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The Memory Hierarchy

Picking the best algorithm for the task will make your code run fast. Using memory properly will make it even faster.
Posted 3 September 2019 at 12:10 PM
By Joseph Mellor

In this article, we're going to explain how your computer uses memory and how you can work with your computer to write code that runs fast. To be clear, using memory properly will not make poorly written code easy to understand nor will it make a bad algorithm a good algorithm. Understanding how memory works, however, will help you find potential bottlenecks in your program and give you a way to solve them.

I once had to clean up some poorly written code for a job I had. When I say poorly written, I don't mean something minor like it didn't use my preferred coding style or even that it was poorly organized. I mean that there were significant parts of the code would just do the wrong thing, other parts that were redundant to the point of hiding bugs, etc. I once replaced around seventy-five lines of dense code with a single line of code because I understood how ASCII worked and because I was able to decouple some nested conditionals (specifically some poorly used switch statements). Before I fixed the code, the program took hours. After I fixed the code, the program took around ten minutes.

I used a lot of tricks (most were variations on using the language properly) to speed up the code, but two tricks in particular might sound weird because they wouldn't seem to do anything.

First, I had a large matrix of numbers (about 10 000 rows by 100 000 columns, I think), but I only ever used two columns of the matrix at a time, so I got rid of the large matrix and replaced it with just the last two columns. Second, I had another large matrix of numbers of the same size as the first matrix, and I noticed that the numbers never went too far above 300. In fact, I was able to guarantee that the numbers would be around 10 000 at an absolute maximum. At the time, I was using four bytes to represent each of the numbers, but you only need two bytes to represent numbers less than 65 536, so I switched the numbers from four bytes to two bytes.

In total, these two operations reduced the memory to a quarter of what I was using and made the program run about four times as fast.

Amdahl's Law in my Program

Let's say you're doing three tasks: one task takes five minutes, another task takes five minutes, and the last task takes fifty minutes. The total time it takes to complete all three tasks is an hour. Now, let's say you can reduce the last task to take only twenty minutes, meaning that it now only takes you half an hour to complete all three tasks. By making the longest part of the program take 40% of the time, you made the entire process take half the time.

Now, let's say instead of making the fifty minute task take 40% of the time, you make one of the five minute tasks take 40% of the time. In this case, you would only shave three minutes off the time, meaning the process still takes fifty seven minutes.

The idea that the net speedup of the program depends on the speedup of an individual part and how much time the program spends on that part is known as Amdahl's Law. If you want, try coming up with the math equations for what I wrote above.

In the program I'm discussing, the two matrices took up around 99% of the program, so any speedup here sped up the entire program by just under the same amount.

How would using a fourth of the memory make the program run about four times as fast?

The Allegory of the Professor

I won't try to dumb down how your computer manages memory, but I think the allegory I'm about to tell should prime your mind for the explanation.

The Professor Arrives

After earning her degree, a woman decided to become a professor at her alma mater to help pay for grad school. The university offered her a job and a small office, which she gladly accepted. She, being methodical, organized everything in her office into boxes and shelves with specific labels. She then put a list of all the labels she was using on the wall along with where she could find it. For example, her markers were on the second shelf from the top. Nothing strange happened until her first day teaching.

The First Day of Class

Just as she was about to leave her office, she realized that she had forgotten to buy a new backpack, as her old one fell apart on her last day of classes. She was going to be late to class if she didn't leave immediately, so she figured that she would just go to class without anything and run back to her office to pick up whatever she needed. Because she couldn't trust herself to remember to pick up things she had put down and she didn't want to lose anything she brought to class, she decided that she wouldn't put anything down anywhere except in her office. Since she had to keep her labels consistent, she would have to either put her stuff back where the label said she could find it or she would have to change the label.

With the rules she had set up, she left for the classroom. When she entered the classroom, she gave a brief introduction to the students and then told them that she was going to discuss the syllabus with them. Just as she was about to hand out the syllabus, she realized that she had left the syllabus back in her office. About five minutes later, she was back at the classroom with copies of the syllabus for everyone. Having distributed the syllabus out to everyone, she started reading off her copy when she felt that it would be good to write something on the board, but her markers were back at the office. Another five minutes passed before she returned to the classroom with the markers, but since she couldn't carry both the markers and the syllabus (her dress didn't have any pockets), she had to leave the syllabus back at her office. A few minutes later, she needed to refer to the syllabus again, and, not wanting to steal a syllabus from the students, she decided to go back to her office.

