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Fast Topic Matching

A common problem in messaging middleware is that of efficiently matching message topics with interested subscribers. For example, assume we have a set of subscribers, numbered 1 to 3:

Subscriber Match Request
1 forex.usd
2 forex.*
3 stock.nasdaq.msft

And we have a stream of messages, numbered 1 to N:

Message Topic
1 forex.gbp
2 stock.nyse.ibm
3 stock.nyse.ge
4 forex.eur
5 forex.usd
… …
N stock.nasdaq.msft

We are then tasked with routing messages whose topics match the respective subscriber requests, where a “*” wildcard matches any word. This is frequently a bottleneck for message-oriented middleware like ZeroMQ, RabbitMQ, ActiveMQ, TIBCO EMS, et al. Because of this, there are a number of well-known solutions to the problem. In this post, I’ll describe some of these solutions, as well as a novel one, and attempt to quantify them through benchmarking. As usual, the code is available on GitHub.

The Naive Solution

The naive solution is pretty simple: use a hashmap that maps topics to subscribers. Subscribing involves adding a new entry to the map (or appending to a list if it already exists). Matching a message to subscribers involves scanning through every entry in the map, checking if the match request matches the message topic, and returning the subscribers for those that do.

Inserts are approximately O(1) and lookups approximately O(n*m) where n is the number of subscriptions and m is the number of words in a topic. This means the performance of this solution is heavily dependent upon how many subscriptions exist in the map and also the access patterns (rate of reads vs. writes). Since most use cases are heavily biased towards searches rather than updates, the naive solution—unsurprisingly—is not a great option.

The microbenchmark below compares the performance of subscribe, unsubscribe, and lookup (matching) operations, first using an empty hashmap (what we call cold) and then with one containing 1,000 randomly generated 5-word topic subscriptions (what we call hot). With the populated subscription map, lookups are about three orders of magnitude slower, which is why we have to use a log scale in the chart below.

subscribe unsubscribe lookup
cold 172ns 51.2ns 787ns
hot 221ns 55ns 815,787ns


Inverted Bitmap

The inverted bitmap technique builds on the observation that lookups are more frequent than updates and assumes that the search space is finite. Consequently, it shifts some of the cost from the read path to the write path. It works by storing a set of bitmaps, one per topic, or criteria, in the search space. Subscriptions are then assigned an increasing number starting at 0. We analyze each subscription to determine the matching criteria and set the corresponding bits in the criteria bitmaps to 1. For example, assume our search space consists of the following set of topics:

  • forex.usd
  • forex.gbp
  • forex.jpy
  • forex.eur
  • stock.nasdaq
  • stock.nyse

We then have the following subscriptions:

  • 0 = forex.* (matches forex.usd, forex.gbp, forex.jpy, and forex.eur)
  • 1 = stock.nyse (matches stock.nyse)
  • 2 = *.* (matches everything)
  • 3 = stock.* (matches stock.nasdaq and stock.nyse)

When we index the subscriptions above, we get the following set of bitmaps:

 Criteria 0 1 2 3
forex.usd 1 0 1 0
forex.gbp 1 0 1 0
forex.jpy 1 0 1 0
forex.eur 1 0 1 0
stock.nasdaq 0 0 1 1
stock.nyse 0 1 1 1

When we match a message, we simply need to lookup the corresponding bitmap and check the set bits. As we see below, subscribe and unsubscribe are quite expensive with respect to the naive solution, but lookups now fall well below half a microsecond, which is pretty good (the fact that the chart below doesn’t use a log scale like the one above should be an indictment of the naive hashmap-based solution).

subscribe unsubscribe lookup
cold 3,795ns 198ns 380ns
hot 3,863ns 198ns 395ns

The inverted bitmap is a better option than the hashmap when we have a read-heavy workload. One limitation is it requires us to know the search space ahead of time or otherwise requires reindexing which, frankly, is prohibitively expensive.

Optimized Inverted Bitmap

The inverted bitmap technique works well enough, but only if the topic space is fairly static. It also falls over pretty quickly when the topic space and number of subscriptions are large, say, millions of topics and thousands of subscribers. The main benefit of topic-based routing is it allows for faster matching algorithms in contrast to content-based routing, which can be exponentially slower. The truth is, to be useful, your topics probably consist of stock.nyse.ibm, stock.nyse.ge, stock.nasdaq.msft, stock.nasdaq.aapl, etc., not stock.nyse and stock.nasdaq. We could end up with an explosion of topics and, even with efficient bitmaps, the memory consumption tends to be too high despite the fact that most of the bitmaps are quite sparse.

Fortunately, we can reduce the amount of memory we consume using a fairly straightforward optimization. Rather than requiring the entire search space a priori, we simply require the max topic size, in terms of words, e.g. stock.nyse.ibm has a size of 3. We can handle topics of the max size or less, e.g. stock.nyse.bac, stock.nasdaq.txn, forex.usd, index, etc. If we see a message with more words than the max, we can safely assume there are no matching subscriptions.

The optimized inverted bitmap works by splitting topics into their constituent parts. Each constituent position has a set of bitmaps, and we use a technique similar to the one described above on each part. We end up with a bitmap for each constituent which we perform a logical AND on to give a resulting bitmap. Each 1 in the resulting bitmap corresponds to a subscription. This means if the max topic size is n, we only AND at most n bitmaps. Furthermore, if we come across any empty bitmaps, we can stop early since we know there are no matching subscribers.

Let’s say our max topic size is 2 and we have the following subscriptions:

  • 0 = forex.*
  • 1 = stock.nyse
  • 2 = index
  • 3 = stock.*

The inverted bitmap for the first constituent looks like the following:

forex.* stock.nyse index stock.*
null 0 0 0 0
forex 1 0 0 0
stock 0 1 0 1
index 0 0 1 0
other 0 0 0 0

And the second constituent bitmap:

forex.* stock.nyse index stock.*
null 0 0 1 0
nyse 0 1 0 0
other 1 0 0 1

The “null” and “other” rows are worth pointing out. “Null” simply means the topic has no corresponding constituent.  For example, “index” has no second constituent, so “null” is marked. “Other” allows us to limit the number of rows needed such that we only need the ones that appear in subscriptions.  For example, if messages are published on forex.eur, forex.usd, and forex.gbp but I merely subscribe to forex.*, there’s no need to index eur, usd, or gbp. Instead, we just mark the “other” row which will match all of them.

Let’s look at an example using the above bitmaps. Imagine we want to route a message published on forex.eur. We split the topic into its constituents: “forex” and “eur.” We get the row corresponding to “forex” from the first constituent bitmap, the one corresponding to “eur” from the second (other), and then AND the rows.

forex.* stock.nyse index stock.*
1 = forex 1 0 0 0
2 = other 1 0 0 1
AND 1 0 0 0

The forex.* subscription matches.

Let’s try one more example: a message published on stock.nyse.

forex.* stock.nyse index stock.*
1 = stock 0 1 0 1
2 = nyse 0 1 0 1
AND 0 1 0 1

In this case, we also need to OR the “other” row for the second constituent. This gives us a match for stock.nyse and stock.*.

Subscribe operations are significantly faster with the space-optimized inverted bitmap compared to the normal inverted bitmap, but lookups are much slower. However, the optimized version consumes roughly 4.5x less memory for every subscription. The increased flexibility and improved scalability makes the optimized version a better choice for all but the very latency-sensitive use cases.

subscribe unsubscribe lookup
cold 1,053ns 330ns 2,724ns
hot 1,076ns 371ns 3,337ns

Trie

The optimized inverted bitmap improves space complexity, but it does so at the cost of lookup efficiency. Is there a way we can reconcile both time and space complexity? While inverted bitmaps allow for efficient lookups, they are quite wasteful for sparse sets, even when using highly compressed bitmaps like Roaring bitmaps.

Tries can often be more space efficient in these circumstances. When we add a subscription, we descend the trie, adding nodes along the way as necessary, until we run out of words in the topic. Finally, we add some metadata containing the subscription information to the last node in the chain. To match a message topic, we perform a similar traversal. If a node doesn’t exist in the chain, we know there are no subscribers. One downside of this method is, in order to support wildcards, we must backtrack on a literal match and check the “*” branch as well.

For the given set of subscriptions, the trie would look something like the following:

  • forex.*
  • stock.nyse
  • index
  • stock.*

You might be tempted to ask: “why do we even need the “*” nodes? When someone subscribes to stock.*, just follow all branches after “stock” and add the subscriber.” This would indeed move the backtracking cost from the read path to the write path, but—like the first inverted bitmap we looked at—it only works if the search space is known ahead of time. It would also largely negate the memory-usage benefits we’re looking for since it would require pre-indexing all topics while requiring a finite search space.

It turns out, this trie technique is how systems like ZeroMQ and RabbitMQ implement their topic matching due to its balance between space and time complexity and overall performance predictability.

subscribe unsubscribe lookup
cold 406ns 221ns 2,145ns
hot 443ns 257ns 2,278ns

We can see that, compared to the optimized inverted bitmap, the trie performs much more predictably with relation to the number of subscriptions held.

Concurrent Subscription Trie

One thing we haven’t paid much attention to so far is concurrency. Indeed, message-oriented middleware is typically highly concurrent since they have to deal with heavy IO (reading messages from the wire, writing messages to the wire, reading messages from disk, writing messages to disk, etc.) and CPU operations (like topic matching and routing). Subscribe, unsubscribe, and lookups are usually all happening in different threads of execution. This is especially important when we want to talk advantage of multi-core processors.

