat': 25, 'Jeff': 41}
print(f"Jeff's age is {name_ages['Jeff']}")  # O(1)! Wow!

It might not be obvious how cool this is until you see how to use it in problems. Cracking the Coding Interview has lots of good problems on hashmaps, but I’ll just reproduce some of the classics here. I think it’s worth knowing these, because they really can give you an intuitive sense of when and how hashmaps are valuable.

The first classic hashmap algorithm is counting frequencies of items in a list. That is, given a list, you want to know how many times each item appears. You can do this via the following:

def get_freqs(l):
    freqs = {}
    for item in l:  # happens O(n) times
        if item not in freqs:  # This check is O(1)! Wow!
            freqs[item] = 1
        else:
            freqs[item] += 1  # Also O(1)! Wow!
    return freqs

Try and think of how you’d do this without hashmaps. Probably, you’d sort the list, and then look at adjacent values. But sorting is, at best $\mathcal{O}(\log n)$. This solution does it in $\mathcal{O}(n)$!

Another classic problem that is solved with hashmaps is to find all repeated elements in a list. This is really just a variant of the last, where you look for elements that have frequency greater than 1.

def get_repeated(l):
    f = get_freqs(l)
    return [item for item in f if f[item] > 1]

Now, if you only need one repeated element, you can be efficient and just terminate on the first one you find. For this, we’ll use a set, which is just a dict with values of None. That is to say, sets are also hashmaps. The important thing to know is that adding to them and checking if something is in them are both $\mathcal{O}(1)$.

def get_repeated(l):
    items = set()
    for item in l:  # happens O(n) times
        if item not in items:  # This check is O(1)! Wow!
            items.add(item)
        else:
            return(item)
    return None  # if this happens, all elements are unique

The last one we’ll do is a bit trickier. You’re given a list of numbers, and a “target”, and your task is to find a pair of numbers in the list that add up to the target. Try and think for yourself how you’d do this - the fact you use hashmaps is a big hint. You should be able to do it in $\mathcal{O}(n)$.

Have you thought about it? When I first encountered this one I had to look up the answer. But here’s how you do it in $\mathcal{O}(n)$:

def get_sum_pair(l, target):
    nums_set = set()
    for num in l:
        other_num = target-num
        if other_num in nums_set: 
            return (num, other_num)
        nums_set.add(num)   # no-op if num is already there
    return None

Note that other_num = target-num is the number that you would need to complete the sum pair; using a hashmap, you can check in $\mathcal{O}(1)$ if you’ve already seen it! Wow!

Hopefully you get it - hashmaps are cool. Go on LeetCode, or pop open your favorite data structures book, or even Cracking the Coding Interview, and get some practice with them.

Sorting & Searching

Sorting and searching are two of the basic tasks you have to be familiar with for any coding interview. You can go into a lot of depth with these, but I’ll stick to the basics here, because that’s what I find most helpful.

Sorting

Sorting is a nice problem in that the statement of the problem is fairly straightforward; given a list of numbers, reorder the list so that every element is less than or equal to the next. There are a number of approaches to sorting. The naive approach is called insertion sort; for example, it is what most people do when sorting a hand of cards. It has some advantages, but is $\mathcal{O}(n^2)$ in time, and so is not the most efficient available.

The two most common fast sorting algorithms are quicksort and mergesort. They are both $\mathcal{O}(n \log n)$ in time,¹¹ and so scale close-to-linearly with the size of the list. I won’t go into the implementation details here; there are plenty of good discussions of them available on the internet.

When thinking about sorting, it’s also worth considering space complexity - can you sort without needing to carry around a second sorted copy of the list? If so, that’s a significant advantage, especially for larger lists. It’s also worth thinking about worst-case vs. average performance - how does the algorithm perform on a randomly shuffled list, and how does it perform on a list specifically designed to take the maximum number of steps for that algorithm to sort? Quicksort, for example, is actually $\mathcal{O}(n^2)$ in the worst case, but is $\mathcal{O}(n \log n)$ on average. Again, you can look to the big-O cheat sheet to make sure you’re remembering all your complexities correctly.

Searching

The problem of searching is often stated as given a sorted list l and an object x, find the index at which an element x lives. (You should immediately ask: What should I return if x is not in l?)The name of the game here is binary search. You basically split the list, then if the number is greater than the split, search the top; otherwise, search the bottom. This is an example of a recursive algorithm, so the way it’s written can be a bit opaque to those not used to looking at recursive code. Once I can wrap my head around it, I find it quite elegant. The important thing to know is that this search is $\mathcal{O}(\log n)$, which means that you don’t touch every element in the list - it’s very fast, even for a large list. The key to this is that the list is already sorted - if it’s not sorted, then you’re out of luck; you’ve got to check every element to find x.

There are tons of examples of binary search in Python online, so I won’t put one here. That said, I have found it interesting to see how thinking in terms of binary search can help you in a variety of areas.

For example, suppose you had some eggs, and worked in a 40-story building, and wanted to know the highest floor you could drop the egg off of without it breaking (it’s kind of a dumb example cause the egg would probably break even on the first floor, but pretend it’s a super-tough egg.) You could drop it from the first floor, and see what happens. Say it doesn’t break. Then drop it from the 40th, and see what happens. Say it does break. Then, you bisect and use the midpoint - drop from the 20th floor. If it breaks here, you next try the 10th - if it doesn’t you next try the 30th. This allows you to find the correct floor much faster than trying each floor in succession.