She then came back with the syllabus and started discussing the textbook, which was back in her office, so she went back to her office to get the textbook so she could read a few example problems. Once she read a problem, she wanted to show the class how to work the problem, which required a marker, so she went back to her office to get the markers. After coming back, she started writing down the problem, but because she didn't have the textbook, she kept having to go back to her office to get the textbook. Before she had finished writing down the first problem, class ended, and she felt terrible. She immediately went out to buy a backpack before her second class started.

The Second Class

Now that the professor had her backpack, she realized that she wouldn't lose anything she brought to class if she put it in her backpack immediately after she finished using it, so she updated her old rule of only putting things down in her office to the new rule of either putting things down in her office or putting things in her backpack when she stopped using them.

She also updated her labeling scheme, where she had a list of everything in her backpack and where she could find it in her backpack. If she took something from her backpack out, she would have to either put it back where her list said it was or change the list. If she took something from her backpack and put it in her office, she would remove it from the list.

First, she reasoned that putting everything in her office into her backpack would be efficient since she would never have to go back to her office, but then she realized that she wasn't a forklift and trying to lift her entire office would probably injure her. Plus, it's not like she could fit everything into her backpack anyway. She had to get a small backpack since larger backpacks cost way more than smaller backpacks. And even if she could fit everything into her backpack and carry everything to the classroom, finding anything in the backpack would be a nightmare.

Anyway, she decided that she would pack everything she needed for her next two classes since she could fit everything she needed in her backpack with a little room to spare. If she forgot anything, she would go back to her office and get it.

Her second class went much better than her first class. She got through the entire syllabus, but she knew that she could be more efficient with her time. For example, because she didn't want to lose anything, she kept having to go back to her backpack to switch between different colors of markers. When she wanted to switch between the textbook and the syllabus, she had to go back to her backpack.

The Third Class

For the last class of the day, she decided that she would go buy a few magnets for the board so she could stick anything light like a syllabus on the board. She also bought a magnetic marker tray so she could keep all her markers in a little basket on the board for easy access (She didn't use their marker trays since they were covered with markers of varying quality.). When she finished using a marker, she would put them back in the marker tray, and when she finished using anything she had put on the board, she would put it right back where she took it from. Since she had reached her allotted budget for her classes, she decided she would buy nothing else. Since she would have to erase the board at the end of class, she would certainly remember to pick up her belongings, so she could now leave things in her office, in her backpack, and on the board.

Her third class finished in half the time of her second class. Once she took all the markers and her syllabus out of the backpack, the only time she ever needed to go into her backpack was to take out the textbook.

After Her First Day

As she got more and more stuff, she eventually had to move some stuff out of her office and into her home. Just as she couldn't fit everything in her office into her backpack, she couldn't fit everything she owned into her office, so she decided that she would make sure that her office had everything it needed for the week. And if she needed something that wasn't in her office, she could drive home, pick it up, and bring it to the office.

From then on, she continued to teach efficiently, much to the delight of her students, who often got out of class five minutes early.

Explaining the Allegory

Designing how your computer manages memory is almost exactly like figuring out an efficient way to have everything you need for a class. I'm going to list all the analogous concepts between the allegory and the computer, and then I'm going to explain them immediately. If you're confused by the computer terms, think entirely in terms of the allegory, then just replace the allegory terms with the computer terms one step at a time.

Allegory Computer
The Professor CPU
Her hands Registers
Classes Programs
Labels and Labeling System Memory Addresses
Her Office RAM
Her Backpack L2 Cache
The Magnets and Marker Basket L1 Cache
Her House Secondary Storage

CPU

The Central Processing Unit (CPU for short) is the brain of a computer. It decides what to do next, how to organize things, etc. I could say a lot about it, but treat it just like the professor.

Registers

Let's say you want to make a computer. First, to do anything on the computer, you have to have some data stored inside the CPU itself, otherwise the CPU can't interact with data at all. Just as the professor needed some way to interact with what she brought to class, so your computer needs some way to interact with its data. You decide that you're going to give the CPU some registers, each of which consists of a few bytes of memory. Accessing memory from registers is almost instantaneous, but you can't have too many registers for reasons involving the specifics hardware. In general, you will have anywhere from four to sixty four (or some other power of two) registers.

Programs

The actual content of the classes determines what stuff you'll need to bring to each class, just as a program determines what the CPU needs to do.