It wasn’t shown, but all of the preceding algorithms used global locks to ensure thread safety between read and write operations, making the data structures safe for concurrent use. However, the microbenchmarks don’t really show the impact of this, which we will see momentarily.

Lock-freedom, which I’ve written about, allows us to increase throughput at the expense of increased tail latency.

Lock-free concurrency means that while a particular thread of execution may be blocked, all CPUs are able to continue processing other work. For example, imagine a program that protects access to some resource using a mutex. If a thread acquires this mutex and is subsequently preempted, no other thread can proceed until this thread is rescheduled by the OS. If the scheduler is adversarial, it may never resume execution of the thread, and the program would be effectively deadlocked. A key point, however, is that the mere lack of a lock does not guarantee a program is lock-free. In this context, “lock” really refers to deadlock, livelock, or the misdeeds of a malevolent scheduler.

The concurrent subscription trie, or CS-trie,  is a new take on the trie-based solution described earlier. It combines the idea of the topic-matching trie with that of a Ctrie, or concurrent trie, which is a non-blocking concurrent hash trie.

The fundamental problem with the trie, as it relates to concurrency, is it requires a global lock, which severely limits throughput. To address this, the CS-trie uses indirection nodes, or I-nodes, which remain present in the trie even as the nodes above and below change. Subscriptions are then added or removed by creating a copy of the respective node, and performing a CAS on its parent I-node. This allows us to add, remove, and lookup subscriptions concurrently and in a lock-free, linearizable manner.

For the given set of subscribers, labeled x, y, and z, the CS-trie would look something like the following:

  • x = foo, bar, bar.baz
  • y = foo, bar.qux
  • z = bar.*

Lookups on the CS-trie perform, on average, better than the standard trie, and the CS-trie scales better with respect to concurrent operations.

subscribe unsubscribe lookup
cold 412ns 245ns 1,615ns
hot 471ns 280ns 1,637ns

Latency Comparison

The chart below shows the topic-matching operation latencies for all of the algorithms side-by-side. First, we look at the performance of a cold start (no subscriptions) and then the performance of a hot start (1,000 subscriptions).

Throughput Comparison

So far, we’ve looked at the latency of individual topic-matching operations. Next, we look at overall throughput of each of the algorithms and their memory footprint.

 algorithm msg/sec
naive  4,053.48
inverted bitmap  1,052,315.02
optimized inverted bitmap  130,705.98
trie  248,762.10
cs-trie  340,910.64

On the surface, the inverted bitmap looks like the clear winner, clocking in at over 1 million matches per second. However, we know the inverted bitmap does not scale and, indeed, this becomes clear when we look at memory consumption, underscored by the fact that the below chart uses a log scale.

Scalability with Respect to Concurrency

Lastly, we’ll look at how each of these algorithms scales with respect to concurrency. We do this by performing concurrent operations and varying the level of concurrency and number of operations. We start with a 50-50 split between reads and writes. We vary the number of goroutines from 2 to 16 (the benchmark was run using a 2.6 GHz Intel Core i7 processor with 8 logical cores). Each goroutine performs 1,000 reads or 1,000 writes. For example, the 2-goroutine benchmark performs 1,000 reads and 1,000 writes, the 4-goroutine benchmark performs 2,000 reads and 2,000 writes, etc. We then measure the total amount of time needed to complete the workload.

We can see that the tries hardly even register on the scale above, so we’ll plot them separately.

The tries are clearly much more efficient than the other solutions, but the CS-trie in particular scales well to the increased workload and concurrency.

Since most workloads are heavily biased towards reads over writes, we’ll run a separate benchmark that uses a 90-10 split reads and writes. This should hopefully provide a more realistic result.

The results look, more or less, like what we would expect, with the reduced writes improving the inverted bitmap performance. The CS-trie still scales quite well in comparison to the global-lock trie.

Conclusion

As we’ve seen, there are several approaches to consider to implement fast topic matching. There are also several aspects to look at: read/write access patterns, time complexity, space complexity, throughput, and latency.

The naive hashmap solution is generally a poor choice due to its prohibitively expensive lookup time. Inverted bitmaps offer a better solution. The standard implementation is reasonable if the search space is finite, small, and known a priori, especially if read latency is critical. The space-optimized version is a better choice for scalability, offering a good balance between read and write performance while keeping a small memory footprint. The trie is an even better choice, providing lower latency than the optimized inverted bitmap and consuming less memory. It’s particularly good if the subscription tree is sparse and topics are not known a priori. Lastly, the concurrent subscription trie is the best option if there is high concurrency and throughput matters. It offers similar performance to the trie but scales better. The only downside is an increase in implementation complexity.

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So You Wanna Go Fast?

I originally proposed this as a GopherCon talk on writing “high-performance Go”, which is why it may seem rambling, incoherent, and—at times—not at all related to Go. The talk was rejected (probably because of the rambling and incoherence), but I still think it’s a subject worth exploring. The good news is, since it was rejected, I can take this where I want. The remainder of this piece is mostly the outline of that talk with some parts filled in, some meandering stories which may or may not pertain to the topic, and some lessons learned along the way. I think it might make a good talk one day, but this will have to do for now.

We work on some interesting things at Workiva—graph traversal, distributed and in-memory calculation engines, low-latency messaging systems, databases optimized for two-dimensional data computation. It turns out, when you want to build a complicated financial-reporting suite with the simplicity and speed of Microsoft Office, and put it entirely in the cloud, you can’t really just plumb some crap together and call it good. It also turns out that when you try to do this, performance becomes kind of important, not because of the complexity of the data—after all, it’s mostly just numbers and formulas—but because of the scale of it. Now, distribute that data in the cloud, consider the security and compliance implications associated with it, add in some collaboration and control mechanisms, and you’ve got yourself some pretty monumental engineering problems.

As I hinted at, performance starts to be really important, whether it’s performing a formula evaluation, publishing a data-change event, or opening up a workbook containing a million rows of data (accountants are weird). A lot of the backend systems powering all of this are, for better or worse, written in Go. Go is, of course, a garbage-collected language, and it compares closely to Java (though the latter has over 20 years invested in it, while the former has about seven).

At this point, you might be asking, “why not C?” It’s honestly a good question to ask, but the reality is there is always history. The first solution was written in Python on Google App Engine (something about MVPs, setting your customers’ expectations low, and giving yourself room to improve?). This was before Go was even a thing, though Java and C were definitely things, but this was a startup. And it was Python. And it was on App Engine. I don’t know exactly what led to those combination of things—I wasn’t there—but, truthfully, App Engine probably played a large role in the company’s early success. Python and App Engine were fast. Not like “this code is fucking fast” fast—what we call performance—more like “we need to get this shit working so we have jobs tomorrow” fast—what we call delivery. I don’t envy that kind of fast, but when you’re a startup trying to disrupt, speed to market matters a hell of a lot more than the speed of your software.

I’ve talked about App Engine at length before. Ultimately, you hit the ceiling of what you can do with it, and you have to migrate off (if you’re a business that is trying to grow, anyway). We hit that migration point at a really weird, uncomfortable time. This was right when Docker was starting to become a thing, and microservices were this thing that everybody was talking about but nobody was doing. Google had been successfully using containers for years, and Netflix was all about microservices. Everybody wanted to be like them, but no one really knew how—but it was the future (unikernels are the new future, by the way).

The problem is—coming from a PaaS like App Engine that does your own laundry—you don’t have the tools, skills, or experience needed to hit the ground running, so you kind of drunkenly stumble your way there. You don’t even have a DevOps team because you didn’t need one! Nobody knew how to use Docker, which is why at the first Dockercon, five people got on stage and presented five solutions to the same problem. It was the blind leading the blind. I love this article by Jesper L. Andersen, How to build stable systems, which contains a treasure trove of practical engineering tips. The very last paragraph of the article reads:

Docker is not mature (Feb 2016). Avoid it in production for now until it matures. Currently Docker is a time sink not fulfilling its promises. This will change over time, so know when to adopt it.

Trying to build microservices using Docker while everyone is stumbling over themselves was, and continues to be, a painful process, exacerbated by the heavy weight suddenly lifted by leaving App Engine. It’s not great if you want to go fast. App Engine made scaling easy by restricting you in what you could do, but once that burden was removed, it was off to the races. What people might not have realized, however, was that App Engine also made distributed systems easy by restricting you in what you could do. Some seem to think the limitations enforced by App Engine are there to make their lives harder or make Google richer (trust me, they’d bill you more if they could), so why would we have similar limitations in our own infrastructure? App Engine makes these limitations, of course, so that it can actually scale. Don’t take that for granted.

App Engine was stateless, so the natural tendency once you’re off it was to make everything stateful. And we did. What I don’t think we realized was that we were, in effect, trading one type of fast for the other—performance for delivery. We can build software that’s fast and runs on your desktop PC like in the 90’s, but now you want to put that in the cloud and make it scale? It takes a big infrastructure investment. It also takes a big time investment. Neither of which are good if you want to go fast, especially when you’re using enough microservices, Docker, and Go to rattle the Hacker News fart chamber. You kind of get caught in this endless rut of innovation that you almost lose your balance. Leaving the statelessness of App Engine for more stateful pastures was sort of like an infant learning to walk. You look down and it dawns on you—you have legs! So you run with it, because that’s amazing, and you stumble spectacularly a few times along the way. Finally, you realize maybe running full speed isn’t the best idea for someone who just learned to walk.

We were also making this transition while Go had started reaching critical mass. Every other headline in the tech aggregators was “why we switched to Go and you should too.” And we did. I swear this post has a point.

