by Niall Douglas. Last updated . This page has been accessed 41,317 times since the 19th June 2010.
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Computer science theory classes teach students that associative
containers (i.e. containers whose items are placed in some form of
faster than O(n) access index) come in two main forms: (i) ordered, in
which case it has O(log N) average lookup complexity and (ii) unordered,
in which case it has O(1) average lookup complexity. The C++ Standard
Template Library comes with
precisely this reason - there is supposedly no other choice except for
esoteric requirements such as indexing a few terabytes of data where,
Wikipedia will show you, you'll find all sorts of crazy indexation
So what if I told you that for the very common case of a pointer sized key lookup that the standard assumptions are wrong? What if there was an algorithm which provides one of the big advantages of ordered indexation which is close fit finds, except it has nearly O(1) complexity rather than O(log N) and is therefore 100% faster? What if, in fact, this algorithm is a good 20% faster than the typical O(1) hash table implementation for medium sized collections and is no slower even at 10,000 items?
You don't believe me? Well I didn't believe me either when I was developing the user mode page allocator for my memory allocator, nedmalloc. But seeing is believing, so here's some scaling graphs for a 2.67Ghz x64 Intel Core 2 Quad:
As you can see, nedtries utterly destroys the traditional red-black binary tree algorithm except for nearest fit finds where it has similar complexity - in fact, unless you have items whose difference cannot be converted into a stable size_t value (which is rare) then for most use cases you should probably never use red-black trees ever again, despite that they are the canonical algorithm for anything which needs to maintain how items relate to one another. You will also notice that for less than 64k items nedtries handily beats the venerable O(1) hash table where it is considerably faster at insertion and finding - only in deletion does the hash table shine. Note that unlike hash tables but like red-black trees, nedtries makes no use of dynamic memory unless you use the STL container emulations trie_map<> and trie_multimap<>.
You will also note the most interesting characteristic of bitwise tries of all: insertion, deletion and finds (including close fit finds) all take about the same amount of time. Almost every other search algorithm will be good at one or two things but bad at the others e.g. hash tables are fast at searching and deletion but slow at inserting. If your code, like a lot of common code, equally does inserts and deletes as it does finds then nedtries will do wonders for your software project.
How does it work?
It works by constructing a binary tree based on individual bit difference, so the higher the entropy (disorder) between key bits, the faster nedtries is relative to other algorithms (equally, the more similar the keys at bit level the slower it goes). It hence has a complexity for a given key of O(1/DKL(key||average key)) which is the inverse of the Kullback–Leibler divergence (i.e. relative entropy) between the entropy of the key you are searching for and the average entropy of the key in the tree. Its worst case complexity - where all keys are almost identical - is O(log2 N). This implies that for well distributed keys that average complexity will always be a lot better than red-black trees, but somewhat worse than O(1) because for obvious reasons, more keys means more unique information and therefore there is always some scaling relation to the number of keys in the tree i.e. the Kolmogorov complexity, or the compressed length, of all keys.
If that hurt your head, it's actually very easy to understand: make sure your keys are well distributed and nedtries will fly, but it isn't quite pure O(1) like a hash table. As you can see in the scaling graphs above, it's much more sensitive to your L2 cache size than a hash table, so once you exceed that number of items a hash table will always win. Therefore, in practice nedtries ought to be used instead of a hash table only up to about 10,000 items after which a hash table is better, though hash tables do require additional dynamic memory allocation for their buckets while nedtries requires no dynamic memory at all. nedtries performs much better in every situation than red-black trees (as RB trees write out a lot of memory as they rebalance themselves, invalidating caches across a SMP config which also makes them particularly ill suited for multiple thread use) except for one case: if you need guaranteed nearest match finds, red-black trees are unbeatable.
I hold that almost always where you're currently using a red-black tree or something very similar, you'll benefit if you replace that with nedtries, and where you're using a hash table with less than 10,000 items you probably will benefit from nedtries. If you can't use dynamic memory, nedtries is a no-brainer.
What implementations are there?
There is a C macro based implementation, a C++ template-through-the-C-macros implementation (this makes debugging vastly easier and lets the C++ compiler optimiser perform better) and two C++ STL API compatible containers which can be slotted in to replace any existing usage of std::map<>, std::unordered_map<>, std::multimap<> or std::unordered_multimap<> (or any of their very similar kin).
The code is 100% C and C++ standard code. If you have a C++11 supporting compiler it will make use of any move constructors you might have.
If you don't understand the algorithm and decide you need to, I'd suggest compiling it as C++ and stepping through it using a debugger.
In more recent years there are competing implementations of bitwise tries. TommyDS contains one, also an in-place implementation, which appears to perform well. Note that the benchmarks he has there compare the C version of nedtries which is about 5-15% slower than the C++-through-the-C-macros version, and I am also unsure if he told the compiler it can use the conditional instructions (CMOV) available since the Pentium Pro as he is compiling nedtries for 32-bit x86. CMOV makes a big difference to nedtries' performance.
Where do I get it from?
You can also pull a full GIT clone from either of:
How do I report bugs, suggest enhancements or get support?
Please report bugs or enhancements to the github issue tracker. You can get non-commercial support from the email address at the bottom of this page, or you can get commercial support from my company ned Productions Consulting Ltd.
v1.02 Final (9th July 2012):
v1.01 RC2 (unreleased):
v1.01 RC1 (19th June 2011):
v1.00 beta 1 (18th June 2010):