To set some ground, let's define the use case where this applies.We are talking about string matches, not regular expressions. In the case of liblognorm, we are also talking about exact string matches, and let me assume this for rsyslog's lookup table functionality as well (at least in "string" mode). Note that this may be extended, and should be no problem, but I have yet to properly evaluate the effect on the parse tree. So let's assume exact matches in this blog post. So basically, we have a set of strings and would like to see if there is an exact match (and pick up some value associated with that match). For example, we may have the strings "Berlin", "Boston", "Bogota" and "London". That example is obviously extremely simplified, yet it will help us understand.
Whenever we talk about time complexity, we need to make sure what exact n we are talking about. When it comes to matching values, n is usually the number of different values. In our example, this n would be four. Of course, one could also look at string lengths, as the length obviously influences the complexity of the comparison operation. Let's call this m, then in the above sample m would be six (length of longest string). If we take the unusual step to include that m, a linear list would have time complexity O(n*m), a binary tree O(log n*m) and rsyslog's parse tree O(m). HOWEVER, there usually is an upper bound on string length, and for that reason m is taken as a constant in almost all cases. By definition, it will then no longer be mention in O notation, and so we end up with the usual view that a linear list is O(n), a binary tree O(log n) ... and rsyslog's parse tree O(1).
This is possible because rsyslog's parse tree is based on the radix tree idea and kind of "compresses" the strings rather than including them in every node. The conceptual in-memory representation of a tree based on above values is as follows:
[root] - B --- O --- GOTA | | + --- STON | +---- ERLIN + ---- LONDON
As you can see, the tree is actually based on single characters that are different between the strings. So we do not have each string at the tree level, but only the rest of what is different. On this works a simple state machine. On a conceptual level, the search string is processed character by character, so each character represents a state transition. For obvious reasons, there can be only as many state transitions as there are characters inside the longest string. This is what we called m above. However, the number of different strings is totally irrelevant for the search, because it does not influence the number of state transitions required. That number is what we called n. So in conclusion, the time complexity is only depending on m, which is kind of constant. Consequently, we have O(1) complexity.
Just to stress the point once again: we do not iterate over the individual strings in a tree-like manner, instead we iterate over the index string characters themselves. This creates a very broad and shallow tree, even for a large number of index values.
On real hardware, the number of strings of course has some slight effect, especially when it comes to cache hit performance. While a small tree may fit into the available cache, a much larger tree obviously will not. However, from a theoretical point of view, this does not matter, we still have constant time for the lookup. From a practical perspective, this also does not matter so much: any larger tree will have a larger memory requirement and thus a lower cache hit rate. In fact, rsyslog's parse tree will probably perform better because of its compressed structure, which makes cache presence of at least top-level elements more likely.
For rsyslog, that's all I need to tell. For liblognorm, the story is a bit longer. It is only "almost" O(1). While the string matches themselves can be carried out in constant time, liblognorm also supports parsers to pull out specific parts of a log message. The current parsers are all O(1) as well, so this does not change the picture. However, state transitions involving a parser are no longer simple operations where the transition is immediately clear. If, for a given prefix, multiple parsers may apply, liblognorm must iterate through them until a matching one is found. This is no longer O(1), but depends on the number of parsers. Of course, we can claim that the number of parsers is still a (even small) constant, and so we can still say all of this is O(1) without violating the definition. While this is true, there unfortunately comes another complexity with parsers: With constant string, backtracking is never required, as we immediately know the exact state transition. With parsers, we have dynamic elements, and later text matches may turn out to be a non-match. In that case, we need to go back to parser detection and see if another parser also fits. If so, we need to give it a try and need to re-evaluate the parse tree from that point on. This process is called backtracking and can ruin the runtime to become as worse as O(n^2), just like regular expressions. Obviously, the problem exists primarily of very generic parsers are used, like "word". If, for example, parsers as specific as IPv4 are used, there is no chance for more than one parser matching. So the problem cannot occur. This is also why I caution against using too-generic parsers. Thankfully, very generic parsers are not used too frequently, and so we usually do not see this problem in practice. This is why I call liblognorm to be "almost" O(1), even though it may degrade to O(n^2) in extreme cases. Heuristically said, these extreme cases will not occur in practice.
The rsyslog parse tree does not use parsers inside the parse tree, just the fixed state transitions. As such, it actually is O(1). Oh, and yes: we should probably call the rsyslog "parse tree" better a "lookup tree" or "search tree" but for historical reasons it currently is named the way it is...