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
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.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. 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


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.


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.

Solving the Referential Integrity Problem

“A man with a watch knows what time it is. A man with two watches is never sure.”

I’ve been developing my open source Android framework, Infinitum, for the better part of 10 months now. It has brought about some really interesting problems that I’ve had to tackle, which is one of the many reasons I enjoy working on it so much.

Chicken or the Egg

Although it’s much more now, Infinitum began as an object-relational mapper which was loosely modeled after Hibernate. One of the first major issues I faced while developing the ORM component was loading object graphs. To illustrate what I mean by this, suppose we’re developing some software for a department store. The domain model for this software might look something like this:

As you can see, an Employee works in one Department, and, conversely, a Department has one or more Employees working in it, forming a many-to-one relationship and resulting in the class below.

Pretty straightforward, right? Now, let’s say we want to retrieve the Employee with, say, the ID 4028 from the database. Thinking about it at a high level and ignoring any notion of lazy loading, this appears to be rather simple.

1. Perform a query on the Employee table.

2. Instantiate a new Employee object.
3. Populate the Employee object’s fields from the query result.

But there’s some handwaving going on in those three steps, specifically the last one. One of the Employee fields is an entity, namely department. Okay, this shouldn’t be a problem. We just need to perform a second query to retrieve the Department associated with the Employee (the result of the first query is going to include the Department foreign key — let’s assume its 14).

Then we just create the Department object, populate it and assign it to the respective field in the Employee.

Once again, there’s a problem. To understand why, it’s helpful to see what the Department class actually looks like.

Do you see what the issue is? In order to construct our Employee, we need to construct his Department. In order to construct his Department, we need to construct the Employee. Our object graph has a cycle that’s throwing us for a (infinite) loop.

Breaking the Cycle

Fortunately, there’s a pretty easy solution for this chicken-or-the-egg problem. We’ll make use of a HashMap to keep tabs on our object graph as we incrementally build it. This will make more sense in just a bit.

We’re going to use a HashMap keyed off of an integer hash where the map values will be the entities in the object graph.

The integer hash will be a unique value computed for each entity we need to load to fulfill the object graph. The idea is that we will store the partially populated entity in the HashMap to have its remaining fields populated later. Loading an entity will take the following steps:

  1. Perform query on the entity table.
  2. Instantiate a new entity object.
  3. Populate the entity object fields which do not belong to a relationship from the query result.
  4. Compute the hash for the partial entity object.
  5. Check if the HashMap contains the computed hash.
  6. If the HashMap contains the hash, return the associated entity object (this breaks any potential cycle).
  7. Otherwise, store the entity object in the HashMap using the hash as its key.
  8. Load related entities by recursively calling this sequence.

Going back to our Employee problem, retrieving an Employee from the database will take these steps:

  1. Perform query on the Employee table.
  2. Instantiate a new Employee object.
  3. Populate the Employee object fields which do not belong to a relationship from the query result.
  4. Compute the hash for the partial Employee object.
  5. Check if the HashMap contains the computed hash (it won’t).
  6. Store the Employee object in the HashMap using the hash as its key.
  7. Perform query on the Department table.
  8. Instantiate a new Department object.
  9. Populate the Department object fields which do not belong to a relationship from the query results.
  10. Compute the hash for the partial Department object.
  11. Check if the HashMap contains the computed hash (again, it won’t).
  12. Store the Department object in the HashMap using the hash as its key.
  13. The cycle will terminate and the two objects in the HashMap, the Employee and the Department, will be fully populated and referencing each other.

Considering the HashMap is not specific to any entity type (i.e. it will hold Employees, Departments, and any other domain types we come up with), how do we compute a unique hash for objects of various types? Moreover, we’re computing hashes for incomplete objects, so what gives?

Obviously, we can’t make use of hashCode() since not every field is guaranteed to be populated. Fortunately, we can take advantage of the fact that every entity must have a primary key, but, unless we’re using a policy where every primary key is unique across every table, this won’t get us very far. We will include the entity type as a factor in our hash code. Here’s the code Infinitum currently uses to compute this hash:

This hash allows us to uniquely identify entities even if they have not been fully populated. Our cycle problem is solved!

Maintaining Referential Integrity

The term “referential integrity” is typically used to refer to a property of relational databases. However, when I say referential integrity, I’m referring to the notion of object references in an object graph. This referential integrity is something ORMs must keep track of or otherwise you run into some big problems.

To illustrate this, say our department store only has one department and two employees who work in said department (this might defeat the purpose of a department store, but just roll with it). Now, let’s say we retrieve one Employee, Bill, from the database. Once again ignoring lazy loading, this should implicitly load an object graph consisting of the Employee, the Department, and the Employees assigned to that Department. Next, let’s subsequently retrieve the second Employee, Frank, from the database. Again, this will load the object graph.

Bill and Frank both work in the same Department, but if referential integrity is not enforced, objects can become out of sync.

The underlying problem is that there are two different copies of the Department object, but we must abide by the Highlander Principle in that “there can be only one.” Bill and Frank should reference the same instance so that, regardless of how the Department is dereferenced, it stays synced between every object in the graph.

In plain terms, when we’re retrieving objects from the database, we must be cautious not to load the same one twice. Otherwise, we’ll have two objects corresponding to a single database row and things will get out of sync.

Enter Identity Map

This presents an interesting problem. Knowing what we learned earlier with regard to the chicken-or-the-egg problem, can we apply a similar solution? The answer is yes! In fact, the solution we discussed earlier was actually masquerading as a fairly common design pattern known as the Identity Map, originally cataloged by Martin Fowler in his book Patterns of Enterprise Application Architecture.

The idea behind the Identity Map pattern is that, every time we read a record from the database, we first check the Identity Map to see if the record has already been retrieved. This allows us to simply return a new reference to the in-memory record rather than creating a new object, maintaining referential integrity.

A secondary benefit to the Identity Map is that, since it acts as a cache, it reduces the number of database calls needed to retrieve objects, which yields a performance enhancement.

An Identity Map is normally tied to some sort of transactional context such as a session. This works exceedingly well for Infinitum because its ORM is built around the notion of a Session object, which  can be configured as a scoped unit of work. The Infinitum Session contains a cache which functions as an Identity Map, solving both the cycle and the referential integrity issues.

It’s worth pointing out, however, that while an Identity Map maintains referential integrity within the context of a session, it doesn’t do anything to prevent incongruities between different sessions. This is a complex problem that usually requires a locking strategy, which is beyond the scope of this blog post.

Under the Hood

It may be helpful to see how Infinitum uses an Identity Map to solve the cycle problem. The method createFromCursor takes a database result cursor and transforms it into an instance of the given type. It makes use of a recursive method that goes through the process I outlined earlier. The call to loadRelationships will result in this recursion.

Entities are stored in the Session cache as they are retrieved, allowing us to enforce referential integrity while also preventing any infinite loops that might occur while building up the object graph.

So that’s it! We’ve learned to make use of the Identity Map pattern to solve some pretty interesting problems. We looked at how we can design an ORM to load object graphs that contain cycles as well as maintain this critical notion of referential integrity. We also saw how the Identity Map helps to give us some performance gain through caching. Infinitum’s ORM module makes use of this pattern in its session caching and many other frameworks use it as well. In a future blog entry, I will talk about lazy loading and how it can be used to avoid loading large object graphs.