2 ms·

These are binary tries with some interesting optimizations, in particular using special instructions supported on all modern cpus to skip common prefixes at the

These are binary tries with some interesting optimizations, in particular using special instructions supported on all modern cpus to skip common prefixes at the start of looking up. According to the author's benchmarks they offer very consistent performance between insert and access operations, which seems to be key to why they perform better than R-B trees. Since tries don't need to re balance on insert we avoid memory operations R-B trees spend reorganizing interior nodes. Even though these are amortized in the R-B, we're still saving work. Likewise, these are faster than small hash tables since we avoid resizing-rehashing overhead. Searching the binary trie is also likely faster than following a collision linked list we might find in a naive hash table implementation. However, for larger containers hashes begin to catch up, since at these sizes the trie involves increasingly more memory accesses. I'd like to see these tries compared to generalized cuckoo hashing. I'd also like to see them compared to the bagwell array mapped tries used by clojure and others.

saurik 16y ago
(For the record, the reason why I have stared at this as much as I have just now is because I study these kinds of data structures and therefore want to understand what is new and compelling about this specific one.) So, tries have the property of being only as balanced as the distribution of their alphabets. It is very easy to end up in a situation where worst case performance of a trie is linear in the number of items in the trie. There are some researchers who were doing work on a concept called "stratified indexes" that allow you to index into deeply skew'd tries that you may find interesting. Regardless, the claims in this article are somewhat off. For example: they state that red-black trees "write out a lot of memory". Any full binary tree with N leaf nodes has N-1 non-leaf nodes. That statement is just as true of red-black trees as it is of this data structure. It is definitely true that the STL red-black tree implementation is inefficient (allocating memory for empty nodes), but that should not be held against the algorithm. The documentation is even weirder. "Unlike either red-black trees or most hash tables, nedtries can store as many items with identical keys as you like." That is certainly not true, and isn't even true of the STL, which has a red-black tree without unique restriction called std::multimap. The documentation also claims that hashtables can store arbitrarily large keys and nedtries can't, which is only as true of hashtables as it is for nedtries. As hashtables aren't ordered, and in fact require fixed-length keys, you can (and do: normally this statement would be going in the other direction of causality) use hash algorithms to hash the information to fit into the key length and then verify the value when you find it. The same thing could be used for nedtries. I'm also having a difficult time understanding why red-black trees are not an "in place algorithm" and nedtries are. I'm not even certain I understand what it would mean for a search tree to be "in place", but the documentation makes claims about "dynamic memory allocation" being its key distinction, and I see no reason why a red-black tree need require that for search or deletion. Insert requires a single allocation, but so does a trie; and, even with the STL most people use a custom pool allocator for that purpose. To be clear, I am not at all stating that "nedtries aren't faster than other implementations", just that that has nothing to do with their algorithm. The algorithm seems to just be an array of tries, where the array is designed to help with the common case of "there are a large number of zeros at the top of the key" (caused by using small numbers as keys). Unfortunately, that makes the distribution among the bin-level tries drastically uneven, making that first "which bucket do I end up in" question severely unfairly unbalanced. That specific optimization, then, doesn't seem to offer anything over a patricia trie, which itself would dramatically decrease the number of non-leaf node traversals (as nedtries does not describe using any kind of path compression at all). These path-compressed tries are what people normally use for high-performance lookup like routing.
- jasonwatkinspdx 16y ago
  I agree that his prose is a little confusing. Here's my understanding of his intent: "red-black trees write out a lot of memory" I think this refers to how R-B trees require re balancing on insert, particularly accessing siblings of ancestors. This is doubly bad on modern processors, as it introduces a chain of control hazards dependent on these accesses. That can burn a lot of cycles on cache misses. In contrast, the trie structures only access nodes along the direct line down to the appropriate leaf. This has a control hazard (ie when to end the loop) and a data dependency between iterations. The control hazard can be easily predicted (assume search recurses) and modern processors are quite good at unrolling the data hazards across the shadow registers (or equivalent). Basically, modern processors and compilers are much happier with the sort of inner loops you see in tries vs restructuring trees (and are happiest with the inner loop of accessing a hash table). "The documentation also claims that hashtables can store arbitrarily large keys and nedtries can't, which is only as true of hashtables as it is for nedtries." I believe his implementation assumes size_t sized keys. This limit comes from his root table, which needs as many entries as bits in the key. While you could code up a variant that used arbitrary length keys I think you'd lose most of the performance advantages he measures (remember, we're mostly talking about constant factors here, not complexity class). He doesn't care about this limit, as his primary application is storing free lists in an allocator, which will always have size_t keys. "I'm also having a difficult time understanding why red-black trees are not an "in place algorithm" and nedtries are." I'm also confused by this, but I'm assuming he's looking at concurrent access. Tries only ever push down toward leaves, so with careful ordering it's possible to ignore many locking concerns. Or put a different way, you're only ever allocating new nodes or filling in previously nil slots in existing nodes. Trees with structural re balancing on insert like R-B could invalidate a concurrent traversal with a rotation. "That specific optimization, then, doesn't seem to offer anything over a patricia trie, which itself would dramatically decrease the number of non-leaf node traversals (as nedtries does not describe using any kind of path compression at all)." I think for his purposes there are some assumptions about distribution: namely if he's storing memory addresses in a nedtrie, it's likely he's allocating them with brk() or the like, and so they're clustered in one or two slots of the root table. Likewise, if he's storing block sizes, then the root table acts as a segregated fit similar to dlmalloc and its many offspring. I'd agree that the way he over generalizes the advantages is a bit confusing, but I do think there's something interesting here vs vanilla patricia tries.