Tips for Writing High-Performance Go

By now, I’ve forgotten what I was writing about, but I promised this post was about Go. It is, and it’s largely about performance fast, not delivery fast—the two are often at odds with each other. Everything up until this point was mostly just useless context and ranting. But it also shows you that we are solving some hard problems and why we are where we are. There is always history.

I work with a lot of smart people. Many of us have a near obsession with performance, but the point I was attempting to make earlier is we’re trying to push the boundaries of what you can expect from cloud software. App Engine had some rigid boundaries, so we made a change. Since adopting Go, we’ve learned a lot about how to make things fast and how to make Go work in the world of systems programming.

Go’s simplicity and concurrency model make it an appealing choice for backend systems, but the larger question is how does it fare for latency-sensitive applications? Is it worth sacrificing the simplicity of the language to make it faster? Let’s walk through a few areas of performance optimization in Go—namely language features, memory management, and concurrency—and try to make that determination. All of the code for the benchmarks presented here are available on GitHub.

Channels

Channels in Go get a lot of attention because they are a convenient concurrency primitive, but it’s important to be aware of their performance implications. Usually the performance is “good enough” for most cases, but in certain latency-critical situations, they can pose a bottleneck. Channels are not magic. Under the hood, they are just doing locking. This works great in a single-threaded application where there is no lock contention, but in a multithreaded environment, performance significantly degrades. We can mimic a channel’s semantics quite easily using a lock-free ring buffer.

The first benchmark looks at the performance of a single-item-buffered channel and ring buffer with a single producer and single consumer. First, we look at the performance in the single-threaded case (GOMAXPROCS=1).

BenchmarkChannel 3000000 512 ns/op
BenchmarkRingBuffer 20000000 80.9 ns/op

As you can see, the ring buffer is roughly six times faster (if you’re unfamiliar with Go’s benchmarking tool, the first number next to the benchmark name indicates the number of times the benchmark was run before giving a stable result). Next, we look at the same benchmark with GOMAXPROCS=8.

BenchmarkChannel-8 3000000 542 ns/op
BenchmarkRingBuffer-8 10000000 182 ns/op

The ring buffer is almost three times faster.

Channels are often used to distribute work across a pool of workers. In this benchmark, we look at performance with high read contention on a buffered channel and ring buffer. The GOMAXPROCS=1 test shows how channels are decidedly better for single-threaded systems.

BenchmarkChannelReadContention 10000000 148 ns/op
BenchmarkRingBufferReadContention 10000 390195 ns/op

However, the ring buffer is faster in the multithreaded case:

BenchmarkChannelReadContention-8 1000000 3105 ns/op
BenchmarkRingBufferReadContention-8 3000000 411 ns/op

Lastly, we look at performance with contention on both the reader and writer. Again, the ring buffer’s performance is much worse in the single-threaded case but better in the multithreaded case.

BenchmarkChannelContention 10000 160892 ns/op
BenchmarkRingBufferContention 2 806834344 ns/op
BenchmarkChannelContention-8 5000 314428 ns/op
BenchmarkRingBufferContention-8 10000 182557 ns/op

The lock-free ring buffer achieves thread safety using only CAS operations. We can see that deciding to use it over the channel depends largely on the number of OS threads available to the program. For most systems, GOMAXPROCS > 1, so the lock-free ring buffer tends to be a better option when performance matters. Channels are a rather poor choice for performant access to shared state in a multithreaded system.

Defer

Defer is a useful language feature in Go for readability and avoiding bugs related to releasing resources. For example, when we open a file to read, we need to be careful to close it when we’re done. Without defer, we need to ensure the file is closed at each exit point of the function.

This is really error-prone since it’s easy to miss a return point. Defer solves this problem by effectively adding the cleanup code to the stack and invoking it when the enclosing function returns.

At first glance, one would think defer statements could be completely optimized away by the compiler. If I defer something at the beginning of a function, simply insert the closure at each point the function returns. However, it’s more complicated than this. For example, we can defer a call within a conditional statement or a loop. The first case might require the compiler to track the condition leading to the defer. The compiler would also need to be able to determine if a statement can panic since this is another exit point for a function. Statically proving this seems to be, at least on the surface, an undecidable problem.

The point is defer is not a zero-cost abstraction. We can benchmark it to show the performance overhead. In this benchmark, we compare locking a mutex and unlocking it with a defer in a loop to locking a mutex and unlocking it without defer.

BenchmarkMutexDeferUnlock-8 20000000 96.6 ns/op
BenchmarkMutexUnlock-8 100000000 19.5 ns/op

Using defer is almost five times slower in this test. To be fair, we’re looking at a difference of 77 nanoseconds, but in a tight loop on a critical path, this adds up. One trend you’ll notice with these optimizations is it’s usually up to the developer to make a trade-off between performance and readability. Optimization rarely comes free.

Reflection and JSON

Reflection is generally slow and should be avoided for latency-sensitive applications. JSON is a common data-interchange format, but Go’s encoding/json package relies on reflection to marshal and unmarshal structs. With ffjson, we can use code generation to avoid reflection and benchmark the difference.

BenchmarkJSONReflectionMarshal-8 200000 7063 ns/op
BenchmarkJSONMarshal-8 500000 3981 ns/op

BenchmarkJSONReflectionUnmarshal-8 200000 9362 ns/op
BenchmarkJSONUnmarshal-8 300000 5839 ns/op

Code-generated JSON is about 38% faster than the standard library’s reflection-based implementation. Of course, if we’re concerned about performance, we should really avoid JSON altogether. MessagePack is a better option with serialization code that can also be generated. In this benchmark, we use the msgp library and compare its performance to JSON.

BenchmarkMsgpackMarshal-8 3000000 555 ns/op
BenchmarkJSONReflectionMarshal-8 200000 7063 ns/op
BenchmarkJSONMarshal-8 500000 3981 ns/op

BenchmarkMsgpackUnmarshal-8 20000000 94.6 ns/op
BenchmarkJSONReflectionUnmarshal-8 200000 9362 ns/op
BenchmarkJSONUnmarshal-8 300000 5839 ns/op

The difference here is dramatic. Even when compared to the generated JSON serialization code, MessagePack is significantly faster.

If we’re really trying to micro-optimize, we should also be careful to avoid using interfaces, which have some overhead not just with marshaling but also method invocations. As with other kinds of dynamic dispatch, there is a cost of indirection when performing a lookup for the method call at runtime. The compiler is unable to inline these calls.

BenchmarkJSONReflectionUnmarshal-8 200000 9362 ns/op
BenchmarkJSONReflectionUnmarshalIface-8 200000 10099 ns/op

We can also look at the overhead of the invocation lookup, I2T, which converts an interface to its backing concrete type. This benchmark calls the same method on the same struct. The difference is the second one holds a reference to an interface which the struct implements.

BenchmarkStructMethodCall-8 2000000000 0.44 ns/op
BenchmarkIfaceMethodCall-8 1000000000 2.97 ns/op

Sorting is a more practical example that shows the performance difference. In this benchmark, we compare sorting a slice of 1,000,000 structs and 1,000,000 interfaces backed by the same struct. Sorting the structs is nearly 92% faster than sorting the interfaces.

BenchmarkSortStruct-8 10 105276994 ns/op
BenchmarkSortIface-8 5 286123558 ns/op

To summarize, avoid JSON if possible. If you need it, generate the marshaling and unmarshaling code. In general, it’s best to avoid code that relies on reflection and interfaces and instead write code that uses concrete types. Unfortunately, this often leads to a lot of duplicated code, so it’s best to abstract this with code generation. Once again, the trade-off manifests.

Memory Management

Go doesn’t actually expose heap or stack allocation directly to the user. In fact, the words “heap” and “stack” do not appear anywhere in the language specification. This means anything pertaining to the stack and heap are technically implementation-dependent. In practice, of course, Go does have a stack per goroutine and a heap. The compiler does escape analysis to determine if an object can live on the stack or needs to be allocated in the heap.

Unsurprisingly, avoiding heap allocations can be a major area of optimization. By allocating on the stack, we avoid expensive malloc calls, as the benchmark below shows.

BenchmarkAllocateHeap-8 20000000 62.3 ns/op 96 B/op 1 allocs/op
BenchmarkAllocateStack-8 100000000 11.6 ns/op 0 B/op 0 allocs/op

Naturally, passing by reference is faster than passing by value since the former requires copying only a pointer while the latter requires copying values. The difference is negligible with the struct used in these benchmarks, though it largely depends on what has to be copied. Keep in mind there are also likely some compiler optimizations being performed in this synthetic benchmark.

BenchmarkPassByReference-8 1000000000 2.35 ns/op
BenchmarkPassByValue-8 200000000 6.36 ns/op

However, the larger issue with heap allocation is garbage collection. If we’re creating lots of short-lived objects, we’ll cause the GC to thrash. Object pooling becomes quite important in these scenarios. In this benchmark, we compare allocating structs in 10 concurrent goroutines on the heap vs. using a sync.Pool for the same purpose. Pooling yields a 5x improvement.

BenchmarkConcurrentStructAllocate-8 5000000 337 ns/op
BenchmarkConcurrentStructPool-8 20000000 65.5 ns/op

It’s important to point out that Go’s sync.Pool is drained during garbage collection. The purpose of sync.Pool is to reuse memory between garbage collections. One can maintain their own free list of objects to hold onto memory across garbage collection cycles, though this arguably subverts the purpose of a garbage collector. Go’s pprof tool is extremely useful for profiling memory usage. Use it before blindly making memory optimizations.