Memory Addresses

You'll also need a way to identify which data you want, so you need to come up with a labeling scheme like the professor. The labeling scheme should consist of a list of what you want to find and where it is. To keep track of where it is, you use a memory address. A memory address is a number that corresponds to a specific location in memory. On a computer, each memory address is a specific number that uniquely identifies a byte of memory.

RAM

Now, you can't store an entire program in the registers alone in the same way the professor couldn't just hold everything she owned in her hands, so you need to make a place to store information such as files and the values of variables. You decide to make a component of the computer that can store a lot of memory, which you call Random Access Memory or RAM for short. Since computers have to be way more organized than an office, you're going to make RAM one continuous list of bytes and use a memory address to access specific bytes.

Random Access?

In STEM, random means you cannot tell me with certainty what's going to happen next. For example, if I flip a coin, you cannot tell me if it's going to be heads or tails, even if you could tell me the likelihood of each outcome. In another example, if I were to roll two dice, I can tell you that the sum will be seven one out of every six rolls, but I can't tell you what the next roll will be.

The term Random Access X just means that the computer does not know what element of X will be accessed next. Random Access Memory means that the computer doesn't know what memory address you're going to read from or write to next.

The earliest modern computers just had registers and RAM for memory. In fact, a lot of computers like calculators (at least up to the TI-89 Titanium) and early video games just had RAM, which meant that you would lose your data if you turned them off. Old games like Metroid used to have passwords that would indicate your progress specifically because they couldn't save the data (The Legend of Zelda was an early exception because it had save files.).

The Caches

The biggest problem with RAM is that it's slow. Just like going to the office to get something took a while for the professor, getting data from RAM takes a while (for computers, anyway). If you think back to the allegory, her original problem was that she had nowhere to store things she had already taken from the office, so she decided that she would create a place to store it that would have faster access at the expense of not being able to store everything.

In computer science, a cache consists of a place where you store things you've already fetched from a larger and slower source of data.

In terms of real world experience, say you're working on a project or a paper or something else that requires you to do research. Now, you will end up making a bunch of searches that don't get you anything useful, but say you find a source that you really like. You know that you're probably going to refer back to it later, so you'll want to somehow save a reference to it, whether it includes bookmarking the page, copying the link, or whatever. In saving something you have already fetched from a larger (the internet has way more memory than your computer) and slower (compare all the time you spent searching for the source to the time it takes to find it in a list of bookmarks) source of data, you have cached the source.

For a CS example, let's say you're running Twitter and you notice that a lot of people are searching for a specific hashtag, #memoryhierarchy for instance. Instead of having each user searching for #memoryhierarchy send a request to go through all your servers searching for tweets with #memoryhierarchy, have one user do it, store the results of that search on a bunch of servers, then whenever someone searches for #memoryhierarchy, just send them the results you've already found without going back through all the servers. Don't waste time finding something that's already been found or doing something that's already been done.

In the example, the backpack and the magnets were two different levels of cache. Getting stuff you need from the backpack takes less time than getting stuff from the office, but the backpack can't hold nearly as much as the office. Likewise, getting stuff you need from the magnets takes less time than getting stuff from the backpack, but the magnets can't hold nearly as much as the backpack.

Accessing memory from the L1 Cache is faster than accessing memory from the L2 Cache, but you can't store as much in the L1 Cache. Likewise, accessing memory from the L2 Cache is faster than accessing memory from RAM, but you can't store as much in the L2 Cache. On modern computers, you often have an L3 Cache between the L2 Cache and RAM, which follows the same pattern of having more memory than the L2 Cache but with a slower access time.

People have already written articles about caches and performance that contain more specific information, so I'll just link an article for you to read. I would focus on the relative speed of the different levels of memory in that article in particular.

Secondary Storage

What's the only type of memory I haven't mentioned yet? That's right, your hard drive or solid state drive. RAM consists of data that your program has requested and your secondary storage consists of data that your program might request in the future, such as essays you've written, textbooks that you downloaded legally, pictures, etc. If it's a file on your computer, it's in secondary storage. Technically, we can extend the idea of secondary storage to anything that isn't RAM, caches, or registers such as a server, but you get the idea. Anyway, secondary storage corresponds to the professor's house or anywhere else she might store stuff that she could bring to her office but won't for sake of space.

Why Have a Memory Hierarchy?