saurik 16y ago

(For the record, the reason why I have stared at this as much as I have just now is because I study these kinds of data structures and therefore want to understand what is new and compelling about this specific one.) So, tries have the property of being only as balanced as the distribution of their alphabets. It is very easy to end up in a situation where worst case performance of a trie is linear in the number of items in the trie. There are some researchers who were doing work on a concept called "stratified indexes" that allow you to index into deeply skew'd tries that you may find interesting. Regardless, the claims in this article are somewhat off. For example: they state that red-black trees "write out a lot of memory". Any full binary tree with N leaf nodes has N-1 non-leaf nodes. That statement is just as true of red-black trees as it is of this data structure. It is definitely true that the STL red-black tree implementation is inefficient (allocating memory for empty nodes), but that should not be held against the algorithm. The documentation is even weirder. "Unlike either red-black trees or most hash tables, nedtries can store as many items with identical keys as you like." That is certainly not true, and isn't even true of the STL, which has a red-black tree without unique restriction called std::multimap. The documentation also claims that hashtables can store arbitrarily large keys and nedtries can't, which is only as true of hashtables as it is for nedtries. As hashtables aren't ordered, and in fact require fixed-length keys, you can (and do: normally this statement would be going in the other direction of causality) use hash algorithms to hash the information to fit into the key length and then verify the value when you find it. The same thing could be used for nedtries. I'm also having a difficult time understanding why red-black trees are not an "in place algorithm" and nedtries are. I'm not even certain I understand what it would mean for a search tree to be "in place", but the documentation makes claims about "dynamic memory allocation" being its key distinction, and I see no reason why a red-black tree need require that for search or deletion. Insert requires a single allocation, but so does a trie; and, even with the STL most people use a custom pool allocator for that purpose. To be clear, I am not at all stating that "nedtries aren't faster than other implementations", just that that has nothing to do with their algorithm. The algorithm seems to just be an array of tries, where the array is designed to help with the common case of "there are a large number of zeros at the top of the key" (caused by using small numbers as keys). Unfortunately, that makes the distribution among the bin-level tries drastically uneven, making that first "which bucket do I end up in" question severely unfairly unbalanced. That specific optimization, then, doesn't seem to offer anything over a patricia trie, which itself would dramatically decrease the number of non-leaf node traversals (as nedtries does not describe using any kind of path compression at all). These path-compressed tries are what people normally use for high-performance lookup like routing.

jasonwatkinspdx 16y ago
I agree that his prose is a little confusing. Here's my understanding of his intent: "red-black trees write out a lot of memory" I think this refers to how R-B trees require re balancing on insert, particularly accessing siblings of ancestors. This is doubly bad on modern processors, as it introduces a chain of control hazards dependent on these accesses. That can burn a lot of cycles on cache misses. In contrast, the trie structures only access nodes along the direct line down to the appropriate leaf. This has a control hazard (ie when to end the loop) and a data dependency between iterations. The control hazard can be easily predicted (assume search recurses) and modern processors are quite good at unrolling the data hazards across the shadow registers (or equivalent). Basically, modern processors and compilers are much happier with the sort of inner loops you see in tries vs restructuring trees (and are happiest with the inner loop of accessing a hash table). "The documentation also claims that hashtables can store arbitrarily large keys and nedtries can't, which is only as true of hashtables as it is for nedtries." I believe his implementation assumes size_t sized keys. This limit comes from his root table, which needs as many entries as bits in the key. While you could code up a variant that used arbitrary length keys I think you'd lose most of the performance advantages he measures (remember, we're mostly talking about constant factors here, not complexity class). He doesn't care about this limit, as his primary application is storing free lists in an allocator, which will always have size_t keys. "I'm also having a difficult time understanding why red-black trees are not an "in place algorithm" and nedtries are." I'm also confused by this, but I'm assuming he's looking at concurrent access. Tries only ever push down toward leaves, so with careful ordering it's possible to ignore many locking concerns. Or put a different way, you're only ever allocating new nodes or filling in previously nil slots in existing nodes. Trees with structural re balancing on insert like R-B could invalidate a concurrent traversal with a rotation. "That specific optimization, then, doesn't seem to offer anything over a patricia trie, which itself would dramatically decrease the number of non-leaf node traversals (as nedtries does not describe using any kind of path compression at all)." I think for his purposes there are some assumptions about distribution: namely if he's storing memory addresses in a nedtrie, it's likely he's allocating them with brk() or the like, and so they're clustered in one or two slots of the root table. Likewise, if he's storing block sizes, then the root table acts as a segregated fit similar to dlmalloc and its many offspring. I'd agree that the way he over generalizes the advantages is a bit confusing, but I do think there's something interesting here vs vanilla patricia tries.