False Sharing

When performance really matters, you have to start thinking at the hardware level. Formula One driver Jackie Stewart is famous for once saying, “You don’t have to be an engineer to be be a racing driver, but you do have to have mechanical sympathy.” Having a deep understanding of the inner workings of a car makes you a better driver. Likewise, having an understanding of how a computer actually works makes you a better programmer. For example, how is memory laid out? How do CPU caches work? How do hard disks work?

Memory bandwidth continues to be a limited resource in modern CPU architectures, so caching becomes extremely important to prevent performance bottlenecks. Modern multiprocessor CPUs cache data in small lines, typically 64 bytes in size, to avoid expensive trips to main memory. A write to a piece of memory will cause the CPU cache to evict that line to maintain cache coherency. A subsequent read on that address requires a refresh of the cache line. This is a phenomenon known as false sharing, and it’s especially problematic when multiple processors are accessing independent data in the same cache line.

Imagine a struct in Go and how it’s laid out in memory. Let’s use the ring buffer from earlier as an example. Here’s what that struct might normally look like:

The queue and dequeue fields are used to determine producer and consumer positions, respectively. These fields, which are both eight bytes in size, are concurrently accessed and modified by multiple threads to add and remove items from the queue. Since these two fields are positioned contiguously in memory and occupy only 16 bytes of memory, it’s likely they will stored in a single CPU cache line. Therefore, writing to one will result in evicting the other, meaning a subsequent read will stall. In more concrete terms, adding or removing things from the ring buffer will cause subsequent operations to be slower and will result in lots of thrashing of the CPU cache.

We can modify the struct by adding padding between fields. Each padding is the width of a single CPU cache line to guarantee the fields end up in different lines. What we end up with is the following:

How big a difference does padding out CPU cache lines actually make? As with anything, it depends. It depends on the amount of multiprocessing. It depends on the amount of contention. It depends on memory layout. There are many factors to consider, but we should always use data to back our decisions. We can benchmark operations on the ring buffer with and without padding to see what the difference is in practice.

First, we benchmark a single producer and single consumer, each running in a goroutine. With this test, the improvement between padded and unpadded is fairly small, about 15%.

BenchmarkRingBufferSPSC-8 10000000 156 ns/op
BenchmarkRingBufferPaddedSPSC-8 10000000 132 ns/op

However, when we have multiple producers and multiple consumers, say 100 each, the difference becomes slightly more pronounced. In this case, the padded version is about 36% faster.

BenchmarkRingBufferMPMC-8 100000 27763 ns/op
BenchmarkRingBufferPaddedMPMC-8 100000 17860 ns/op

False sharing is a very real problem. Depending on the amount of concurrency and memory contention, it can be worth introducing padding to help alleviate its effects. These numbers might seem negligible, but they start to add up, particularly in situations where every clock cycle counts.

Lock-Freedom

Lock-free data structures are important for fully utilizing multiple cores. Considering Go is targeted at highly concurrent use cases, it doesn’t offer much in the way of lock-freedom. The encouragement seems to be largely directed towards channels and, to a lesser extent, mutexes.

That said, the standard library does offer the usual low-level memory primitives with the atomic package. Compare-and-swap, atomic pointer access—it’s all there. However, use of the atomic package is heavily discouraged:

We generally don’t want sync/atomic to be used at all…Experience has shown us again and again that very very few people are capable of writing correct code that uses atomic operations…If we had thought of internal packages when we added the sync/atomic package, perhaps we would have used that. Now we can’t remove the package because of the Go 1 guarantee.

How hard can lock-free really be though? Just rub some CAS on it and call it a day, right? After a sufficient amount of hubris, I’ve come to learn that it’s definitely a double-edged sword. Lock-free code can get complicated in a hurry. The atomic and unsafe packages are not easy to use, at least not at first. The latter gets its name for a reason. Tread lightly—this is dangerous territory. Even more so, writing lock-free algorithms can be tricky and error-prone. Simple lock-free data structures, like the ring buffer, are pretty manageable, but anything more than that starts to get hairy.

The Ctrie, which I wrote about in detail, was my foray into the world of lock-free data structures beyond your standard fare of queues and lists. Though the theory is reasonably understandable, the implementation is thoroughly complex. In fact, the complexity largely stems from the lack of a native double compare-and-swap, which is needed to atomically compare indirection nodes (to detect mutations on the tree) and node generations (to detect snapshots taken of the tree). Since no hardware provides such an operation, it has to be simulated using standard primitives (and it can).

The first Ctrie implementation was actually horribly broken, and not even because I was using Go’s synchronization primitives incorrectly. Instead, I had made an incorrect assumption about the language. Each node in a Ctrie has a generation associated with it. When a snapshot is taken of the tree, its root node is copied to a new generation. As nodes in the tree are accessed, they are lazily copied to the new generation (à la persistent data structures), allowing for constant-time snapshotting. To avoid integer overflow, we use objects allocated on the heap to demarcate generations. In Go, this is done using an empty struct. In Java, two newly constructed Objects are not equivalent when compared since their memory addresses will be different. I made a blind assumption that the same was true in Go, when in fact, it’s not. Literally the last paragraph of the Go language specification reads:

A struct or array type has size zero if it contains no fields (or elements, respectively) that have a size greater than zero. Two distinct zero-size variables may have the same address in memory.

Oops. The result was that two different generations were considered equivalent, so the double compare-and-swap always succeeded. This allowed snapshots to potentially put the tree in an inconsistent state. That was a fun bug to track down. Debugging highly concurrent, lock-free code is hell. If you don’t get it right the first time, you’ll end up sinking a ton of time into fixing it, but only after some really subtle bugs crop up. And it’s unlikely you get it right the first time. You win this time, Ian Lance Taylor.

But wait! Obviously there’s some payoff with complicated lock-free algorithms or why else would one subject themselves to this? With the Ctrie, lookup performance is comparable to a synchronized map or a concurrent skip list. Inserts are more expensive due to the increased indirection. The real benefit of the Ctrie is its scalability in terms of memory consumption, which, unlike most hash tables, is always a function of the number of keys currently in the tree. The other advantage is its ability to perform constant-time, linearizable snapshots. We can compare performing a “snapshot” on a synchronized map concurrently in 100 different goroutines with the same test using a Ctrie:

BenchmarkConcurrentSnapshotMap-8 1000 9941784 ns/op
BenchmarkConcurrentSnapshotCtrie-8 20000 90412 ns/op

Depending on access patterns, lock-free data structures can offer better performance in multithreaded systems. For example, the NATS message bus uses a synchronized map-based structure to perform subscription matching. When compared with a Ctrie-inspired, lock-free structure, throughput scales a lot better. The blue line is the lock-based data structure, while the red line is the lock-free implementation.

matchbox_bench_1_1

Avoiding locks can be a boon depending on the situation. The advantage was apparent when comparing the ring buffer to the channel. Nonetheless, it’s important to weigh any benefit against the added complexity of the code. In fact, sometimes lock-freedom doesn’t provide any tangible benefit at all!

A Note on Optimization

As we’ve seen throughout this post, performance optimization almost always comes with a cost. Identifying and understanding optimizations themselves is just the first step. What’s more important is understanding when and where to apply them. The famous quote by C. A. R. Hoare, popularized by Donald Knuth, has become a longtime adage of programmers:

The real problem is that programmers have spent far too much time worrying about efficiency in the wrong places and at the wrong times; premature optimization is the root of all evil (or at least most of it) in programming.

Though the point of this quote is not to eliminate optimization altogether, it’s to learn how to strike a balance between speeds—speed of an algorithm, speed of delivery, speed of maintenance, speed of a system. It’s a highly subjective topic, and there is no single rule of thumb. Is premature optimization the root of all evil? Should I just make it work, then make it fast? Does it need to be fast at all? These are not binary decisions. For example, sometimes making it work then making it fast is impossible if there is a fundamental problem in the design.

However, I will say focus on optimizing along the critical path and outward from that only as necessary. The further you get from that critical path, the more likely your return on investment is to diminish and the more time you end up wasting. It’s important to identify what adequate performance is. Do not spend time going beyond that point. This is an area where data-driven decisions are key—be empirical, not impulsive. More important, be practical. There’s no use shaving nanoseconds off of an operation if it just doesn’t matter. There is more to going fast than fast code.

Wrapping Up

If you’ve made it this far, congratulations, there might be something wrong with you. We’ve learned that there are really two kinds of fast in software—delivery and performance.  Customers want the first, developers want the second, and CTOs want both. The first is by far the most important, at least when you’re trying to go to market. The second is something you need to plan for and iterate on. Both are usually at odds with each other.

Perhaps more interestingly, we looked at a few ways we can eke out that extra bit of performance in Go and make it viable for low-latency systems. The language is designed to be simple, but that simplicity can sometimes come at a price. Like the trade-off between the two fasts, there is a similar trade-off between code lifecycle and code performance. Speed comes at the cost of simplicity, at the cost of development time, and at the cost of continued maintenance. Choose wisely.

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Breaking and Entering: Lose the Lock While Embracing Concurrency

This article originally appeared on Workiva’s engineering blog as a two-part series.

Providing robust message routing was a priority for us at Workiva when building our distributed messaging infrastructure. This encompassed directed messaging, which allows us to route messages to specific endpoints based on service or client identifiers, but also topic fan-out with support for wildcards and pattern matching.

Existing message-oriented middleware, such as RabbitMQ, provide varying levels of support for these but don’t offer the rich features needed to power Wdesk. This includes transport fallback with graceful degradation, tunable qualities of service, support for client-side messaging, and pluggable authentication middleware. As such, we set out to build a new system, not by reinventing the wheel, but by repurposing it.