I hope at this point you see the general pattern: more memory means less speed. Of course, the memory-speed trade-off isn't a fundamental law of nature, but then your computer would be way more expensive since faster memory is more expensive (time is money, after all). The memory hierarchy is a nice compromise between cost, memory, and speed. If you write good programs that use the memory hierarchy to your advantage, you can effectively simulate having a lot of high speed memory.

How to Use the Cache

As a programmer, you do not move things back and forth between caches and RAM, your computer handles that for you. The computer, however, does it in a reliable way that you can use to your advantage. In the following subsections, we're going to discuss the three ways you can optimize your cache use.

These next two principles are known as the Principles of Locality.

Temporal Locality

Let's go back to the allegory before the professor bought her backpack. She spent most of her time going back and forth between her office because she kept switching between things she needed. She used the syllabus first, then her markers, then the syllabus, then the textbook, then the markers, then she kept switching back and forth between the markers and the textbook until her class ended. We can come up with a way she could have sped up her class by adding another rule: once she takes something from her office, she cannot use anything else until she has completely finished using what she currently has. In her case, she could have discussed the syllabus in its entirety without stopping to draw anything on the board, then gone back to her office to trade the syllabus with the textbook, then discussed everything relevant from the textbook, then gone back to her office to get the markers, then used the markers. If she had, she would have only gone back to her office twice, meaning she would have only wasted ten minutes.

In the same way, using the cache to your advantage is quite simple: once you access some data, do everything you can with the data before getting new data.

Now, you might come up with the reasonable objection that she probably would have needed to switch more often than just when she finished using something. I have assumed that she could do what she needed to do with her markers without having to refer back to the syllabus or the textbook and vice versa. The general principle of doing everything you can with what you have still applies, but it just turns out that she couldn't do everything she needed to do with a resource until she finished. If you as a programmer just had registers and RAM, you would also probably need to switch between resources. Since we have more than just RAM and registers, let's add the cache into the mix.

Whenever you access data, the computer will copy it into the fastest memory it can because it thinks you're probably going to use it again in the future, which is known as Temporal Locality. For example, if you're calculating an average value, you'll want the memory for the running total to be accessed quickly, so you'll probably store it in either a register or the L1 cache. Once you're done calculating the average, you might not need the total anymore, so your computer is going to leave it in the L1 cache until the program needs the space for something more important. It will then demote the data for the total down to the L2 cache. If the running total gets used again, it will move back into the L1 cache, otherwise, it will stay in the L2 cache until the program needs the space for something more important. It will then move the data for the total down to the L3 cache and so on until it reaches RAM, where it will either be used again and moved back up to the L1 cache or the program will tell the computer it's done using the memory.

In other words, if you use data once, it will be in the high speed memory until something else replaces it, meaning you will pay no speed penalty if you use the data soon after.

As a real world example, let's say we have an AI that will find text and faces in a set of images. Furthermore, let's say we have a hundred images to check and we can only fit one 4 KB image in the L1 cache at a time.

If we were to write this code:

for image in images:
    find_text(image)

for image in images:
    find_faces(image)

we would end up loading each image once for each for loop, meaning we load an image into the cache 200 times. On the other hand, if we were to write this code:

for image in images:
    find_text(image)
    find_faces(image)

when we run find_faces, we still have the image in the cache from when we accessed the image in find_text, meaning we would only load an image into the cache 100 times.

Spatial Locality

When the professor first started with her labeling scheme, she originally didn't organize her office at all. For example, to find everything she needed for her third class, she would have to check in the leftmost drawer under her desk for her markers, take the third book from the left on the top shelf of her bookshelf, take her computer off her desk, and then get the documents from her class from the fifth binder from the top of her pile of binders. Her other classes had similar requirements. Since she was quite descriptive, it wasn't hard for her to find everything that she needed, but it still took a while to find out where to go in her office, getting what she needed, and put it in her backpack for each resource. Since she wanted to speed up the process of going to class, she decided that she would group together all the resources she needed by class, not by type. She decided to put her documents, the textbook, and anything else she needed for each class next to each other in a box so that she could pick up the box, move it next to her backpack, then take what she needed from the box and put it into her backpack.

Of course, this system wasn't perfect. Sometimes, different classes needed different resources but she couldn't put them in both boxes at the same time, so she decided to put the resource into the box for the class that used the resource most often. For example, her third class used her computer way more often than her other two classes, so she put it in the box for her third class. Other times, she had to put some resources for different classes into the same box to save space. All things considered though, the boxes made her more efficient.