Sorting and searching are fundamental algorithms, and have been well studied for decades. Having a basic fluency in them shows a familiarity with the field of computers science that many employers like to see. In my opinion, you should be able to quickly and easily implement the three sorting algorithms above, and binary search, in Python, or whatever your language of choice is.

Working with SQL

Finally, let’s talk a bit about SQL. SQL is a tool used to interact with so-called “relational” databases, which just means that each row in a table has certain values (columns), and that those values have the same type for each row (that is, the schema is uniform throughout the table).¹² It is not exactly a language, it’s more like a family of languages. There are many “dialects” which all have slight differences, but they behave the same with regards to core functionality; for example, you can do

SELECT column FROM table WHERE columns = 'value'

in any SQL-like language.¹³ Modern data-storage and -access solutions like Spark and Presto are very different from older databases in their underlying architecture, but still use a SQL dialect for accessing data.

Solving problems in SQL involves thinking in a quite different way than solving a similar problem on an array in Python. There is no real notion of iteration, or at least it’s not easily accessible, so most of the complicated action happens via table joins. I used SQLZoo, and particularly the “assessments”, to practice my SQL and get it up to snuff. LeetCode also has a SQL section (I think they call it “database”).

It’s essential to know SQL as a working data scientist. You’ll almost certainly use it in your day-to-day activities. That said, it’s not always asked in the interviews, so you might clarify with the company whether they will ask you SQL questions.

A Note on Dialects

There are many dialects of SQL, and changing the dialect changes things like (for example) how you work with dates. It’s worth asking the company you’re interviewing with what dialect they want you to know, if they have one in mind. If you’re just writing SQL on a whiteboard, then I would be surprised if they were picky about this; I would just say something like “here I’d use DATE(table.dt_str) or whatever the string-to-date conversion function is in your dialect”. In this case it’s just details that move around, but the big picture is generally the same for different dialects.

Conclusion

Coding interviews are stressful. From what I can tell, that’s just the way it is. For me, the best antidote to that is being well-prepared. I think companies are moving more towards constructive, cooperative interview formats, and away from the classic Google brain-teaser kind of questions, which helps with this, but you can still expect to be challenged during these interviews.

Remember to be kind to yourself. You’ll probably fail many times before you succeed. That’s fine, and is what happens to almost everyone. Just keep practicing, and keep learning from your mistakes. Good luck!

1. You should be using Python 3 at this point, but also be familiar with the differences between 2 and 3, and be able to write code in Python 2 if need be. ↩

2. For “big data” stored in the cloud, an efficient format called Parquet is the standard. In my experience, however, it’s uncommon to work with parquet files directly in Pandas; you often read them into a distributed framework like Spark and work with them in that context. ↩

3. The correct answer is, of course, emacs. ↩

4. pyplot is an API within matplotlib that was designed in order to mimic the MATLAB plotting API. It is generally what I use; I begin most of my matplotlib work with from matplotlib import pyplot as plt. I only rarely need to import matplotlib direct, and that’s generally for configuration work. ↩

5. I read the book when preparing for a software engineer interview at Google, so I picked up a lot more than was necessary for a data science interview. I still find the material helpful, however, and it’s nice to be able to demonstrate that you have gone above and beyond in a realm that data scientists sometimes neglect (efficient software design). ↩

6. It goes well beyond what you’ll need for a data science interview, however - it gets into tree structures, graphs (and graph traversal algorithms), and other more advanced topics. I’d recommend focusing on complexity analysis, arrays, and hashmaps as the most important data structures that a data scientist will use day-to-day. ↩

7. This is only approximately true, or rather it is is asymptotically true; this scaling law holds in the limit as $n\rightarrow\infty$. ↩

8. It’s a bit weird to use both $n$ and $k$ in your complexity - mathematically, what this means is that we consider them separate variables , and we can take the limit of either one independently from the other. If, for example, you knew that $k = n/4$, so you always wanted the top quarter of the list, then this would be $\mathcal{O}(n^2)$, since $n/4 = \mathcal{O}(n)$. ↩

9. I’m glossing over some details here - the numbers I quote above are for a fixed-size array. So, if you build up an array by adding elements at the end, it may seem like you get to just do a bunch of $\mathcal{O}(1)$ .appends, but in reality, you have to occasionally resize the array to make more space, which slows things down to an average append time of $\mathcal{O}(n)$. If you want a list-like type where inserting elements is easy ($\mathcal{O}(1)$) but accessing elements is difficult ($\mathcal{O}(n)$), then you want a linked list. Linked lists aren’t as important for data scientists to use, so I won’t get into them much here. ↩

10. You might wonder why we would ever use an array over a hashmap if hashmaps are strictly superior with respect to their complexity. It’s a good question. The answer is that arrays take up less space (they don’t have to store the keys, only the values) and they are much easier to work with in code (they look cleaner, and are more intuitive for unordered data). Furthermore, if you had a hashmap that linked integers 0 through 10 to strings, and you wanted to change the element at key 5, then you’d have to go through what is currently at keys 5 through 10, and increment their keys by one, so you would end up back at an inefficient insertion algorithm like you have with arrays. ↩

11. This is true on average; see the section below for a discussion of average vs. worst-case complexity. ↩

12. Non-relational database formats, like HBase and NoSQL, basically function like giant hashmaps; they have a single “key”, and then the “value” can contain arbitrary data - you don’t have to have certain columns in there. The advantage of this is flexibility, but the disadvantage is that sorting and filtering are slower because the database doesn’t have a pre-defined schema. ↩

13. Technically, SQL is an ANSI Standard that many different dialects implement - so, to call yourself a SQL dialect, you must have features defined by this standard, like the SELECT, FROM, and WHERE clauses shown above. ↩