Eventually, we settled on Apache Kafka as our wheel, or perhaps more accurately, our log. Kafka demonstrates a telling story of speed, scalability, and fault tolerance—each a requisite component of any reliable messaging system—but it’s only half the story. Pub/sub is a critical messaging pattern for us and underpins a wide range of use cases, but Kafka’s topic model isn’t designed for this purpose. One of the key engineering challenges we faced was building a practical routing mechanism by which messages are matched to interested subscribers. On the surface, this problem appears fairly trivial and is far from novel, but it becomes quite interesting as we dig deeper.

Back to Basics

Topic routing works by matching a published message with interested subscribers. A consumer might subscribe to the topic “foo.bar.baz,” in which any message published to this topic would be delivered to them. We also must support * and # wildcards, which match exactly one word and zero or more words, respectively. In this sense, we follow the AMQP spec:

The routing key used for a topic exchange MUST consist of zero or more words delimited by dots. Each word may contain the letters A–Z and a–z and digits 0–9. The routing pattern follows the same rules as the routing key with the addition that * matches a single word, and # matches zero or more words. Thus the routing pattern *.stock.# matches the routing keys usd.stock and eur.stock.db but not stock.nasdaq.

This problem can be modeled using a trie structure. RabbitMQ went with this approach after exploring other options, like caching topics and indexing the patterns or using a deterministic finite automaton. The latter options have greater time and space complexities. The former requires backtracking the tree for wildcard lookups.

The subscription trie looks something like this:

subscription_trie

Even in spite of the backtracking required for wildcards, the trie ends up being a more performant solution due to its logarithmic complexity and tendency to fit CPU cache lines. Most tries have hot paths, particularly closer to the root, so caching becomes indispensable. The trie approach is also vastly easier to implement.

In almost all cases, this subscription trie needs to be thread-safe as clients are concurrently subscribing, unsubscribing, and publishing. We could serialize access to it with a reader-writer lock. For some, this would be the end of the story, but for high-throughput systems, locking is a major bottleneck. We can do better.

Breaking the Lock

We considered lock-free techniques that could be applied. Lock-free concurrency means that while a particular thread of execution may be blocked, all CPUs are able to continue processing other work. For example, imagine a program that protects access to some resource using a mutex. If a thread acquires this mutex and is subsequently preempted, no other thread can proceed until this thread is rescheduled by the OS. If the scheduler is adversarial, it may never resume execution of the thread, and the program would be effectively deadlocked. A key point, however, is that the mere lack of a lock does not guarantee a program is lock-free. In this context, “lock” really refers to deadlock, livelock, or the misdeeds of a malevolent scheduler.

In practice, what lock-free concurrency buys us is increased system throughput at the expense of increased tail latencies. Looking at a transactional system, lock-freedom allows us to process many concurrent transactions, any of which may block, while guaranteeing systemwide progress. Depending on the access patterns, when a transaction does block, there are always other transactions which can be processed—a CPU never idles. For high-throughput databases, this is essential.

Concurrent Timelines and Linearizability

Lock-freedom can be achieved using a number of techniques, but it ultimately reduces to a small handful of fundamental patterns. In order to fully comprehend these patterns, it’s important to grasp the concept of linearizability.

It takes approximately 100 nanoseconds for data to move from the CPU to main memory. This means that the laws of physics govern the unavoidable discrepancy between when you perceive an operation to have occurred and when it actually occurred. There is the time from when an operation is invoked to when some state change physically occurs (call it tinv), and there is the time from when that state change occurs to when we actually observe the operation as completed (call it tcom). Operations are not instantaneous, which means the wall-clock history of operations is uncertain. tinv and tcom vary for every operation. This is more easily visualized using a timeline diagram like the one below:

timeline

This timeline shows several reads and writes happening concurrently on some state. Physical time moves from left to right. This illustrates that even if a write is invoked before another concurrent write in real time, the later write could be applied first. If there are multiple threads performing operations on shared state, the notion of physical time is meaningless.

We use a linearizable consistency model to allow some semblance of a timeline by providing a total order of all state updates. Linearizability requires that each operation appears to occur atomically at some point between its invocation and completion. This point is called the linearization point. When an operation completes, it’s guaranteed to be observable by all threads because, by definition, the operation occurred before its completion time. After this point, reads will only see this value or a later one—never anything before. This gives us a proper sequencing of operations which can be reasoned about. Linearizability is a correctness condition for concurrent objects.

Of course, linearizability comes at a cost. This is why most memory models aren’t linearizable by default. Going back to our subscription trie, we could make operations on it appear atomic by applying a global lock. This kills throughput, but it ensures linearization.

lock trie

In reality, the trie operations do not occur at a specific instant in time as the illustration above depicts. However, mutual exclusion gives it the appearance and, as a result, linearizability holds at the expense of systemwide progress. Acquiring and releasing the lock appear instantaneous in the timeline because they are backed by atomic hardware operations like test-and-set. Linearizability is a composable property, meaning if an object is composed of linearizable objects, it is also linearizable. This allows us to construct abstractions from linearizable hardware instructions to data structures, all the way up to linearizable distributed systems.

Read-Modify-Write and CAS

Locks are expensive, not just due to contention but because they completely preclude parallelism. As we saw, if a thread which acquires a lock is preempted, any other threads waiting for the lock will continue to block.

Read-modify-write operations like compare-and-swap offer a lock-free approach to ensuring linearizable consistency. Such techniques loosen the bottleneck by guaranteeing systemwide throughput even if one or more threads are blocked. The typical pattern is to perform some speculative work then attempt to publish the changes with a CAS. If the CAS fails, then another thread performed a concurrent operation, and the transaction needs to be retried. If it succeeds, the operation was committed and is now visible, preserving linearizability. The CAS loop is a pattern used in many lock-free data structures and proves to be a useful primitive for our subscription trie.

CAS is susceptible to the ABA problem. These operations work by comparing values at a memory address. If the value is the same, it’s assumed that nothing has changed. However, this can be problematic if another thread modifies the shared memory and changes it back before the first thread resumes execution. The ABA problem is represented by the following sequence of events:

  1. Thread T1 reads shared-memory value A
  2. T1 is preempted, and T2 is scheduled
  3. T2 changes A to B then back to A
  4. T2 is preempted, and T1 is scheduled
  5. T1 sees the shared-memory value is A and continues

In this situation, T1 assumes nothing has changed when, in fact, an invariant may have been violated. We’ll see how this problem is addressed later.

At this point, we’ve explored the subscription-matching problem space, demonstrated why it’s an area of high contention, and examined why locks pose a serious problem to throughput. Linearizability provides an important foundation of understanding for lock-freedom, and we’ve looked at the most fundamental pattern for building lock-free data structures, compare-and-swap. Next, we will take a deep dive on applying lock-free techniques in practice by building on this knowledge. We’ll continue our narrative of how we applied these same techniques to our subscription engine and provide some further motivation for them.

Lock-Free Applied

Let’s revisit our subscription trie from earlier. Our naive approach to making it linearizable was to protect it with a lock. This proved easy, but as we observed, severely limited throughput. For a message broker, access to this trie is a critical path, and we usually have multiple threads performing inserts, removals, and lookups on it concurrently. Intuition tells us we can implement these operations without coarse-grained locking by relying on a CAS to perform mutations on the trie.

If we recall, read-modify-write is typically applied by copying a shared variable to a local variable, performing some speculative work on it, and attempting to publish the changes with a CAS. When inserting into the trie, our speculative work is creating an updated copy of a node. We commit the new node by updating the parent’s reference with a CAS. For example, if we want to add a subscriber to a node, we would copy the node, add the new subscriber, and CAS the pointer to it in the parent.

This approach is broken, however. To see why, imagine if a thread inserts a subscription on a node while another thread concurrently inserts a subscription as a child of that node. The second insert could be lost due to the sequencing of the reference updates. The diagram below illustrates this problem. Dotted lines represent a reference updated with a CAS.

trie cas add

The orphaned nodes containing “x” and “z” mean the subscription to “foo.bar” was lost. The trie is in an inconsistent state.

We looked to existing research in the field of non-blocking data structures to help illuminate a path. “Concurrent Tries with Efficient Non-Blocking Snapshots” by Prokopec et al. introduces the Ctrie, a non-blocking, concurrent hash trie based on shared-memory, single-word CAS instructions.

A hash array mapped trie (HAMT) is an implementation of an associative array which, unlike a hashmap, is dynamically allocated. Memory consumption is always proportional to the number of keys in the trie. A HAMT works by hashing keys and using the resulting bits in the hash code to determine which branches to follow down the trie. Each node contains a table with a fixed number of branch slots. Typically, the number of branch slots is 32. On a 64-bit machine, this would mean it takes 256 bytes (32 branches x 8-byte pointers) to store the branch table of a node.

The size of L1-L3 cache lines is 64 bytes on most modern processors. We can’t fit the branch table in a CPU cache line, let alone the entire node. Instead of allocating space for all branches, we use a bitmap to indicate the presence of a branch at a particular slot. This reduces the size of an empty node from roughly 264 bytes to 12 bytes. We can safely fit a node with up to six branches in a single cache line.

The Ctrie is a concurrent, lock-free version of the HAMT which ensures progress and linearizability. It solves the CAS problem described above by introducing indirection nodes, or I-nodes, which remain present in the trie even as nodes above and below change. This invariant ensures correctness on inserts by applying the CAS operation on the I-node instead of the internal node array.