The computer uses the same exact idea to make sure the memory you want to use is in the cache: if you load in some memory, it will try to load in the neighboring memory too in what is known as a cache line, which means that if you want to use the cache to your advantage, keep related data together. A typical cache line is 64 bytes.

In my description of the professor and her boxes, I modified slightly what the computer does to make the allegory make sense. Your computer will not look through the boxes and figure out what you need to put in the cache. Instead, it would put the entire box into the backpack regardless of whether the professor needed everything in the box. Knowing this fact, you should make sure that you fill your boxes with exactly what you need.

As a real world example, let's say that you're programming 2D video game and you want to find out if two objects are colliding. For example, if a character is colliding with a wall, he, she, or it should stop moving. If a character collides with an enemy projectile, the character should take damage. To tell if two objects are colliding, all you need to know is where the collision area or hitbox of the objects are in relation to each other. While you could store the hitbox within the object it represents (such as a player or a projectile) alongside other data like health, stamina, inventory, etc., the hitbox is only relevant to collision detection. Instead, you can store all the hitboxes together. Since a 2D hitbox only contains four numbers for the left, right, top, and bottom sides of the box and each number can be stored in four bytes, you can store sixteen hitboxes in a single 64 byte cache line. If you were to store the hitbox within the collision objects, you would end up having to load way more memory since all the hitboxes are in different cache lines.

Finite Amount of Memory

For the professor, she should try to fill the box with smaller objects like thinner binders and excerpts from the textbook.

Lastly, the computer can only fit so much in the cache, so reducing the amount of memory you use means you can fit more useful memory in the cache. This last principle is just "Don't waste fast memory on irrelevant data".

Swap Memory

What happens when the professor can't keep everything she needs for the day in her office? She has to store some of the stuff in her home, so she decides to store the stuff she won't use for now back at home and pick it up later when she needs it. Since she doesn't want her job to be spread out all over the house and waste time finding it, she assigns one room of her house near the door to let her pick up what she needs to pick up quickly.

On you computer, you have swap memory, which functions just like the room in the professor's house near the door. It's pretty much auxiliary RAM that's way slower to use. If you ever require more memory than your RAM has space for, you will end up using swap memory. Your computer is smart about using swap, meaning it will only put things in swap that you're barely using (such as tabs in your browse that you haven't looked at in a week) and it will only use swap when it runs out of RAM.

What Happens if I Run Out of Swap?

Don't.

Seriously, What Happens if I Run Out of Swap?

In general, your computer screen will probably freeze since the computer is spending all its processing power on whatever you're doing, so it will stop caring about trivial things like changing the pixels of the screen or processing input. Your two options are to wait until it finishes or shut off the computer by holding the power button, as you can't shut down the computer the normal way.

The answer is still "Don't" because you would have to intentionally try to run out of swap or deal with such massive amounts of memory that you can expect to run out of swap. For example, the entire human genome needs about 3 GB of memory to store all the base pairs (technically less, but whatever), meaning you could load the entire human genome into memory and still not even run out of RAM. The only practical way to run out of swap is to have way more programs running at the same time as your computer or twenty tabs of Google Chrome.

In terms of the professor, running out of swap means she's hoarding and needs an intervention.

How Did You Get a 4x Speedup?

If you remember the beginning of the article, I said I was able to achieve a massive speedup by first reducing one of the large matrices to just the last two columns then by using two bytes to represent the numbers instead of four bytes, reducing the memory the program used to about a quarter of what it was. In doing so, I was able to fit more numbers into the cache, which made my code faster. Plus, by halving the size of each individual number, the computer could store twice as many numbers in a single cache line, meaning it only had to fetch memory half as often.

I also guarantee that I did everything with the data I could before getting new data throughout the program, but I would have to make this article even longer, so I'll pass on the example.

Summary

Once again, it's up to you to find out what's making your program slow and make that part faster. Furthermore, using memory properly will not make poorly written code easy to understand nor will it make a bad algorithm a good algorithm. Using memory properly, however, will make a good program into a great program.

If you're coming from the Making Sense of C series, the next article in the series is Types in C, but feel free to make a detour to Representing Integers in Binary if you haven't already read it.

A picture of Joseph Mellor, the author.

Joseph Mellor is a Junior at TU majoring in Physics, Computer Science, and Math. He is also the chief editor of the website and the author of the tumd markdown compiler. If you want to see more of his work, check out his github repository.
Credit to Allison Pennybaker for the picture.