An I-node may point to a Ctrie node, or C-node, which is an internal node containing a bitmap and array of references to branches. A branch is either an I-node or a singleton node (S-node) containing a key-value pair. The S-node is a leaf in the Ctrie. A newly initialized Ctrie starts with a root pointer to an I-node which points to an empty C-node. The diagram below illustrates a sequence of inserts on a Ctrie.

ctrie insert

An insert starts by atomically reading the I-node’s reference. Next, we copy the C-node and add the new key, recursively insert on an I-node, or extend the Ctrie with a new I-node. The new C-node is then published by performing a CAS on the parent I-node. A failed CAS indicates another thread has mutated the I-node. We re-linearize by atomically reading the I-node’s reference again, which gives us the current state of the Ctrie according to its linearizable history. We then retry the operation until the CAS succeeds. In this case, the linearization point is a successful CAS. The following figure shows why the presence of I-nodes ensures consistency.

ctrie insert correctness

In the above diagram, (k4,v4) is inserted into a Ctrie containing (k1,v1), (k2,v2), and (k3,v3). The new key-value pair is added to node C1 by creating a copy, C1‘, with the new entry. A CAS is then performed on the pointer at I1, indicated by the dotted line. Since C1‘ continues pointing to I2, any concurrent updates which occur below it will remain present in the trie. C1 is then garbage collected once no more threads are accessing it. Because of this, Ctries are much easier to implement in a garbage-collected language. It turns out that this deferred reclamation also solves the ABA problem described earlier by ensuring memory addresses are recycled only when it’s safe to do so.

The I-node invariant is enough to guarantee correctness for inserts and lookups, but removals require some additional invariants in order to avoid update loss. Insertions extend the Ctrie with additional levels, while removals eliminate the need for some of these levels. This is because we want to keep the Ctrie as compact as possible while still remaining correct. For example, a remove operation could result in a C-node with a single S-node below it. This state is valid, but the Ctrie could be made more compact and lookups on the lone S-node more efficient if it were moved up into the C-node above. This would allow the I-node and C-node to be removed.

The problem with this approach is it will cause insertions to be lost. If we move the S-node up and replace the dangling I-node reference with it, another thread could perform a concurrent insert on that I-node just before the compression occurs. The insert would be lost because the pointer to the I-node would be removed.

This issue is solved by introducing a new type of node called the tomb node (T-node) and an associated invariant. The T-node is used to ensure proper ordering during removals. The invariant is as follows: if an I-node points to a T-node at some time t0, then for all times greater than t0, the I-node points to the same T-node. More concisely, a T-node is the last value assigned to an I-node. This ensures that no insertions occur at an I-node if it is being compressed. We call such an I-node a tombed I-node.

If a removal results in a non-root-level C-node with a single S-node below it, the C-node is replaced with a T-node wrapping the S-node. This guarantees that every I-node except the root points to a C-node with at least one branch. This diagram depicts the result of removing (k2,v2) from a Ctrie:

ctrie removal

Removing (k2,v2) results in a C-node with a single branch, so it’s subsequently replaced with a T-node. The T-node provides a sequencing mechanism by effectively acting as a marker. While it solves the problem of lost updates, it doesn’t give us a compacted trie. If two keys have long matching hash code prefixes, removing one of the keys would result in a long chain of C-nodes followed by a single T-node at the end.

An invariant was introduced which says once an I-node points to a T-node, it will always point to that T-node. This means we can’t change a tombed I-node’s pointer, so instead we replace the I-node with its resurrection. The resurrection of a tombed I-node is the S-node wrapped in its T-node. When a T-node is produced during a removal, we ensure that it’s still reachable, and if it is, resurrect its tombed I-node in the C-node above. If it’s not reachable, another thread has already performed the compression. To ensure lock-freedom, all operations which read a T-node must help compress it instead of waiting for the removing thread to complete. Compression on the Ctrie from the previous diagram is illustrated below.

ctrie compression

The resurrection of the tombed I-node ensures the Ctrie is optimally compressed for arbitrarily long chains while maintaining integrity.

With a 32-bit hash code space, collisions are rare but still nontrivial. To deal with this, we introduce one final node, the list node (L-node). An L-node is essentially a persistent linked list. If there is a collision between the hash codes of two different keys, they are placed in an L-node. This is analogous to a hash table using separate chaining to resolve collisions.

One interesting property of the Ctrie is support for lock-free, linearizable, constant-time snapshots. Most concurrent data structures do not support snapshots, instead opting for locks or requiring a quiescent state. This allows Ctries to have O(1) iterator creation, clear, and size retrieval (amortized).

Constant-time snapshots are implemented by writing the Ctrie as a persistent data structure and assigning a generation count to each I-node. A persistent hash trie is updated by rewriting the path from the root of the trie down to the leaf the key belongs to while leaving the rest of the trie intact. The generation demarcates Ctrie snapshots. To create a new snapshot, we copy the root I-node and assign it a new generation. When an operation detects that an I-node’s generation is older than the root’s generation, it copies the I-node to the new generation and updates the parent. The path from the root to some node is only updated the first time it’s accessed, making the snapshot a O(1) operation.

The final piece needed for snapshots is a special type of CAS operation. There is a race condition between the thread creating a snapshot and the threads which have already read the root I-node with the previous generation. The linearization point for an insert is a successful CAS on an I-node, but we need to ensure that both the I-node has not been modified and its generation matches that of the root. This could be accomplished with a double compare-and-swap, but most architectures do not support such an operation.

The alternative is to use a RDCSS double-compare-single-swap originally described by Harris et al. We implement an operation with similar semantics to RDCSS called GCAS, or generation compare-and-swap. The GCAS allows us to atomically compare both the I-node pointer and its generation to the expected values before committing an update.

After researching the Ctrie, we wrote a Go implementation in order to gain a deeper understanding of the applied techniques. These same ideas would hopefully be adaptable to our problem domain.

Generalizing the Ctrie

The subscription trie shares some similarities to the hash array mapped trie but there are some key differences. First, values are not strictly stored at the leaves but can be on internal nodes as well. Second, the decomposed topic is used to determine how the trie is descended rather than a hash code. Wildcards complicate lookups further by requiring backtracking. Lastly, the number of branches on a node is not a fixed size. Applying the Ctrie techniques to the subscription trie, we end up with something like this:

matchbox

Much of the same logic applies. The main distinctions are the branch traversal based on topic words and rules around wildcards. Each branch is associated with a word and set of subscribers and may or may not point to an I-node. The I-nodes still ensure correctness on inserts. The behavior of T-nodes changes slightly. With the Ctrie, a T-node is created from a C-node with a single branch and then compressed. With the subscription trie, we don’t introduce a T-node until all branches are removed. A branch is pruned if it has no subscribers and points to nowhere or it has no subscribers and points to a tombed I-node. The GCAS and snapshotting remain unchanged.

We implemented this Ctrie derivative in order to build our concurrent pattern-matching engine, matchbox. This library provides an exceptionally simple API which allows a client to subscribe to a topic, unsubscribe from a topic, and lookup a topic’s subscribers. Snapshotting is also leveraged to retrieve the global subscription tree and the topics to which clients are currently subscribed. These are useful to see who currently has subscriptions and for what.

In Practice

Matchbox has been pretty extensively benchmarked, but to see how it behaves, it’s critical to observe its performance under contention. Many messaging systems opt for a mutex which tends to result in a lot of lock contention. It’s important to know what the access patterns look like in practice, but for our purposes, it’s heavily parallel. We don’t want to waste CPU cycles if we can help it.

To see how matchbox compares to lock-based subscription structures, I benchmarked it against gnatsd, a popular high-performance messaging system also written in Go. Gnatsd uses a tree-like structure protected by a mutex to manage subscriptions and offers similar wildcard semantics.

The benchmarks consist of one or more insertion goroutines and one or more lookup goroutines. Each insertion goroutine inserts 1000 subscriptions, and each lookup goroutine looks up 1000 subscriptions. We scale these goroutines up to see how the systems behave under contention.

The first benchmark is a 1:1 concurrent insert-to-lookup workload. A lookup corresponds to a message being published and matched to interested subscribers, while an insert occurs when a client subscribes to a topic. In practice, lookups are much more frequent than inserts, so the second benchmark is a 1:3 concurrent insert-to-lookup workload to help simulate this. The timings correspond to the complete insert and lookup workload. GOMAXPROCS was set to 8, which controls the number of operating system threads that can execute simultaneously. The benchmarks were run on a machine with a 2.6 GHz Intel Core i7 processor.

matchbox_bench_1_1

matchbox_bench_1_3

It’s quite clear that the lock-free approach scales a lot better under contention. This follows our intuition because lock-freedom allows system-wide progress even when a thread is blocked. If one goroutine is blocked on an insert or lookup operation, other operations may proceed. With a mutex, this isn’t possible.

Matchbox performs well, particularly in multithreaded environments, but there are still more optimizations to be made. This includes improvements both in memory consumption and runtime performance. Applying the Ctrie techniques to this type of trie results in a fairly non-compact structure. There may be ways to roll up branches—either eagerly or after removals—and expand them lazily as necessary. Other optimizations might include placing a cache or Bloom filter in front of the trie to avoid descending it. The main difficulty with these will be managing support for wildcards.

Conclusion

To summarize, we’ve seen why subscription matching is often a major concern for message-oriented middleware and why it’s frequently a bottleneck. Concurrency is crucial for high-performance systems, and we’ve looked at how we can achieve concurrency without relying on locks while framing it within the context of linearizability. Compare-and-swap is a fundamental tool used to implement lock-free data structures, but it’s important to be conscious of the pitfalls. We introduce invariants to protect data consistency. The Ctrie is a great example of how to do this and was foundational in our lock-free subscription-matching implementation. Finally, we validated our work by showing that lock-free data structures scale dramatically better with multithreaded workloads under contention.

My thanks to Steven Osborne and Dustin Hiatt for reviewing this article.

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Stream Processing and Probabilistic Methods: Data at Scale

Stream processing and related abstractions have become all the rage following the rise of systems like Apache Kafka, Samza, and the Lambda architecture. Applying the idea of immutable, append-only event sourcing means we’re storing more data than ever before. However, as the cost of storage continues to decline, it’s becoming more feasible to store more data for longer periods of time. With immutability, how the data lives isn’t interesting anymore. It’s all about how it moves.

The shifting landscape of data architecture parallels the way we’re designing systems today. Specifically, the evolution of monolithic to service-oriented architecture necessitates a change in the way we do data integration. The traditional normalization approach doesn’t cut it. Our systems are composed of databases, caches, search indexes, data warehouses, and a multitude of other components. Moreover, there’s an increasing demand for online, real-time processing of this data that’s tantamount to the growing popularity of large-scale, offline processing along the lines of Hadoop. This presents an interesting set of new challenges, namely, how do we drink from the firehose without getting drenched?

The answer most likely lies in frameworks like Samza, Storm, and Spark Streaming. Similarly, tools like StatsD solve the problem of collecting real-time analytics. However, this discussion is meant to explore some of the primitives used in stream processing. The ideas extend far beyond event sourcing, generalizing to any type of data stream, unbounded or not.

Batch vs. Streaming

With offline or batch processing, we often have some heuristics which provide insight into our data set, and we can typically afford multiple passes of the data. Without a strict time constraint, data structures are less important. We can store the entire data set on disk (or perhaps across a distributed file system) and process it in batches.

With streaming data, however, there’s a need for near real-time processing—collecting analytics, monitoring and alerting, updating search indexes and caches, etc. With web crawlers, we process a stream of URLs and documents and produce indexed content. With websites, we process a stream of page views and update various counters and gauges. With email, we process a stream of text and produce a filtered, spam-free inbox. These cases involve massive, often limitless data sets. Processing that data online can’t be done with the conventional ETL or MapReduce-style methods, and unless you’re doing windowed processing, it’s entirely impractical to store that data in memory.

Framing the Problem

As a concrete example of stream processing, imagine we want to count the number of distinct document views across a large corpus, say, Wikipedia. A naive solution would be to use a hash table which maps a document to a count. Wikipedia has roughly 35 million pages. Let’s assume each document is identified by a 16-byte GUID, and the counters are stored as 8-byte integers. This means we need in the ball park of a gigabyte of memory. Given today’s hardware, this might not sound completely unreasonable, but now let’s say we want to track views per unique IP address. We see that this approach quickly becomes intractable.

To illustrate further, consider how to count the cardinality of IP addresses which access our website. Instead of a hash table, we can use a bitmap or sparse bit array to count addresses. There are over four billion possible distinct IPv4 addresses. Sure, we could allocate half a gigabyte of RAM, but if you’re developing performance-critical systems, large heap sizes are going to kill you, and this overhead doesn’t help. ((Granted, if you’re sensitive to garbage-collection pauses, you may be better off using something like C or Rust, but that’s not always the most practical option.))

A Probabilistic Approach

Instead, we turn to probabilistic ways of solving these problems. Probabilistic algorithms and data structures seem to be oft-overlooked, or maybe purposely ignored, by system designers despite the theory behind them having been around for a long time in many cases. The goal should be to move these from the world of academia to industry because they are immensely useful and widely neglected.

Probabilistic solutions trade space and performance for accuracy. While a loss in precision may be a cause for concern to some, the fact is with large data streams, we have to trade something off to get real-time insight. Much like the CAP theorem, we must choose between consistency (accuracy) and availability (online). With CAP, we can typically adjust the level in which that trade-off occurs. This space/performance-accuracy exchange behaves very much the same way.

The literature can get pretty dense, so let’s look at what some of these approaches are and the problems they solve in a way that’s (hopefully) understandable.

Bloom Filters

The Bloom filter is probably the most well-known and, conceptually, simplest probabilistic data structure. It also serves as a good foundation because there are a lot of twists you can put on it. The theory provides a kernel from which many other probabilistic solutions are derived, as we will see.

Bloom filters answer a simple question: is this element a member of a set? A normal set requires linear space because it stores every element. A Bloom filter doesn’t store the actual elements, it merely stores the “membership” of them. It uses sub-linear space opening the possibility for false positives, meaning there’s a non-zero probability it reports an item is in the set when it’s actually not. This has a wide range of applications. For example, a Bloom filter can be placed in front of a database. When a query for a piece of data comes in and the filter doesn’t contain it, we completely bypass the database.

The Bloom filter consists of a bit array of length m and k hash functions. Both of these parameters are configurable, but we can optimize them based on a desired rate of false positives. Each bit in the array is initially unset. When an element is “added” to the filter, it’s hashed by each of the k functions, h1…hk, and modded by m, resulting in k indices into the bit array. Each bit at the respective index is set.

To query the membership of an element, we hash it and check if each bit is set. If any of them are zero, we know the item isn’t in the set. What happens when two different elements hash to the same index? This is where false positives are introduced. If the bits aren’t set, we know the element wasn’t added, but if they are, it’s possible that some other element or group of elements hashed to the same indices. Bloom filters have false positives, but false negatives are impossible—a member will never be reported incorrectly as a non-member. Unfortunately, this also means items can’t be removed from the filter.

It’s clear that the likelihood of false positives increases with each element added to the filter. We can target a specific probability of false positives by selecting an optimal value for m and k for up to n insertions. ((The math behind determining optimal m and k is left as an exercise for the reader.)) While this implementation works well in practice, it has a drawback in that some elements are potentially more sensitive to false positives than others. We can solve this problem by partitioning the m bits among the k hash functions such that each one produces an index over its respective partition. As a result, each element is always described by exactly k bits. This prevents any one element from being especially sensitive to false positives. Since calculating k hashes for every element is computationally expensive, we can actually perform a single hash and derive k hashes from it for a significant speedup. ((Less Hashing, Same Performance: Building a Better Bloom Filter discusses the use of two hash functions to simulate additional hashes. We can use a 64-bit hash, such as FNV, and use the upper and lower 32-bits as two different hashes.))

A Bloom filter eventually reaches a point where all bits are set, which means every query will indicate membership, effectively making the probability of false positives one. The problem with this is it requires a priori knowledge of the data set in order to select optimal parameters and avoid “overfilling.” Consequently, Bloom filters are ideal for offline processing and not so great for dealing with streams. Next, we’ll look at some variations of the Bloom filter which attempt to deal with this issue.

Scalable Bloom Filter

As we saw earlier, traditional Bloom filters are a great way to deal with set-membership problems in a space-efficient way, but they require knowing the size of the data set ahead of time in order to be effective. The Scalable Bloom Filter (SBF) was introduced by Almeida et al. as a way to cope with the capacity dilemma.

The Scalable Bloom Filter dynamically adapts to the size of the data set while enforcing a tight upper bound on the rate of false positives. Like the classic Bloom filter, false negatives are impossible. The SBF is essentially an array of Bloom filters with geometrically decreasing false-positive rates. New elements are added to the last filter. When this filter becomes “full”—more specifically, when it reaches a target fill ratio—a new filter is added with a tightened error probability. A tightening ratio, r, controls the growth of new filters.

Testing membership of an element consists of checking each of the filters. The geometrically decreasing false-positive rate of each filter is an interesting property. Since the fill ratio determines when a new filter is added, it turns out this progression can be modeled as a Taylor series which allows us to provide a tight upper bound on false positives using an optimal fill ratio of 0.5 (once the filter is 50% full, a new one is added). The compounded probability over the whole series converges to a target value, even accounting for an infinite series.

We can limit the error rate, but obviously the trade-off is that we end up allocating memory proportional to the size of the data set since we’re continuously adding filters. We also pay some computational cost on adds. Amortized, the cost of filter insertions is negligible, but for every add we must compute the current fill ratio of the filter to determine if a new filter needs to be added. Fortunately, we can optimize this by computing an estimated fill ratio. If we keep a running count of the items added to the filter, n, the approximate current fill ratio, p, can be obtained from the Taylor series expansion with p \approx 1-e^{-n/m} where m is the number of bits in the filter. Calculating this estimate turns out to be quite a bit faster than computing the actual fill ratio—fewer memory reads.

To provide some context around performance, adds in my Go implementation of a classic Bloom filter take about 166 ns on my MacBook Pro, and membership tests take 118 ns. With the SBF, adds take 422 ns and tests 113 ns. This isn’t a completely fair comparison since, as the SBF grows over time, tests require scanning through each filter, but certainly the add numbers are intuitive.

Scalable Bloom Filters are useful for cases where the size of the data set isn’t known a priori and memory constraints aren’t of particular concern. It’s an effective variation of a regular Bloom filter which generally requires space allocation orders-of-magnitude larger than the data set to allow enough headroom.

Stable Bloom Filter

Classic Bloom filters aren’t great for streaming data, and Scalable Bloom Filters, while a better option, still present a memory problem. Another derivative, the Stable Bloom Filter (SBF), was proposed by Deng and Rafiei as a technique for detecting duplicates in unbounded data streams with limited space. Most solutions work by dividing the stream into fixed-size windows and solving the problem within that discrete space. This allows the use of traditional methods like the Bloom filter.

The SBF attempts to approximate duplicates without the use of windows. Clearly, we can’t store the entire history of an infinite stream in limited memory. Instead, the thinking is more recent data has more value than stale data in many scenarios. As an example, web crawlers may not care about redundantly fetching a web page that was crawled a long time ago compared to fetching a page that was crawled more recently. With this impetus, the SBF works by representing more recent data while discarding old information.

The Stable Bloom Filter tweaks the classic Bloom filter by replacing the bit array with an array of m cells, each allocated d bits. The cells are simply counters, initially set to zero. The maximum value, Max, of a cell is 2^d-1. When an element is added, we first have to ensure space for it. This is done by selecting P random cells and decrementing their counters by one, clamping to zero. Next, we hash the element to k cells and set their counters to Max. Testing membership consists of probing the k cells. If any of them are zero, it’s not a member. The Bloom filter is actually a special case of an SBF where d is one and P is zero.

Like traditional Bloom filters, an SBF has a non-zero probability of false positives, which is controlled by several parameters. Unlike the Bloom filter, an SBF has a tight upper bound on the rate of false positives while introducing a non-zero rate of false negatives. The false-positive rate of a classic Bloom filter eventually reaches one, after which all queries result in a false positive. The stable-point property of an SBF means the false-positive rate asymptotically approaches a configurable fixed constant.

Stable Bloom Filters lend themselves to situations where the size of the data set isn’t known ahead of time and memory is bounded. For example, an SBF can be used to deduplicate events from an unbounded event stream with a specified upper bound on false positives and minimal false negatives. In particular, if the stream is not uniformly distributed, meaning duplicates are likely to be grouped closer together, the rate of false positives becomes immaterial.

Counting Bloom Filter

Conventional Bloom filters don’t allow items to be removed. The Stable Bloom Filter, on the other hand, works by evicting old data, but its API doesn’t expose a way to remove specific elements. The Counting Bloom Filter (CBF) is yet another twist on this structure. Introduced by Fan et al. in Summary Cache: A Scalable Wide-Area Web Cache Sharing Protocol, the CBF provides a way of deleting data from a filter.

It’s actually a very straightforward approach. The filter comprises an array of n-bit buckets similar to the Stable Bloom Filter. When an item is added, the corresponding counters are incremented, and when it’s removed, the counters are decremented.

Obviously, a CBF takes n-times more space than a regular Bloom filter, but it also has a scalability limit. Unless items are removed frequently enough, a counting filter’s false-positive probability will approach one. We need to dimension it accordingly, and this normally requires inferring from our data set. Likewise, since the CBF allows removals, it exposes an opportunity for false negatives. The multi-bit counters diminish the rate, but it’s important to be cognizant of this property so the CBF can be applied appropriately.

Inverse Bloom Filter

One of the distinguishing features of Bloom filters is the guarantee of no false negatives. This can be a valuable invariant, but what if we want the opposite of that? Is there an efficient data structure that can offer us no false positives with the possibility of false negatives? The answer is deceptively simple.

Jeff Hodges describes a rather trivial—yet strikingly pragmatic—approach which he calls the “opposite of a Bloom filter.” Since that name is a bit of a mouthful, I’m referring to this as the Inverse Bloom Filter (IBF).

The Inverse Bloom Filter may report a false negative but can never report a false positive. That is, it may indicate that an item has not been seen when it actually has, but it will never report an item as seen which it hasn’t come across. The IBF behaves in a similar manner to a fixed-size hash map of m buckets which doesn’t handle conflicts, but it provides lock-free concurrency using an underlying CAS.

The test-and-add operation hashes an element to an index in the filter. We take the value at that index while atomically storing the new value, then the old value is compared to the new one. If they differ, the new value wasn’t a member. If the values are equivalent, it was in the set.

The Inverse Bloom Filter is a nice option for dealing with unbounded streams or large data sets due to its limited memory usage. If duplicates are close together, the rate of false negatives becomes vanishingly small with an adequately sized filter.

HyperLogLog

Let’s revisit the problem of counting distinct IP addresses visiting a website. One solution might be to allocate a sparse bit array proportional to the number of IPv4 addresses. Alternatively, we could combine a Bloom filter with a counter. When an address comes in and it’s not in the filter, increment the count. While these both work, they’re not likely to scale very well. Rather, we can apply a probabilistic data structure known as the HyperLogLog (HLL). First presented by Flajolet et al. in 2007, HyperLogLog is an algorithm which approximately counts the number of distinct elements, or cardinality, of a multiset (a set which allows multiple occurrences of its elements).

Imagine our stream as a series of coin tosses. Someone tells you the most heads they flipped in a row was three. You can guess that they didn’t flip the coin very many times. However, if they flipped 20 heads in a row, it probably took them quite a while. This seems like a really unconvincing way of estimating the number of tosses, but from this kernel of thought grows a fruitful idea.

HLL replaces heads and tails with zeros and ones. Instead of counting the longest run of heads, it counts the longest run of leading zeros in a binary hash. Using a good hash function means that the values are, more or less, statistically independent and uniformly distributed. If the maximum run of leading zeros is n, the number of distinct elements in the set is approximately 2^n. But wait, what’s the math behind this? If we think about it, half of all binary numbers start with 1. Each additional bit divides the probability of a run further in half; that is, 25% start with 01, 12.5% start with 001, 6.25% start with 0001, ad infinitum. Thus, the probability of a run of length n is 2^{-(n+1)}.

Like flipping a coin, there’s potential for an enormous amount of variance with this technique. For instance, it’s entirely possible—though unlikely—that you flip 20 heads in a row on your first try. One experiment doesn’t provide enough data, so instead you flip 10 coins. HyperLogLog uses a similar strategy to reduce variance by splitting the stream up across a set of buckets, or registers, and applying the counting algorithm on the values in each one. It tracks the longest run of zeros in every register, and the total cardinality is computed by taking the harmonic mean across all registers. The harmonic mean is used to discount outliers since the distribution of values tends to skew towards the right. We can increase accuracy by adding more registers, which of course comes at the expense of performance and memory.

HyperLogLog is a remarkable algorithm. It almost feels like magic, and it’s exceptionally useful for working with data streams. HLL has a fraction of the memory footprint of other solutions. In fact, it’s usually several orders of magnitude while remaining surprisingly accurate.

Count-Min Sketch

HyperLogLog is a great option to efficiently count the number of distinct elements in a stream using a minimal amount of space, but it only gives us cardinality. What about counting the frequencies of specific elements? Cormode and Muthukrishnan developed the Count-Min sketch (CM sketch) to approximate the occurrences of different event types from a stream of events.

In contrast to a hash table, the Count-Min sketch uses sub-linear space to count frequencies. It consists of a matrix with w columns and d rows. These parameters determine the trade-off between space/time constraints and accuracy. Each row has an associated hash function. When an element arrives, it’s hashed for each row. The corresponding index in the rows are incremented by one. In this regard, the CM sketch shares some similarities with the Bloom filter.

count_min_sketch

The frequency of an element is estimated by taking the minimum of all the element’s respective counter values. The thought process here is that there is possibility for collisions between elements, which affects the counters for multiple items.  Taking the minimum count results in a closer approximation.

The Count-Min sketch is an excellent tool for counting problems for the same reasons as HyperLogLog. It has a lot of similarities to Bloom filters, but where Bloom filters effectively represent sets, the CM sketch considers multisets.

MinHash

The last probabilistic technique we’ll briefly look at is MinHash. This algorithm, invented by Andrei Broder, is used to quickly estimate the similarity between two sets. This has a variety of uses, such as detecting duplicate bodies of text and clustering or comparing documents.

MinHash works, in part, by using the Jaccard coefficient, a statistic which represents the size of the intersection of two sets divided by the size of the union:

J(A,B) = {{|A \cap B|}\over{|A \cup B|}}

This provides a measure of how similar the two sets are, but computing the intersection and union is expensive. Instead, MinHash essentially works by comparing randomly selected subsets through element hashing. It resembles locality-sensitive hashing (LSH), which means that similar items have a greater tendency to map to the same hash values. This varies from most hashing schemes, which attempt to avoid collisions. Instead, LSH is designed to maximize collisions of similar elements.

MinHash is one of the more difficult algorithms to fully grasp, and I can’t even begin to provide a proper explanation. If you’re interested in how it works, read the literature but know that there are a number of variations of it. The important thing is to know it exists and what problems it can be applied to.

In Practice

We’ve gone over several fundamental primitives for processing large data sets and streams while solving a few different types of problems. Bloom filters can be used to reduce disk reads and other expensive operations. They can also be purposed to detect duplicate events in a stream or prune a large decision tree. HyperLogLog excels at counting the number of distinct items in a stream while using minimal space, and the Count-Min sketch tracks the frequency of particular elements. Lastly, the MinHash method provides an efficient mechanism for comparing the similarity between documents. This has a number of applications, namely around identifying duplicates and characterizing content.

While there are no doubt countless implementations for most of these data structures and algorithms, I’ve been putting together a Go library geared towards providing efficient techniques for stream processing called Boom Filters. It includes implementations for all of the probabilistic data structures outlined above. As streaming data and real-time consumption grow more and more prominent, it will become important that we have the tools and understanding to deal with them. If nothing else, it’s valuable to be aware of probabilistic methods because they often provide accurate results at a fraction of the cost. These things aren’t just academic research, they form the basis for building online, high-performance systems at scale.